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| author | nfenwick <nfenwick@pglaf.org> | 2025-01-22 07:58:32 -0800 |
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| committer | nfenwick <nfenwick@pglaf.org> | 2025-01-22 07:58:32 -0800 |
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| tree | a2bf0efd8f9991db9dfd74cb802dcb089195ccda /old/67153-h | |
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diff --git a/old/67153-h/67153-h.htm b/old/67153-h/67153-h.htm deleted file mode 100644 index 0f60786..0000000 --- a/old/67153-h/67153-h.htm +++ /dev/null @@ -1,12043 +0,0 @@ -<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" - "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> -<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"> - <head> - <meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> - <meta http-equiv="Content-Style-Type" content="text/css" /> - <title> - The Project Gutenberg eBook of The Fourth Dimension, by C. 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Howard Hinton</p> -<div style='display:block; margin:1em 0'> -This eBook is for the use of anyone anywhere in the United States and -most other parts of the world at no cost and with almost no restrictions -whatsoever. You may copy it, give it away or re-use it under the terms -of the Project Gutenberg License included with this eBook or online -at <a href="https://www.gutenberg.org">www.gutenberg.org</a>. If you -are not located in the United States, you will have to check the laws of the -country where you are located before using this eBook. -</div> - -<p style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Title: The Fourth Dimension</p> -<p style='display:block; margin-top:1em; margin-bottom:0; margin-left:2em; text-indent:-2em'>Author: C. Howard Hinton</p> -<p style='display:block; text-indent:0; margin:1em 0'>Release Date: January 12, 2022 [eBook #67153]</p> -<p style='display:block; text-indent:0; margin:1em 0'>Language: English</p> - <p style='display:block; margin-top:1em; margin-bottom:0; margin-left:2em; text-indent:-2em; text-align:left'>Produced by: Chris Curnow, Les Galloway and the Online Distributed Proofreading Team at https://www.pgdp.net (This file was produced from images generously made available by The Internet Archive)</p> -<div style='margin-top:2em; margin-bottom:4em'>*** START OF THE PROJECT GUTENBERG EBOOK THE FOURTH DIMENSION ***</div> - -<div class="transnote"> -<h3> Transcriber’s Notes</h3> - -<p>Obvious typographical errors have been silently corrected. All other -spelling and punctuation remains unchanged.</p> - -<p>The cover was prepared by the transcriber and is placed in the public -domain.</p> -</div> -<hr class="chap" /> - - -<div class="half-title">THE FOURTH DIMENSION</div> - - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<h2 class="nobreak" id="SOME_OPINIONS_OF_THE_PRESS">SOME OPINIONS OF THE PRESS</h2> -</div> - - -<p>“<i>Mr. C. H. Hinton discusses the subject of the higher dimensionality of -space, his aim being to avoid mathematical subtleties and technicalities, and -thus enable his argument to be followed by readers who are not sufficiently -conversant with mathematics to follow these processes of reasoning.</i>”—<span class="smcap">Notts -Guardian.</span></p> - -<p>“<i>The fourth dimension is a subject which has had a great fascination for -many teachers, and though one cannot pretend to have quite grasped -Mr. Hinton’s conceptions and arguments, yet it must be admitted that he -reveals the elusive idea in quite a fascinating light. Quite apart from the -main thesis of the book many chapters are of great independent interest. -Altogether an interesting, clever and ingenious book.</i>”—<span class="smcap">Dundee Courier.</span></p> - -<p>“<i>The book will well repay the study of men who like to exercise their wits -upon the problems of abstract thought.</i>”—<span class="smcap">Scotsman.</span></p> - -<p>“<i>Professor Hinton has done well to attempt a treatise of moderate size, -which shall at once be clear in method and free from technicalities of the -schools.</i>”—<span class="smcap">Pall Mall Gazette.</span></p> - -<p>“<i>A very interesting book he has made of it.</i>”—<span class="smcap">Publishers’ Circular.</span></p> - -<p>“<i>Mr. Hinton tries to explain the theory of the fourth dimension so that -the ordinary reasoning mind can get a grasp of what metaphysical -mathematicians mean by it. If he is not altogether successful it is not from -want of clearness on his part, but because the whole theory comes as such an -absolute shock to all one’s preconceived ideas.</i>”—<span class="smcap">Bristol Times.</span></p> - -<p>“<i>Mr. Hinton’s enthusiasm is only the result of an exhaustive study, which -has enabled him to set his subject before the reader with far more than the -amount of lucidity to which it is accustomed.</i>”—<span class="smcap">Pall Mall Gazette.</span></p> - -<p>“<i>The book throughout is a very solid piece of reasoning in the domain of -higher mathematics.</i>”—<span class="smcap">Glasgow Herald.</span></p> - -<p>“<i>Those who wish to grasp the meaning of this somewhat difficult subject -would do well to read</i> The Fourth Dimension. <i>No mathematical knowledge -is demanded of the reader, and any one, who is not afraid of a little hard -thinking, should be able to follow the argument.</i>”—<span class="smcap">Light.</span></p> - -<p>“<i>A splendidly clear re-statement of the old problem of the fourth dimension. -All who are interested in this subject will find the work not only fascinating, -but lucid, it being written in a style easily understandable. The illustrations -make still more clear the letterpress, and the whole is most admirably adapted -to the requirements of the novice or the student.</i>”—<span class="smcap">Two Worlds.</span></p> - -<p>“<i>Those in search of mental gymnastics will find abundance of exercise in -Mr. C. H. Hinton’s</i> Fourth Dimension.”—<span class="smcap">Westminster Review.</span></p> - - -<p><span class="smcap">First Edition</span>, <i>April 1904</i>; <span class="smcap">Second Edition</span>, <i>May 1906</i>.</p> - - -<hr class="chap x-ebookmaker-drop" /> - -<div class="figcenter illowp100" id="i_frontis" style="max-width: 50em;"> - <img src="images/i_frontis.jpg" alt="" /> - <div class="caption">Views of the Tessaract.</div> -</div> - -<div class="chapter"></div> - - -<h1> -<small>THE</small><br /> - -FOURTH DIMENSION</h1> - -<p class="center small">BY</p> - -<p class="center">C. HOWARD HINTON, M.A.<br /> - -<small>AUTHOR OF “SCIENTIFIC ROMANCES”<br /> -“A NEW ERA OF THOUGHT,” ETC., ETC.</small></p> - -<div class="figcenter illowp20" id="colop" style="max-width: 9.375em;"> - <img src="images/colop.png" alt="Colophon" /> -</div> - -<p class="center"><small>LONDON</small><br /> -SWAN SONNENSCHEIN & CO., LIMITED<br /> -25 HIGH STREET, BLOOMSBURY<br /> -<br /> -<small>1906</small><br /> -</p> - - -<p class="center small spaced"> -PRINTED BY<br /> -HAZELL, WATSON AND VINEY, LD.,<br /> -LONDON AND AYLESBURY.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_v">[Pg v]</span></p> - -<h2 class="nobreak" id="PREFACE">PREFACE</h2> -</div> - - -<p>I have endeavoured to present the subject of the higher -dimensionality of space in a clear manner, devoid of -mathematical subtleties and technicalities. In order to -engage the interest of the reader, I have in the earlier -chapters dwelt on the perspective the hypothesis of a -fourth dimension opens, and have treated of the many -connections there are between this hypothesis and the -ordinary topics of our thoughts.</p> - -<p>A lack of mathematical knowledge will prove of no -disadvantage to the reader, for I have used no mathematical -processes of reasoning. I have taken the view -that the space which we ordinarily think of, the space -of real things (which I would call permeable matter), -is different from the space treated of by mathematics. -Mathematics will tell us a great deal about space, just -as the atomic theory will tell us a great deal about the -chemical combinations of bodies. But after all, a theory -is not precisely equivalent to the subject with regard -to which it is held. There is an opening, therefore, from -the side of our ordinary space perceptions for a simple, -altogether rational, mechanical, and observational way<span class="pagenum" id="Page_vi">[Pg vi]</span> -of treating this subject of higher space, and of this -opportunity I have availed myself.</p> - -<p>The details introduced in the earlier chapters, especially -in Chapters VIII., IX., X., may perhaps be found -wearisome. They are of no essential importance in the -main line of argument, and if left till Chapters XI. -and XII. have been read, will be found to afford -interesting and obvious illustrations of the properties -discussed in the later chapters.</p> - -<p>My thanks are due to the friends who have assisted -me in designing and preparing the modifications of -my previous models, and in no small degree to the -publisher of this volume, Mr. Sonnenschein, to whose -unique appreciation of the line of thought of this, as -of my former essays, their publication is owing. By -the provision of a coloured plate, in addition to the other -illustrations, he has added greatly to the convenience -of the reader.</p> - -<p class="psig"> -<span class="smcap">C. Howard Hinton.</span></p> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_vii">[Pg vii]</span></p> - -<h2 class="nobreak" id="CONTENTS">CONTENTS</h2> -</div> - - -<table class="standard" summary=""> -<tr> -<td class="tdr"><small>CHAP</small>.</td> -<td></td> -<td class="tdr"><small>PAGE</small></td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_I">I.</a></td> -<td class="tdh"><span class="smcap">Four-Dimensional Space</span></td> -<td class="tdr">1</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_II">II.</a></td> -<td class="tdh"><span class="smcap">The Analogy of a Plane World</span></td> -<td class="tdr">6</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_III">III.</a></td> -<td class="tdh"><span class="smcap">The Significance of a Four-Dimensional -Existence</span></td> -<td class="tdr">15</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_IV">IV.</a></td> -<td class="tdh"><span class="smcap">The First Chapter in the History of Four -Space</span></td> -<td class="tdr">23</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_V">V.</a></td> -<td class="tdh"><span class="smcap">The Second Chapter in the History Of Four Space</span></td> -<td class="tdr">41</td> -</tr> -<tr> -<td></td> -<td class="tdh"><small>Lobatchewsky, Bolyai, and Gauss<br />Metageometry</small></td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_VI">VI.</a></td> -<td class="tdh"><span class="smcap">The Higher World</span></td> -<td class="tdr">61</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_VII">VII.</a></td> -<td class="tdh"><span class="smcap">The Evidence for a Fourth Dimension</span></td> -<td class="tdr">76</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_VIII">VIII.</a></td> -<td class="tdh"><span class="smcap">The Use of Four Dimensions in Thought</span></td> -<td class="tdr">85</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_IX">IX.</a></td> -<td class="tdh"><span class="smcap">Application to Kant’s Theory of Experience</span></td> -<td class="tdr">107</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_X">X.</a></td> -<td class="tdh"><span class="smcap">A Four-Dimensional Figure</span></td> -<td class="tdr">122</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_XI">XI.</a></td> -<td class="tdh"><span class="smcap">Nomenclature and Analogies</span></td> -<td class="tdr">136<span class="pagenum" id="Page_viii">[Pg viii]</span></td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_XII">XII.</a></td> -<td class="tdh"><span class="smcap">The Simplest Four-Dimensional Solid</span></td> -<td class="tdr">157</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_XIII">XIII.</a></td> -<td class="tdh"><span class="smcap">Remarks on the Figures</span></td> -<td class="tdr">178</td> -</tr> -<tr> -<td class="tdr"><a href="#CHAPTER_XIV">XIV.</a></td> -<td class="tdh"><span class="smcap">A Recapitulation and Extension of the -Physical Argument</span></td> -<td class="tdr">203</td> -</tr> -<tr> -<td class="tdl" colspan="2"><a href="#APPENDIX_I">APPENDIX I.</a>—<span class="smcap">The Models</span></td> -<td class="tdr">231</td> -</tr> -<tr> -<td class="tdl" colspan="2"><a href="#APPENDIX_II">APPENDIX II.</a>—<span class="smcap">A Language of Space</span></td> -<td class="tdr">248</td> -</tr> -</table> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_1">[Pg 1]</span></p> - -<p class="half-title">THE FOURTH DIMENSION</p> - - -<hr class="small" /> - - -<h2 class="nobreak" id="CHAPTER_I">CHAPTER I<br /> - - -<small>FOUR-DIMENSIONAL SPACE</small></h2> -</div> - -<p>There is nothing more indefinite, and at the same time -more real, than that which we indicate when we speak -of the “higher.” In our social life we see it evidenced -in a greater complexity of relations. But this complexity -is not all. There is, at the same time, a contact -with, an apprehension of, something more fundamental, -more real.</p> - -<p>With the greater development of man there comes -a consciousness of something more than all the forms -in which it shows itself. There is a readiness to give -up all the visible and tangible for the sake of those -principles and values of which the visible and tangible -are the representation. The physical life of civilised -man and of a mere savage are practically the same, but -the civilised man has discovered a depth in his existence, -which makes him feel that that which appears all to -the savage is a mere externality and appurtenage to his -true being.</p> - -<p>Now, this higher—how shall we apprehend it? It is -generally embraced by our religious faculties, by our -idealising tendency. But the higher existence has two -sides. It has a being as well as qualities. And in trying<span class="pagenum" id="Page_2">[Pg 2]</span> -to realise it through our emotions we are always taking the -subjective view. Our attention is always fixed on what we -feel, what we think. Is there any way of apprehending -the higher after the purely objective method of a natural -science? I think that there is.</p> - -<p>Plato, in a wonderful allegory, speaks of some men -living in such a condition that they were practically -reduced to be the denizens of a shadow world. They -were chained, and perceived but the shadows of themselves -and all real objects projected on a wall, towards -which their faces were turned. All movements to them -were but movements on the surface, all shapes but the -shapes of outlines with no substantiality.</p> - -<p>Plato uses this illustration to portray the relation -between true being and the illusions of the sense world. -He says that just as a man liberated from his chains -could learn and discover that the world was solid and -real, and could go back and tell his bound companions of -this greater higher reality, so the philosopher who has -been liberated, who has gone into the thought of the -ideal world, into the world of ideas greater and more -real than the things of sense, can come and tell his fellow -men of that which is more true than the visible sun—more -noble than Athens, the visible state.</p> - -<p>Now, I take Plato’s suggestion; but literally, not -metaphorically. He imagines a world which is lower -than this world, in that shadow figures and shadow -motions are its constituents; and to it he contrasts the real -world. As the real world is to this shadow world, so is the -higher world to our world. I accept his analogy. As our -world in three dimensions is to a shadow or plane world, -so is the higher world to our three-dimensional world. -That is, the higher world is four-dimensional; the higher -being is, so far as its existence is concerned apart from its -qualities, to be sought through the conception of an actual<span class="pagenum" id="Page_3">[Pg 3]</span> -existence spatially higher than that which we realise with -our senses.</p> - -<p>Here you will observe I necessarily leave out all that -gives its charm and interest to Plato’s writings. All -those conceptions of the beautiful and good which live -immortally in his pages.</p> - -<p>All that I keep from his great storehouse of wealth is -this one thing simply—a world spatially higher than this -world, a world which can only be approached through the -stocks and stones of it, a world which must be apprehended -laboriously, patiently, through the material things -of it, the shapes, the movements, the figures of it.</p> - -<p>We must learn to realise the shapes of objects in -this world of the higher man; we must become familiar -with the movements that objects make in his world, so -that we can learn something about his daily experience, -his thoughts of material objects, his machinery.</p> - -<p>The means for the prosecution of this enquiry are given -in the conception of space itself.</p> - -<p>It often happens that that which we consider to be -unique and unrelated gives us, within itself, those relations -by means of which we are able to see it as related to -others, determining and determined by them.</p> - -<p>Thus, on the earth is given that phenomenon of weight -by means of which Newton brought the earth into its -true relation to the sun and other planets. Our terrestrial -globe was determined in regard to other bodies of the -solar system by means of a relation which subsisted on -the earth itself.</p> - -<p>And so space itself bears within it relations of which -we can determine it as related to other space. For within -space are given the conceptions of point and line, line and -plane, which really involve the relation of space to a -higher space.</p> - -<p>Where one segment of a straight line leaves off and<span class="pagenum" id="Page_4">[Pg 4]</span> -another begins is a point, and the straight line itself can -be generated by the motion of the point.</p> - -<p>One portion of a plane is bounded from another by a -straight line, and the plane itself can be generated by -the straight line moving in a direction not contained -in itself.</p> - -<p>Again, two portions of solid space are limited with -regard to each other by a plane; and the plane, moving -in a direction not contained in itself, can generate solid -space.</p> - -<p>Thus, going on, we may say that space is that which -limits two portions of higher space from each other, and -that our space will generate the higher space by moving -in a direction not contained in itself.</p> - -<p>Another indication of the nature of four-dimensional -space can be gained by considering the problem of the -arrangement of objects.</p> - -<p>If I have a number of swords of varying degrees of -brightness, I can represent them in respect of this quality -by points arranged along a straight line.</p> - -<div class="figleft illowp25" id="fig_1" style="max-width: 10em;"> - <img src="images/fig_1.png" alt="" /> - <div class="caption">Fig. 1.</div> -</div> - -<p>If I place a sword at <span class="allsmcap">A</span>, <a href="#fig_1">fig. 1</a>, and regard it as having -a certain brightness, then the other swords -can be arranged in a series along the -line, as at <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, etc., according to -their degrees of brightness.</p> - -<div class="figleft illowp25" id="fig_2" style="max-width: 10em;"> - <img src="images/fig_2.png" alt="" /> - <div class="caption">Fig. 2.</div> -</div> - -<p>If now I take account of another quality, say length, -they can be arranged in a plane. Starting from <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, I -can find points to represent different -degrees of length along such lines as -<span class="allsmcap">AF</span>, <span class="allsmcap">BD</span>, <span class="allsmcap">CE</span>, drawn from <span class="allsmcap">A</span> and <span class="allsmcap">B</span> and <span class="allsmcap">C</span>. -Points on these lines represent different -degrees of length with the same degree of -brightness. Thus the whole plane is occupied by points -representing all conceivable varieties of brightness and -length.</p> - -<p><span class="pagenum" id="Page_5">[Pg 5]</span></p> - -<div class="figleft illowp30" id="fig_3" style="max-width: 10em;"> - <img src="images/fig_3.png" alt="" /> - <div class="caption">Fig. 3.</div> -</div> - -<p>Bringing in a third quality, say sharpness, I can draw, -as in <a href="#fig_3">fig. 3</a>, any number of upright -lines. Let distances along these -upright lines represent degrees of -sharpness, thus the points <span class="allsmcap">F</span> and <span class="allsmcap">G</span> -will represent swords of certain -definite degrees of the three qualities -mentioned, and the whole of space will serve to represent -all conceivable degrees of these three qualities.</p> - -<p>If now I bring in a fourth quality, such as weight, and -try to find a means of representing it as I did the other -three qualities, I find a difficulty. Every point in space is -taken up by some conceivable combination of the three -qualities already taken.</p> - -<p>To represent four qualities in the same way as that in -which I have represented three, I should need another -dimension of space.</p> - -<p>Thus we may indicate the nature of four-dimensional -space by saying that it is a kind of space which would -give positions representative of four qualities, as three-dimensional -space gives positions representative of three -qualities.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_6">[Pg 6]</span></p> - -<h2 class="nobreak" id="CHAPTER_II">CHAPTER II<br /> - -<small><i>THE ANALOGY OF A PLANE WORLD</i></small></h2></div> - - -<p>At the risk of some prolixity I will go fully into the -experience of a hypothetical creature confined to motion -on a plane surface. By so doing I shall obtain an analogy -which will serve in our subsequent enquiries, because the -change in our conception, which we make in passing from -the shapes and motions in two dimensions to those in -three, affords a pattern by which we can pass on still -further to the conception of an existence in four-dimensional -space.</p> - -<p>A piece of paper on a smooth table affords a ready -image of a two-dimensional existence. If we suppose the -being represented by the piece of paper to have no -knowledge of the thickness by which he projects above the -surface of the table, it is obvious that he can have no -knowledge of objects of a similar description, except by -the contact with their edges. His body and the objects -in his world have a thickness of which however, he has no -consciousness. Since the direction stretching up from -the table is unknown to him he will think of the objects -of his world as extending in two dimensions only. Figures -are to him completely bounded by their lines, just as solid -objects are to us by their surfaces. He cannot conceive -of approaching the centre of a circle, except by breaking -through the circumference, for the circumference encloses -the centre in the directions in which motion is possible to<span class="pagenum" id="Page_7">[Pg 7]</span> -him. The plane surface over which he slips and with -which he is always in contact will be unknown to him; -there are no differences by which he can recognise its -existence.</p> - -<p>But for the purposes of our analogy this representation -is deficient.</p> - -<p>A being as thus described has nothing about him to -push off from, the surface over which he slips affords no -means by which he can move in one direction rather than -another. Placed on a surface over which he slips freely, -he is in a condition analogous to that in which we should -be if we were suspended free in space. There is nothing -which he can push off from in any direction known to him.</p> - -<p>Let us therefore modify our representation. Let us -suppose a vertical plane against which particles of thin -matter slip, never leaving the surface. Let these particles -possess an attractive force and cohere together into a disk; -this disk will represent the globe of a plane being. He -must be conceived as existing on the rim.</p> - -<div class="figleft illowp25" id="fig_4" style="max-width: 10.9375em;"> - <img src="images/fig_4.png" alt="" /> - <div class="caption">Fig. 4.</div> -</div> - -<p>Let 1 represent this vertical disk of flat matter and 2 -the plane being on it, standing upon its -rim as we stand on the surface of our earth. -The direction of the attractive force of his -matter will give the creature a knowledge -of up and down, determining for him one -direction in his plane space. Also, since -he can move along the surface of his earth, -he will have the sense of a direction parallel to its surface, -which we may call forwards and backwards.</p> - -<p>He will have no sense of right and left—that is, of the -direction which we recognise as extending out from the -plane to our right and left.</p> - -<p>The distinction of right and left is the one that we -must suppose to be absent, in order to project ourselves -into the condition of a plane being.</p> - -<p><span class="pagenum" id="Page_8">[Pg 8]</span></p> - -<p>Let the reader imagine himself, as he looks along the -plane, <a href="#fig_4">fig. 4</a>, to become more and more identified with -the thin body on it, till he finally looks along parallel to -the surface of the plane earth, and up and down, losing -the sense of the direction which stretches right and left. -This direction will be an unknown dimension to him.</p> - -<p>Our space conceptions are so intimately connected with -those which we derive from the existence of gravitation -that it is difficult to realise the condition of a plane being, -without picturing him as in material surroundings with -a definite direction of up and down. Hence the necessity -of our somewhat elaborate scheme of representation, which, -when its import has been grasped, can be dispensed with -for the simpler one of a thin object slipping over a -smooth surface, which lies in front of us.</p> - -<p>It is obvious that we must suppose some means by -which the plane being is kept in contact with the surface -on which he slips. The simplest supposition to make is -that there is a transverse gravity, which keeps him to the -plane. This gravity must be thought of as different to -the attraction exercised by his matter, and as unperceived -by him.</p> - -<p>At this stage of our enquiry I do not wish to enter -into the question of how a plane being could arrive at -a knowledge of the third dimension, but simply to investigate -his plane consciousness.</p> - -<p>It is obvious that the existence of a plane being must -be very limited. A straight line standing up from the -surface of his earth affords a bar to his progress. An -object like a wheel which rotates round an axis would -be unknown to him, for there is no conceivable way in -which he can get to the centre without going through -the circumference. He would have spinning disks, but -could not get to the centre of them. The plane being -can represent the motion from any one point of his space<span class="pagenum" id="Page_9">[Pg 9]</span> -to any other, by means of two straight lines drawn at -right angles to each other.</p> - -<div class="figleft illowp35" id="fig_5" style="max-width: 26.6875em;"> - <img src="images/fig_5.png" alt="" /> - <div class="caption">Fig. 5.</div> -</div> - -<p>Let <span class="allsmcap">AX</span> and <span class="allsmcap">AY</span> be two such axes. He can accomplish -the translation from <span class="allsmcap">A</span> to <span class="allsmcap">B</span> by going along <span class="allsmcap">AX</span> to <span class="allsmcap">C</span>, and -then from <span class="allsmcap">C</span> along <span class="allsmcap">CB</span> parallel to <span class="allsmcap">AY</span>.</p> - -<p>The same result can of course be obtained -by moving to <span class="allsmcap">D</span> along <span class="allsmcap">AY</span> and then parallel -to <span class="allsmcap">AX</span> from <span class="allsmcap">D</span> to <span class="allsmcap">B</span>, or of course by any -diagonal movement compounded by these -axial movements.</p> - -<p>By means of movements parallel to -these two axes he can proceed (except for -material obstacles) from any one point of his space to -any other.</p> - -<div class="figleft illowp35" id="fig_6" style="max-width: 16.875em;"> - <img src="images/fig_6.png" alt="" /> - <div class="caption">Fig. 6.</div> -</div> - -<p>If now we suppose a third line drawn -out from <span class="allsmcap">A</span> at right angles to the plane -it is evident that no motion in either -of the two dimensions he knows will -carry him in the least degree in the -direction represented by <span class="allsmcap">AZ</span>.</p> - -<p>The lines <span class="allsmcap">AZ</span> and <span class="allsmcap">AX</span> determine a -plane. If he could be taken off his plane, and transferred -to the plane <span class="allsmcap">AXZ</span>, he would be in a world exactly -like his own. From every line in his -world there goes off a space world exactly -like his own.</p> - -<div class="figleft illowp25" id="fig_7" style="max-width: 12.5em;"> - <img src="images/fig_7.png" alt="" /> - <div class="caption">Fig. 7.</div> -</div> - -<p>From every point in his world a line can -be drawn parallel to <span class="allsmcap">AZ</span> in the direction -unknown to him. If we suppose the square -in <a href="#fig_7">fig. 7</a> to be a geometrical square from -every point of it, inside as well as on the -contour, a straight line can be drawn parallel -to <span class="allsmcap">AZ</span>. The assemblage of these lines constitute a solid -figure, of which the square in the plane is the base. If -we consider the square to represent an object in the plane<span class="pagenum" id="Page_10">[Pg 10]</span> -being’s world then we must attribute to it a very small -thickness, for every real thing must possess all three -dimensions. This thickness he does not perceive, but -thinks of this real object as a geometrical square. He -thinks of it as possessing area only, and no degree of -solidity. The edges which project from the plane to a -very small extent he thinks of as having merely length -and no breadth—as being, in fact, geometrical lines.</p> - -<p>With the first step in the apprehension of a third -dimension there would come to a plane being the conviction -that he had previously formed a wrong conception -of the nature of his material objects. He had conceived -them as geometrical figures of two dimensions only. -If a third dimension exists, such figures are incapable -of real existence. Thus he would admit that all his real -objects had a certain, though very small thickness in the -unknown dimension, and that the conditions of his -existence demanded the supposition of an extended sheet -of matter, from contact with which in their motion his -objects never diverge.</p> - -<p>Analogous conceptions must be formed by us on the -supposition of a four-dimensional existence. We must -suppose a direction in which we can never point extending -from every point of our space. We must draw a distinction -between a geometrical cube and a cube of real -matter. The cube of real matter we must suppose to -have an extension in an unknown direction, real, but so -small as to be imperceptible by us. From every point -of a cube, interior as well as exterior, we must imagine -that it is possible to draw a line in the unknown direction. -The assemblage of these lines would constitute a higher -solid. The lines going off in the unknown direction from -the face of a cube would constitute a cube starting from -that face. Of this cube all that we should see in our -space would be the face.</p> - -<p><span class="pagenum" id="Page_11">[Pg 11]</span></p> - -<p>Again, just as the plane being can represent any -motion in his space by two axes, so we can represent any -motion in our three-dimensional space by means of three -axes. There is no point in our space to which we cannot -move by some combination of movements on the directions -marked out by these axes.</p> - -<p>On the assumption of a fourth dimension we have -to suppose a fourth axis, which we will call <span class="allsmcap">AW</span>. It must -be supposed to be at right angles to each and every -one of the three axes <span class="allsmcap">AX</span>, <span class="allsmcap">AY</span>, <span class="allsmcap">AZ</span>. Just as the two axes, -<span class="allsmcap">AX</span>, <span class="allsmcap">AZ</span>, determine a plane which is similar to the original -plane on which we supposed the plane being to exist, but -which runs off from it, and only meets it in a line; so in -our space if we take any three axes such as <span class="allsmcap">AX</span>, <span class="allsmcap">AY</span>, and -<span class="allsmcap">AW</span>, they determine a space like our space world. This -space runs off from our space, and if we were transferred -to it we should find ourselves in a space exactly similar to -our own.</p> - -<p>We must give up any attempt to picture this space in -its relation to ours, just as a plane being would have to -give up any attempt to picture a plane at right angles -to his plane.</p> - -<p>Such a space and ours run in different directions from -the plane of <span class="allsmcap">AX</span> and <span class="allsmcap">AY</span>. They meet in this plane but -have nothing else in common, just as the plane space -of <span class="allsmcap">AX</span> and <span class="allsmcap">AY</span> and that of <span class="allsmcap">AX</span> and <span class="allsmcap">AZ</span> run in different -directions and have but the line <span class="allsmcap">AX</span> in common.</p> - -<p>Omitting all discussion of the manner on which a plane -being might be conceived to form a theory of a three-dimensional -existence, let us examine how, with the means -at his disposal, he could represent the properties of three-dimensional -objects.</p> - -<div class="figleft illowp40" id="fig_8" style="max-width: 25em;"> - <img src="images/fig_8.png" alt="" /> - <div class="caption">Fig. 8.</div> -</div> - -<p>There are two ways in which the plane being can think -of one of our solid bodies. He can think of the cube, -<a href="#fig_8">fig. 8</a>, as composed of a number of sections parallel to<span class="pagenum" id="Page_12">[Pg 12]</span> -his plane, each lying in the third dimension a little -further off from his plane than -the preceding one. These sections -he can represent as a -series of plane figures lying in -his plane, but in so representing -them he destroys the coherence -of them in the higher figure. -The set of squares, <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, <span class="allsmcap">D</span>, -represents the section parallel -to the plane of the cube shown in figure, but they are -not in their proper relative positions.</p> - -<p>The plane being can trace out a movement in the third -dimension by assuming discontinuous leaps from one -section to another. Thus, a motion along the edge of -the cube from left to right would be represented in the -set of sections in the plane as the succession of the -corners of the sections <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, <span class="allsmcap">D</span>. A point moving from -<span class="allsmcap">A</span> through <span class="allsmcap">BCD</span> in our space must be represented in the -plane as appearing in <span class="allsmcap">A</span>, then in <span class="allsmcap">B</span>, and so on, without -passing through the intervening plane space.</p> - -<p>In these sections the plane being leaves out, of course, -the extension in the third dimension; the distance between -any two sections is not represented. In order to realise -this distance the conception of motion can be employed.</p> - -<div class="figleft illowp25" id="fig_9" style="max-width: 12.5em;"> - <img src="images/fig_9.png" alt="" /> - <div class="caption">Fig. 9.</div> -</div> - -<p>Let <a href="#fig_9">fig. 9</a> represent a cube passing transverse to the -plane. It will appear to the plane being as a -square object, but the matter of which this -object is composed will be continually altering. -One material particle takes the place of another, -but it does not come from anywhere or go -anywhere in the space which the plane being -knows.</p> - -<p>The analogous manner of representing a higher solid in -our case, is to conceive it as composed of a number of<span class="pagenum" id="Page_13">[Pg 13]</span> -sections, each lying a little further off in the unknown -direction than the preceding.</p> - -<div class="figleft illowp75" id="fig_10" style="max-width: 31.25em;"> - <img src="images/fig_10.png" alt="" /> - <div class="caption">Fig. 10.</div> -</div> - -<p>We can represent these sections as a number of solids. -Thus the cubes <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, <span class="allsmcap">D</span>, -may be considered as -the sections at different -intervals in the unknown -dimension of a higher -cube. Arranged thus their coherence in the higher figure -is destroyed, they are mere representations.</p> - -<p>A motion in the fourth dimension from <span class="allsmcap">A</span> through <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, -etc., would be continuous, but we can only represent it as -the occupation of the positions <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, etc., in succession. -We can exhibit the results of the motion at different -stages, but no more.</p> - -<p>In this representation we have left out the distance -between one section and another; we have considered the -higher body merely as a series of sections, and so left out -its contents. The only way to exhibit its contents is to -call in the aid of the conception of motion.</p> - -<div class="figleft illowp25" id="fig_11" style="max-width: 9.375em;"> - <img src="images/fig_11.png" alt="" /> - <div class="caption">Fig. 11.</div> -</div> - -<p>If a higher cube passes transverse to our space, it will -appear as a cube isolated in space, the part -that has not come into our space and the part -that has passed through will not be visible. -The gradual passing through our space would -appear as the change of the matter of the cube -before us. One material particle in it is succeeded by -another, neither coming nor going in any direction we can -point to. In this manner, by the duration of the figure, -we can exhibit the higher dimensionality of it; a cube of -our matter, under the circumstances supposed, namely, -that it has a motion transverse to our space, would instantly -disappear. A higher cube would last till it had passed -transverse to our space by its whole distance of extension -in the fourth dimension.</p> - -<p><span class="pagenum" id="Page_14">[Pg 14]</span></p> - -<p>As the plane being can think of the cube as consisting -of sections, each like a figure he knows, extending away -from his plane, so we can think of a higher solid as composed -of sections, each like a solid which we know, but -extending away from our space.</p> - -<p>Thus, taking a higher cube, we can look on it as -starting from a cube in our space and extending in the -unknown dimension.</p> - -<div class="figcenter illowp100" id="fig_12" style="max-width: 25em;"> - <img src="images/fig_12.png" alt="" /> - <div class="caption">Fig. 12.</div> -</div> - -<p>Take the face <span class="allsmcap">A</span> and conceive it to exist as simply a -face, a square with no thickness. From this face the -cube in our space extends by the occupation of space -which we can see.</p> - -<p>But from this face there extends equally a cube in the -unknown dimension. We can think of the higher cube, -then, by taking the set of sections <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, <span class="allsmcap">D</span>, etc., and -considering that from each of them there runs a cube. -These cubes have nothing in common with each other, -and of each of them in its actual position all that we can -have in our space is an isolated square. It is obvious that -we can take our series of sections in any manner we -please. We can take them parallel, for instance, to any -one of the three isolated faces shown in the figure. -Corresponding to the three series of sections at right -angles to each other, which we can make of the cube -in space, we must conceive of the higher cube, as composed -of cubes starting from squares parallel to the faces -of the cube, and of these cubes all that exist in our space -are the isolated squares from which they start.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_15">[Pg 15]</span></p> - -<h2 class="nobreak" id="CHAPTER_III">CHAPTER III<br /> - -<small><i>THE SIGNIFICANCE OF A FOUR-DIMENSIONAL -EXISTENCE</i></small></h2></div> - - -<p>Having now obtained the conception of a four-dimensional -space, and having formed the analogy which, without -any further geometrical difficulties, enables us to enquire -into its properties, I will refer the reader, whose interest -is principally in the mechanical aspect, to Chapters VI. -and VII. In the present chapter I will deal with the -general significance of the enquiry, and in the next -with the historical origin of the idea.</p> - -<p>First, with regard to the question of whether there -is any evidence that we are really in four-dimensional -space, I will go back to the analogy of the plane world.</p> - -<p>A being in a plane world could not have any experience -of three-dimensional shapes, but he could have -an experience of three-dimensional movements.</p> - -<p>We have seen that his matter must be supposed to -have an extension, though a very small one, in the third -dimension. And thus, in the small particles of his -matter, three-dimensional movements may well be conceived -to take place. Of these movements he would only -perceive the resultants. Since all movements of an -observable size in the plane world are two-dimensional, -he would only perceive the resultants in two dimensions -of the small three-dimensional movements. Thus, there -would be phenomena which he could not explain by his<span class="pagenum" id="Page_16">[Pg 16]</span> -theory of mechanics—motions would take place which -he could not explain by his theory of motion. Hence, -to determine if we are in a four-dimensional world, we -must examine the phenomena of motion in our space. -If movements occur which are not explicable on the suppositions -of our three-dimensional mechanics, we should -have an indication of a possible four-dimensional motion, -and if, moreover, it could be shown that such movements -would be a consequence of a four-dimensional motion in -the minute particles of bodies or of the ether, we should -have a strong presumption in favour of the reality of -the fourth dimension.</p> - -<p>By proceeding in the direction of finer and finer subdivision, -we come to forms of matter possessing properties -different to those of the larger masses. It is probable that -at some stage in this process we should come to a form -of matter of such minute subdivision that its particles -possess a freedom of movement in four dimensions. This -form of matter I speak of as four-dimensional ether, and -attribute to it properties approximating to those of a -perfect liquid.</p> - -<p>Deferring the detailed discussion of this form of matter -to Chapter VI., we will now examine the means by which -a plane being would come to the conclusion that three-dimensional -movements existed in his world, and point -out the analogy by which we can conclude the existence -of four-dimensional movements in our world. Since the -dimensions of the matter in his world are small in the -third direction, the phenomena in which he would detect -the motion would be those of the small particles of -matter.</p> - -<p>Suppose that there is a ring in his plane. We can -imagine currents flowing round the ring in either of two -opposite directions. These would produce unlike effects, -and give rise to two different fields of influence. If the<span class="pagenum" id="Page_17">[Pg 17]</span> -ring with a current in it in one direction be taken up -and turned over, and put down again on the plane, it -would be identical with the ring having a current in the -opposite direction. An operation of this kind would be -impossible to the plane being. Hence he would have -in his space two irreconcilable objects, namely, the two -fields of influence due to the two rings with currents in -them in opposite directions. By irreconcilable objects -in the plane I mean objects which cannot be thought -of as transformed one into the other by any movement -in the plane.</p> - -<p>Instead of currents flowing in the rings we can imagine -a different kind of current. Imagine a number of small -rings strung on the original ring. A current round these -secondary rings would give two varieties of effect, or two -different fields of influence, according to its direction. -These two varieties of current could be turned one into -the other by taking one of the rings up, turning it over, -and putting it down again in the plane. This operation -is impossible to the plane being, hence in this case also -there would be two irreconcilable fields in the plane. -Now, if the plane being found two such irreconcilable -fields and could prove that they could not be accounted -for by currents in the rings, he would have to admit the -existence of currents round the rings—that is, in rings -strung on the primary ring. Thus he would come to -admit the existence of a three-dimensional motion, for -such a disposition of currents is in three dimensions.</p> - -<p>Now in our space there are two fields of different -properties, which can be produced by an electric current -flowing in a closed circuit or ring. These two fields can -be changed one into the other by reversing the currents, but -they cannot be changed one into the other by any turning -about of the rings in our space; for the disposition of the -field with regard to the ring itself is different when we<span class="pagenum" id="Page_18">[Pg 18]</span> -turn the ring, over and when we reverse the direction of -the current in the ring.</p> - -<p>As hypotheses to explain the differences of these two -fields and their effects we can suppose the following kinds -of space motions:—First, a current along the conductor; -second, a current round the conductor—that is, of rings of -currents strung on the conductor as an axis. Neither of -these suppositions accounts for facts of observation.</p> - -<p>Hence we have to make the supposition of a four-dimensional -motion. We find that a four-dimensional -rotation of the nature explained in a subsequent chapter, -has the following characteristics:—First, it would give us -two fields of influence, the one of which could be turned -into the other by taking the circuit up into the fourth -dimension, turning it over, and putting it down in our -space again, precisely as the two kinds of fields in the -plane could be turned one into the other by a reversal of -the current in our space. Second, it involves a phenomenon -precisely identical with that most remarkable and -mysterious feature of an electric current, namely that it -is a field of action, the rim of which necessarily abuts on a -continuous boundary formed by a conductor. Hence, on -the assumption of a four-dimensional movement in the -region of the minute particles of matter, we should expect -to find a motion analogous to electricity.</p> - -<p>Now, a phenomenon of such universal occurrence as -electricity cannot be due to matter and motion in any -very complex relation, but ought to be seen as a simple -and natural consequence of their properties. I infer that -the difficulty in its theory is due to the attempt to explain -a four-dimensional phenomenon by a three-dimensional -geometry.</p> - -<p>In view of this piece of evidence we cannot disregard -that afforded by the existence of symmetry. In this -connection I will allude to the simple way of producing<span class="pagenum" id="Page_19">[Pg 19]</span> -the images of insects, sometimes practised by children. -They put a few blots of ink in a straight line on a piece of -paper, fold the paper along the blots, and on opening it the -lifelike presentment of an insect is obtained. If we were -to find a multitude of these figures, we should conclude -that they had originated from a process of folding over; -the chances against this kind of reduplication of parts -is too great to admit of the assumption that they had -been formed in any other way.</p> - -<p>The production of the symmetrical forms of organised -beings, though not of course due to a turning over of -bodies of any appreciable size in four-dimensional space, -can well be imagined as due to a disposition in that -manner of the smallest living particles from which they -are built up. Thus, not only electricity, but life, and the -processes by which we think and feel, must be attributed -to that region of magnitude in which four-dimensional -movements take place.</p> - -<p>I do not mean, however, that life can be explained as a -four-dimensional movement. It seems to me that the -whole bias of thought, which tends to explain the -phenomena of life and volition, as due to matter and -motion in some peculiar relation, is adopted rather in the -interests of the explicability of things than with any -regard to probability.</p> - -<p>Of course, if we could show that life were a phenomenon -of motion, we should be able to explain a great deal that is -at present obscure. But there are two great difficulties in -the way. It would be necessary to show that in a germ -capable of developing into a living being, there were -modifications of structure capable of determining in the -developed germ all the characteristics of its form, and not -only this, but of determining those of all the descendants -of such a form in an infinite series. Such a complexity of -mechanical relations, undeniable though it be, cannot<span class="pagenum" id="Page_20">[Pg 20]</span> -surely be the best way of grouping the phenomena and -giving a practical account of them. And another difficulty -is this, that no amount of mechanical adaptation would -give that element of consciousness which we possess, and -which is shared in to a modified degree by the animal -world.</p> - -<p>In those complex structures which men build up and -direct, such as a ship or a railway train (and which, if seen -by an observer of such a size that the men guiding them -were invisible, would seem to present some of the -phenomena of life) the appearance of animation is not -due to any diffusion of life in the material parts of the -structure, but to the presence of a living being.</p> - -<p>The old hypothesis of a soul, a living organism within -the visible one, appears to me much more rational than the -attempt to explain life as a form of motion. And when we -consider the region of extreme minuteness characterised -by four-dimensional motion the difficulty of conceiving -such an organism alongside the bodily one disappears. -Lord Kelvin supposes that matter is formed from the -ether. We may very well suppose that the living -organisms directing the material ones are co-ordinate -with them, not composed of matter, but consisting of -etherial bodies, and as such capable of motion through -the ether, and able to originate material living bodies -throughout the mineral.</p> - -<p>Hypotheses such as these find no immediate ground for -proof or disproof in the physical world. Let us, therefore, -turn to a different field, and, assuming that the human -soul is a four-dimensional being, capable in itself of four -dimensional movements, but in its experiences through -the senses limited to three dimensions, ask if the history -of thought, of these productivities which characterise man, -correspond to our assumption. Let us pass in review -those steps by which man, presumably a four-dimensional<span class="pagenum" id="Page_21">[Pg 21]</span> -being, despite his bodily environment, has come to recognise -the fact of four-dimensional existence.</p> - -<p>Deferring this enquiry to another chapter, I will here -recapitulate the argument in order to show that our -purpose is entirely practical and independent of any -philosophical or metaphysical considerations.</p> - -<p>If two shots are fired at a target, and the second bullet -hits it at a different place to the first, we suppose that -there was some difference in the conditions under which -the second shot was fired from those affecting the first -shot. The force of the powder, the direction of aim, the -strength of the wind, or some condition must have been -different in the second case, if the course of the bullet was -not exactly the same as in the first case. Corresponding -to every difference in a result there must be some difference -in the antecedent material conditions. By tracing -out this chain of relations we explain nature.</p> - -<p>But there is also another mode of explanation which we -apply. If we ask what was the cause that a certain ship -was built, or that a certain structure was erected, we might -proceed to investigate the changes in the brain cells of -the men who designed the works. Every variation in one -ship or building from another ship or building is accompanied -by a variation in the processes that go on in the -brain matter of the designers. But practically this would -be a very long task.</p> - -<p>A more effective mode of explaining the production of -the ship or building would be to enquire into the motives, -plans, and aims of the men who constructed them. We -obtain a cumulative and consistent body of knowledge -much more easily and effectively in the latter way.</p> - -<p>Sometimes we apply the one, sometimes the other -mode of explanation.</p> - -<p>But it must be observed that the method of explanation -founded on aim, purpose, volition, always presupposes<span class="pagenum" id="Page_22">[Pg 22]</span> -a mechanical system on which the volition and aim -works. The conception of man as willing and acting -from motives involves that of a number of uniform processes -of nature which he can modify, and of which he -can make application. In the mechanical conditions of -the three-dimensional world, the only volitional agency -which we can demonstrate is the human agency. But -when we consider the four-dimensional world the -conclusion remains perfectly open.</p> - -<p>The method of explanation founded on purpose and aim -does not, surely, suddenly begin with man and end with -him. There is as much behind the exhibition of will and -motive which we see in man as there is behind the -phenomena of movement; they are co-ordinate, neither -to be resolved into the other. And the commencement -of the investigation of that will and motive which lies -behind the will and motive manifested in the three-dimensional -mechanical field is in the conception of a -soul—a four-dimensional organism, which expresses its -higher physical being in the symmetry of the body, and -gives the aims and motives of human existence.</p> - -<p>Our primary task is to form a systematic knowledge of -the phenomena of a four-dimensional world and find those -points in which this knowledge must be called in to -complete our mechanical explanation of the universe. -But a subsidiary contribution towards the verification of -the hypothesis may be made by passing in review the -history of human thought, and enquiring if it presents -such features as would be naturally expected on this -assumption.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_23">[Pg 23]</span></p> - -<h2 class="nobreak" id="CHAPTER_IV">CHAPTER IV<br /> - -<small><i>THE FIRST CHAPTER IN THE HISTORY -OF FOUR SPACE</i></small></h2></div> - - -<p>Parmenides, and the Asiatic thinkers with whom he is -in close affinity, propound a theory of existence which -is in close accord with a conception of a possible relation -between a higher and a lower dimensional space. This -theory, prior and in marked contrast to the main stream -of thought, which we shall afterwards describe, forms a -closed circle by itself. It is one which in all ages has -had a strong attraction for pure intellect, and is the -natural mode of thought for those who refrain from -projecting their own volition into nature under the guise -of causality.</p> - -<p>According to Parmenides of the school of Elea the all -is one, unmoving and unchanging. The permanent amid -the transient—that foothold for thought, that solid ground -for feeling on the discovery of which depends all our life—is -no phantom; it is the image amidst deception of true -being, the eternal, the unmoved, the one. Thus says -Parmenides.</p> - -<p>But how explain the shifting scene, these mutations -of things!</p> - -<p>“Illusion,” answers Parmenides. Distinguishing between -truth and error, he tells of the true doctrine of the -one—the false opinion of a changing world. He is no -less memorable for the manner of his advocacy than for<span class="pagenum" id="Page_24">[Pg 24]</span> -the cause he advocates. It is as if from his firm foothold -of being he could play with the thoughts under the -burden of which others laboured, for from him springs -that fluency of supposition and hypothesis which forms -the texture of Plato’s dialectic.</p> - -<p>Can the mind conceive a more delightful intellectual -picture than that of Parmenides, pointing to the one, the -true, the unchanging, and yet on the other hand ready to -discuss all manner of false opinion, forming a cosmogony -too, false “but mine own” after the fashion of the time?</p> - -<p>In support of the true opinion he proceeded by the -negative way of showing the self-contradictions in the -ideas of change and motion. It is doubtful if his criticism, -save in minor points, has ever been successfully refuted. -To express his doctrine in the ponderous modern way we -must make the statement that motion is phenomenal, -not real.</p> - -<p>Let us represent his doctrine.</p> - -<div class="figleft illowp35" id="fig_13" style="max-width: 9.375em;"> - <img src="images/fig_13.png" alt="" /> - <div class="caption">Fig. 13.</div> -</div> - -<p>Imagine a sheet of still water into which a slanting stick -is being lowered with a motion vertically -downwards. Let 1, 2, 3 (Fig. 13), -be three consecutive positions of the -stick. <span class="allsmcap">A</span>, <span class="allsmcap">B</span>, <span class="allsmcap">C</span>, will be three consecutive -positions of the meeting of the stick, -with the surface of the water. As -the stick passes down, the meeting will -move from <span class="allsmcap">A</span> on to <span class="allsmcap">B</span> and <span class="allsmcap">C</span>.</p> - -<p>Suppose now all the water to be -removed except a film. At the meeting -of the film and the stick there -will be an interruption of the film. -If we suppose the film to have a property, -like that of a soap bubble, of closing up round any -penetrating object, then as the stick goes vertically -downwards the interruption in the film will move on.</p> - -<p><span class="pagenum" id="Page_25">[Pg 25]</span></p> - -<div class="figleft illowp35" id="fig_14" style="max-width: 10em;"> - <img src="images/fig_14.png" alt="" /> - <div class="caption">Fig. 14.</div> -</div> - -<p>If we pass a spiral through the film the intersection -will give a point moving in a circle shown by the dotted -lines in the figure. Suppose -now the spiral to be still and -the film to move vertically -upwards, the whole spiral will -be represented in the film of -the consecutive positions of the -point of intersection. In the -film the permanent existence -of the spiral is experienced as -a time series—the record of -traversing the spiral is a point -moving in a circle. If now -we suppose a consciousness connected -with the film in such a way that the intersection of -the spiral with the film gives rise to a conscious experience, -we see that we shall have in the film a point moving in a -circle, conscious of its motion, knowing nothing of that -real spiral the record of the successive intersections of -which by the film is the motion of the point.</p> - -<p>It is easy to imagine complicated structures of the -nature of the spiral, structures consisting of filaments, -and to suppose also that these structures are distinguishable -from each other at every section. If we consider -the intersections of these filaments with the film as it -passes to be the atoms constituting a filmar universe, -we shall have in the film a world of apparent motion; -we shall have bodies corresponding to the filamentary -structure, and the positions of these structures with -regard to one another will give rise to bodies in the -film moving amongst one another. This mutual motion -is apparent merely. The reality is of permanent structures -stationary, and all the relative motions accounted for by -one steady movement of the film as a whole.</p> - -<p><span class="pagenum" id="Page_26">[Pg 26]</span></p> - -<p>Thus we can imagine a plane world, in which all the -variety of motion is the phenomenon of structures consisting -of filamentary atoms traversed by a plane of -consciousness. Passing to four dimensions and our -space, we can conceive that all things and movements -in our world are the reading off of a permanent reality -by a space of consciousness. Each atom at every moment -is not what it was, but a new part of that endless line -which is itself. And all this system successively revealed -in the time which is but the succession of consciousness, -separate as it is in parts, in its entirety is one vast unity. -Representing Parmenides’ doctrine thus, we gain a firmer -hold on it than if we merely let his words rest, grand and -massive, in our minds. And we have gained the means also -of representing phases of that Eastern thought to which -Parmenides was no stranger. Modifying his uncompromising -doctrine, let us suppose, to go back to the plane -of consciousness and the structure of filamentary atoms, -that these structures are themselves moving—are acting, -living. Then, in the transverse motion of the film, there -would be two phenomena of motion, one due to the reading -off in the film of the permanent existences as they are in -themselves, and another phenomenon of motion due to -the modification of the record of the things themselves, by -their proper motion during the process of traversing them.</p> - -<p>Thus a conscious being in the plane would have, as it -were, a two-fold experience. In the complete traversing -of the structure, the intersection of which with the film -gives his conscious all, the main and principal movements -and actions which he went through would be the record -of his higher self as it existed unmoved and unacting. -Slight modifications and deviations from these movements -and actions would represent the activity and self-determination -of the complete being, of his higher self.</p> - -<p>It is admissible to suppose that the consciousness in<span class="pagenum" id="Page_27">[Pg 27]</span> -the plane has a share in that volition by which the -complete existence determines itself. Thus the motive -and will, the initiative and life, of the higher being, would -be represented in the case of the being in the film by an -initiative and a will capable, not of determining any great -things or important movements in his existence, but only -of small and relatively insignificant activities. In all the -main features of his life his experience would be representative -of one state of the higher being whose existence -determines his as the film passes on. But in his minute -and apparently unimportant actions he would share in -that will and determination by which the whole of the -being he really is acts and lives.</p> - -<p>An alteration of the higher being would correspond to -a different life history for him. Let us now make the -supposition that film after film traverses these higher -structures, that the life of the real being is read off again -and again in successive waves of consciousness. There -would be a succession of lives in the different advancing -planes of consciousness, each differing from the preceding, -and differing in virtue of that will and activity which in -the preceding had not been devoted to the greater and -apparently most significant things in life, but the minute -and apparently unimportant. In all great things the -being of the film shares in the existence of his higher -self as it is at any one time. In the small things he -shares in that volition by which the higher being alters -and changes, acts and lives.</p> - -<p>Thus we gain the conception of a life changing and -developing as a whole, a life in which our separation and -cessation and fugitiveness are merely apparent, but which -in its events and course alters, changes, develops; and -the power of altering and changing this whole lies in the -will and power the limited being has of directing, guiding, -altering himself in the minute things of his existence.</p> - -<p><span class="pagenum" id="Page_28">[Pg 28]</span></p> - -<p>Transferring our conceptions to those of an existence in -a higher dimensionality traversed by a space of consciousness, -we have an illustration of a thought which has -found frequent and varied expression. When, however, -we ask ourselves what degree of truth there lies in it, we -must admit that, as far as we can see, it is merely symbolical. -The true path in the investigation of a higher -dimensionality lies in another direction.</p> - -<p>The significance of the Parmenidean doctrine lies in -this that here, as again and again, we find that those conceptions -which man introduces of himself, which he does -not derive from the mere record of his outward experience, -have a striking and significant correspondence to the -conception of a physical existence in a world of a higher -space. How close we come to Parmenides’ thought by -this manner of representation it is impossible to say. -What I want to point out is the adequateness of the -illustration, not only to give a static model of his doctrine, -but one capable as it were, of a plastic modification into a -correspondence into kindred forms of thought. Either one -of two things must be true—that four-dimensional conceptions -give a wonderful power of representing the thought -of the East, or that the thinkers of the East must have been -looking at and regarding four-dimensional existence.</p> - -<p>Coming now to the main stream of thought we must -dwell in some detail on Pythagoras, not because of his -direct relation to the subject, but because of his relation -to investigators who came later.</p> - -<p>Pythagoras invented the two-way counting. Let us -represent the single-way counting by the posits <i>aa</i>, -<i>ab</i>, <i>ac</i>, <i>ad</i>, using these pairs of letters instead of the -numbers 1, 2, 3, 4. I put an <i>a</i> in each case first for a -reason which will immediately appear.</p> - -<p>We have a sequence and order. There is no conception -of distance necessarily involved. The difference<span class="pagenum" id="Page_29">[Pg 29]</span> -between the posits is one of order not of distance—only -when identified with a number of equal material -things in juxtaposition does the notion of distance arise.</p> - -<p>Now, besides the simple series I can have, starting from -<i>aa</i>, <i>ba</i>, <i>ca</i>, <i>da</i>, from <i>ab</i>, <i>bb</i>, <i>cb</i>, <i>db</i>, and so on, and forming -a scheme:</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdlp"><i>da</i></td> -<td class="tdlp"><i>db</i></td> -<td class="tdlp"><i>dc</i></td> -<td class="tdlp"><i>dd</i></td> -</tr> -<tr> -<td class="tdlp"><i>ca</i></td> -<td class="tdlp"><i>cb</i></td> -<td class="tdlp"><i>cc</i></td> -<td class="tdlp"><i>cd</i></td> -</tr> -<tr> -<td class="tdlp"><i>ba</i></td> -<td class="tdlp"><i>bb</i></td> -<td class="tdlp"><i>bc</i></td> -<td class="tdlp"><i>bd</i></td> -</tr> -<tr> -<td class="tdlp"><i>aa</i></td> -<td class="tdlp"><i>ab</i></td> -<td class="tdlp"><i>ac</i></td> -<td class="tdlp"><i>ad</i></td> -</tr> -</table> - - -<p>This complex or manifold gives a two-way order. I can -represent it by a set of points, if I am on my guard -against assuming any relation of distance.</p> - -<div class="figleft illowp25" id="fig_15" style="max-width: 10em;"> - <img src="images/fig_15.png" alt="" /> - <div class="caption">Fig. 15.</div> -</div> - -<p>Pythagoras studied this two-fold way of -counting in reference to material bodies, and -discovered that most remarkable property of -the combination of number and matter that -bears his name.</p> - -<p>The Pythagorean property of an extended material -system can be exhibited in a manner which will be of -use to us afterwards, and which therefore I will employ -now instead of using the kind of figure which he himself -employed.</p> - -<p>Consider a two-fold field of points arranged in regular -rows. Such a field will be presupposed in the following -argument.</p> - -<div class="figleft illowp40" id="fig_16" style="max-width: 21.25em;"> - <img src="images/fig_16.png" alt="" /> - <div class="caption">Fig. 16.</div> -</div> - -<p>It is evident that in <a href="#fig_16">fig. 16</a> four -of the points determine a square, -which square we may take as the -unit of measurement for areas. -But we can also measure areas -in another way.</p> - -<p>Fig. 16 (1) shows four points determining a square.</p> - -<p>But four squares also meet in a point, <a href="#fig_16">fig. 16</a> (2).</p> - -<p>Hence a point at the corner of a square belongs equally -to four squares.</p> - -<p><span class="pagenum" id="Page_30">[Pg 30]</span></p> - -<p>Thus we may say that the point value of the square -shown is one point, for if we take the square in <a href="#fig_16">fig. 16</a> (1) -it has four points, but each of these belong equally to -four other squares. Hence one fourth of each of them -belongs to the square (1) in <a href="#fig_16">fig. 16</a>. Thus the point -value of the square is one point.</p> - -<p>The result of counting the points is the same as that -arrived at by reckoning the square units enclosed.</p> - -<p>Hence, if we wish to measure the area of any square -we can take the number of points it encloses, count these -as one each, and take one-fourth of the number of points -at its corners.</p> - -<div class="figleft illowp25" id="fig_17" style="max-width: 12.5em;"> - <img src="images/fig_17.png" alt="" /> - <div class="caption">Fig. 17.</div> -</div> - -<p>Now draw a diagonal square as shown in <a href="#fig_17">fig. 17</a>. It -contains one point and the four corners count for one -point more; hence its point value is 2. -The value is the measure of its area—the -size of this square is two of the unit squares.</p> - -<p>Looking now at the sides of this figure -we see that there is a unit square on each -of them—the two squares contain no points, -but have four corner points each, which gives the point -value of each as one point.</p> - -<p>Hence we see that the square on the diagonal is equal -to the squares on the two sides; or as it is generally -expressed, the square on the hypothenuse is equal to -the sum of the squares on the sides.</p> - -<div class="figleft illowp25" id="fig_18" style="max-width: 12.5em;"> - <img src="images/fig_18.png" alt="" /> - <div class="caption">Fig. 18.</div> -</div> - -<p>Noticing this fact we can proceed to ask if it is always -true. Drawing the square shown in <a href="#fig_18">fig. 18</a>, we can count -the number of its points. There are five -altogether. There are four points inside -the square on the diagonal, and hence, with -the four points at its corners the point -value is 5—that is, the area is 5. Now -the squares on the sides are respectively -of the area 4 and 1. Hence in this case also the square<span class="pagenum" id="Page_31">[Pg 31]</span> -on the diagonal is equal to the sum of the square on -the sides. This property of matter is one of the first -great discoveries of applied mathematics. We shall prove -afterwards that it is not a property of space. For the -present it is enough to remark that the positions in -which the points are arranged is entirely experimental. -It is by means of equal pieces of some material, or the -same piece of material moved from one place to another, -that the points are arranged.</p> - -<p>Pythagoras next enquired what the relation must be -so that a square drawn slanting-wise should be equal to -one straight-wise. He found that a square whose side is -five can be placed either rectangularly along the lines -of points, or in a slanting position. And this square is -equivalent to two squares of sides 4 and 3.</p> - -<p>Here he came upon a numerical relation embodied in -a property of matter. Numbers immanent in the objects -produced the equality so satisfactory for intellectual apprehension. -And he found that numbers when immanent -in sound—when the strings of a musical instrument -were given certain definite proportions of length—were -no less captivating to the ear than the equality of squares -was to the reason. What wonder then that he ascribed -an active power to number!</p> - -<p>We must remember that, sharing like ourselves the -search for the permanent in changing phenomena, the -Greeks had not that conception of the permanent in -matter that we have. To them material things were not -permanent. In fire solid things would vanish; absolutely -disappear. Rock and earth had a more stable existence, -but they too grew and decayed. The permanence of -matter, the conservation of energy, were unknown to -them. And that distinction which we draw so readily -between the fleeting and permanent causes of sensation, -between a sound and a material object, for instance, had<span class="pagenum" id="Page_32">[Pg 32]</span> -not the same meaning to them which it has for us. -Let us but imagine for a moment that material things -are fleeting, disappearing, and we shall enter with a far -better appreciation into that search for the permanent -which, with the Greeks, as with us, is the primary -intellectual demand.</p> - -<p>What is that which amid a thousand forms is ever the -same, which we can recognise under all its vicissitudes, -of which the diverse phenomena are the appearances?</p> - -<p>To think that this is number is not so very wide of -the mark. With an intellectual apprehension which far -outran the evidences for its application, the atomists -asserted that there were everlasting material particles, -which, by their union, produced all the varying forms and -states of bodies. But in view of the observed facts of -nature as then known, Aristotle, with perfect reason, -refused to accept this hypothesis.</p> - -<p>He expressly states that there is a change of quality, -and that the change due to motion is only one of the -possible modes of change.</p> - -<p>With no permanent material world about us, with -the fleeting, the unpermanent, all around we should, I -think, be ready to follow Pythagoras in his identification -of number with that principle which subsists amidst -all changes, which in multitudinous forms we apprehend -immanent in the changing and disappearing substance -of things.</p> - -<p>And from the numerical idealism of Pythagoras there -is but a step to the more rich and full idealism of Plato. -That which is apprehended by the sense of touch we -put as primary and real, and the other senses we say -are merely concerned with appearances. But Plato took -them all as valid, as giving qualities of existence. That -the qualities were not permanent in the world as given -to the senses forced him to attribute to them a different<span class="pagenum" id="Page_33">[Pg 33]</span> -kind of permanence. He formed the conception of a -world of ideas, in which all that really is, all that affects -us and gives the rich and wonderful wealth of our -experience, is not fleeting and transitory, but eternal. -And of this real and eternal we see in the things about -us the fleeting and transient images.</p> - -<p>And this world of ideas was no exclusive one, wherein -was no place for the innermost convictions of the soul and -its most authoritative assertions. Therein existed justice, -beauty—the one, the good, all that the soul demanded -to be. The world of ideas, Plato’s wonderful creation -preserved for man, for his deliberate investigation and -their sure development, all that the rude incomprehensible -changes of a harsh experience scatters and -destroys.</p> - -<p>Plato believed in the reality of ideas. He meets us -fairly and squarely. Divide a line into two parts, he -says; one to represent the real objects in the world, the -other to represent the transitory appearances, such as the -image in still water, the glitter of the sun on a bright -surface, the shadows on the clouds.</p> - -<div class="figcenter illowp100" id="i_033a" style="max-width: 50em;"> - <img src="images/i_033a.png" alt="" /> - <div class="caption"><table class="standard" summary=""> -<col width="30%" /><col width="20%" /><col width="30%" /> -<tr> -<td class="tdc">Real things:<br /> <i>e.g.</i>, the sun.</td> -<td></td> -<td class="tdc">Appearances:<br /> <i>e.g.</i>, the reflection of the sun.</td> -</tr> -</table> -</div> -</div> - -<p>Take another line and divide it into two parts, one -representing our ideas, the ordinary occupants of our -minds, such as whiteness, equality, and the other representing -our true knowledge, which is of eternal principles, -such as beauty, goodness.</p> - -<div class="figcenter illowp100" id="i_033b" style="max-width: 50em;"> - <img src="images/i_033b.png" alt="" /> - <div class="caption"><table class="standard" summary=""> -<col width="30%" /><col width="20%" /><col width="30%" /> -<tr> -<td class="tdc">Eternal principles,<br />as beauty.</td> -<td></td> -<td class="tdc"> Appearances in the mind,<br />as whiteness, equality</td> -</tr> -</table> -</div> -</div> - -<p>Then as A is to B, so is A<sup>1</sup> to B<sup>1</sup></p> - -<p>That is, the soul can proceed, going away from real<span class="pagenum" id="Page_34">[Pg 34]</span> -things to a region of perfect certainty, where it beholds -what is, not the scattered reflections; beholds the sun, not -the glitter on the sands; true being, not chance opinion.</p> - -<p>Now, this is to us, as it was to Aristotle, absolutely -inconceivable from a scientific point of view. We can -understand that a being is known in the fulness of his -relations; it is in his relations to his circumstances that -a man’s character is known; it is in his acts under his -conditions that his character exists. We cannot grasp or -conceive any principle of individuation apart from the -fulness of the relations to the surroundings.</p> - -<p>But suppose now that Plato is talking about the higher -man—the four-dimensional being that is limited in our -external experience to a three-dimensional world. Do not -his words begin to have a meaning? Such a being -would have a consciousness of motion which is not as -the motion he can see with the eyes of the body. He, -in his own being, knows a reality to which the outward -matter of this too solid earth is flimsy superficiality. He -too knows a mode of being, the fulness of relations, in -which can only be represented in the limited world of -sense, as the painter unsubstantially portrays the depths -of woodland, plains, and air. Thinking of such a being -in man, was not Plato’s line well divided?</p> - -<p>It is noteworthy that, if Plato omitted his doctrine of -the independent origin of ideas, he would present exactly -the four-dimensional argument; a real thing as we think -it is an idea. A plane being’s idea of a square object is -the idea of an abstraction, namely, a geometrical square. -Similarly our idea of a solid thing is an abstraction, for in -our idea there is not the four-dimensional thickness which -is necessary, however slight, to give reality. The argument -would then run, as a shadow is to a solid object, so -is the solid object to the reality. Thus A and B´ would -be identified.</p> - -<p><span class="pagenum" id="Page_35">[Pg 35]</span></p> - -<p>In the allegory which I have already alluded to, Plato -in almost as many words shows forth the relation between -existence in a superficies and in solid space. And he -uses this relation to point to the conditions of a higher -being.</p> - -<p>He imagines a number of men prisoners, chained so -that they look at the wall of a cavern in which they are -confined, with their backs to the road and the light. -Over the road pass men and women, figures and processions, -but of all this pageant all that the prisoners -behold is the shadow of it on the wall whereon they gaze. -Their own shadows and the shadows of the things in the -world are all that they see, and identifying themselves -with their shadows related as shadows to a world of -shadows, they live in a kind of dream.</p> - -<p>Plato imagines one of their number to pass out from -amongst them into the real space world, and then returning -to tell them of their condition.</p> - -<p>Here he presents most plainly the relation between -existence in a plane world and existence in a three-dimensional -world. And he uses this illustration as a -type of the manner in which we are to proceed to a -higher state from the three-dimensional life we know.</p> - -<p>It must have hung upon the weight of a shadow which -path he took!—whether the one we shall follow toward -the higher solid and the four-dimensional existence, or -the one which makes ideas the higher realities, and the -direct perception of them the contact with the truer -world.</p> - -<p>Passing on to Aristotle, we will touch on the points -which most immediately concern our enquiry.</p> - -<p>Just as a scientific man of the present day in -reviewing the speculations of the ancient world would -treat them with a curiosity half amused but wholly -respectful, asking of each and all wherein lay their<span class="pagenum" id="Page_36">[Pg 36]</span> -relation to fact, so Aristotle, in discussing the philosophy -of Greece as he found it, asks, above all other things: -“Does this represent the world? In this system is there -an adequate presentation of what is?”</p> - -<p>He finds them all defective, some for the very reasons -which we esteem them most highly, as when he criticises -the Atomic theory for its reduction of all change to motion. -But in the lofty march of his reason he never loses sight -of the whole; and that wherein our views differ from his -lies not so much in a superiority of our point of view, as -in the fact which he himself enunciates—that it is impossible -for one principle to be valid in all branches of -enquiry. The conceptions of one method of investigation -are not those of another; and our divergence lies in our -exclusive attention to the conceptions useful in one way -of apprehending nature rather than in any possibility we -find in our theories of giving a view of the whole transcending -that of Aristotle.</p> - -<p>He takes account of everything; he does not separate -matter and the manifestation of matter; he fires all -together in a conception of a vast world process in -which everything takes part—the motion of a grain of -dust, the unfolding of a leaf, the ordered motion of the -spheres in heaven—all are parts of one whole which -he will not separate into dead matter and adventitious -modifications.</p> - -<p>And just as our theories, as representative of actuality, -fall before his unequalled grasp of fact, so the doctrine -of ideas fell. It is not an adequate account of existence, -as Plato himself shows in his “Parmenides”; -it only explains things by putting their doubles beside -them.</p> - -<p>For his own part Aristotle invented a great marching -definition which, with a kind of power of its own, cleaves -its way through phenomena to limiting conceptions on<span class="pagenum" id="Page_37">[Pg 37]</span> -either hand, towards whose existence all experience -points.</p> - -<p>In Aristotle’s definition of matter and form as the -constituent of reality, as in Plato’s mystical vision of the -kingdom of ideas, the existence of the higher dimensionality -is implicitly involved.</p> - -<p>Substance according to Aristotle is relative, not absolute. -In everything that is there is the matter of which it -is composed, the form which it exhibits; but these are -indissolubly connected, and neither can be thought -without the other.</p> - -<p>The blocks of stone out of which a house is built are the -material for the builder; but, as regards the quarrymen, -they are the matter of the rocks with the form he has -imposed on them. Words are the final product of the -grammarian, but the mere matter of the orator or poet. -The atom is, with us, that out of which chemical substances -are built up, but looked at from another point of view is -the result of complex processes.</p> - -<p>Nowhere do we find finality. The matter in one sphere -is the matter, plus form, of another sphere of thought. -Making an obvious application to geometry, plane figures -exist as the limitation of different portions of the plane -by one another. In the bounding lines the separated -matter of the plane shows its determination into form. -And as the plane is the matter relatively to determinations -in the plane, so the plane itself exists in virtue of the -determination of space. A plane is that wherein formless -space has form superimposed on it, and gives an actuality -of real relations. We cannot refuse to carry this process -of reasoning a step farther back, and say that space itself -is that which gives form to higher space. As a line is -the determination of a plane, and a plane of a solid, so -solid space itself is the determination of a higher space.</p> - -<p>As a line by itself is inconceivable without that plane<span class="pagenum" id="Page_38">[Pg 38]</span> -which it separates, so the plane is inconceivable without -the solids which it limits on either hand. And so space -itself cannot be positively defined. It is the negation -of the possibility of movement in more than three -dimensions. The conception of space demands that of -a higher space. As a surface is thin and unsubstantial -without the substance of which it is the surface, so matter -itself is thin without the higher matter.</p> - -<p>Just as Aristotle invented that algebraical method of -representing unknown quantities by mere symbols, not by -lines necessarily determinate in length as was the habit -of the Greek geometers, and so struck out the path -towards those objectifications of thought which, like -independent machines for reasoning, supply the mathematician -with his analytical weapons, so in the formulation -of the doctrine of matter and form, of potentiality and -actuality, of the relativity of substance, he produced -another kind of objectification of mind—a definition -which had a vital force and an activity of its own.</p> - -<p>In none of his writings, as far as we know, did he carry it -to its legitimate conclusion on the side of matter, but in -the direction of the formal qualities he was led to his -limiting conception of that existence of pure form which -lies beyond all known determination of matter. The -unmoved mover of all things is Aristotle’s highest -principle. Towards it, to partake of its perfection all -things move. The universe, according to Aristotle, is an -active process—he does not adopt the illogical conception -that it was once set in motion and has kept on ever since. -There is room for activity, will, self-determination, in -Aristotle’s system, and for the contingent and accidental -as well. We do not follow him, because we are accustomed -to find in nature infinite series, and do not feel -obliged to pass on to a belief in the ultimate limits to -which they seem to point.</p> - -<p><span class="pagenum" id="Page_39">[Pg 39]</span></p> - -<p>But apart from the pushing to the limit, as a relative -principle this doctrine of Aristotle’s as to the relativity of -substance is irrefragible in its logic. He was the first to -show the necessity of that path of thought which when -followed leads to a belief in a four-dimensional space.</p> - -<p>Antagonistic as he was to Plato in his conception -of the practical relation of reason to the world of -phenomena, yet in one point he coincided with him. -And in this he showed the candour of his intellect. He -was more anxious to lose nothing than to explain everything. -And that wherein so many have detected an -inconsistency, an inability to free himself from the school -of Plato, appears to us in connection with our enquiry -as an instance of the acuteness of his observation. For -beyond all knowledge given by the senses Aristotle held -that there is an active intelligence, a mind not the passive -recipient of impressions from without, but an active and -originative being, capable of grasping knowledge at first -hand. In the active soul Aristotle recognised something -in man not produced by his physical surroundings, something -which creates, whose activity is a knowledge -underived from sense. This, he says, is the immortal and -undying being in man.</p> - -<p>Thus we see that Aristotle was not far from the -recognition of the four-dimensional existence, both -without and within man, and the process of adequately -realising the higher dimensional figures to which we -shall come subsequently is a simple reduction to practice -of his hypothesis of a soul.</p> - -<p>The next step in the unfolding of the drama of the -recognition of the soul as connected with our scientific -conception of the world, and, at the same time, the -recognition of that higher of which a three-dimensional -world presents the superficial appearance, took place many -centuries later. If we pass over the intervening time<span class="pagenum" id="Page_40">[Pg 40]</span> -without a word it is because the soul was occupied with -the assertion of itself in other ways than that of knowledge. -When it took up the task in earnest of knowing this -material world in which it found itself, and of directing -the course of inanimate nature, from that most objective -aim came, reflected back as from a mirror, its knowledge -of itself.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_41">[Pg 41]</span></p> - -<h2 class="nobreak" id="CHAPTER_V">CHAPTER V<br /> - -<small><i>THE SECOND CHAPTER IN THE HISTORY -OF FOUR SPACE</i></small></h2></div> - - -<p><span class="smcap">Lobatchewsky, Bolyai, and Gauss</span> -Before entering on a description of the work of -Lobatchewsky and Bolyai it will not be out of place -to give a brief account of them, the materials for which -are to be found in an article by Franz Schmidt in the -forty-second volume of the <i>Mathematische Annalen</i>, -and in Engel’s edition of Lobatchewsky.</p> - -<p>Lobatchewsky was a man of the most complete and -wonderful talents. As a youth he was full of vivacity, -carrying his exuberance so far as to fall into serious -trouble for hazing a professor, and other freaks. Saved -by the good offices of the mathematician Bartels, who -appreciated his ability, he managed to restrain himself -within the bounds of prudence. Appointed professor at -his own University, Kasan, he entered on his duties under -the regime of a pietistic reactionary, who surrounded -himself with sycophants and hypocrites. Esteeming -probably the interests of his pupils as higher than any -attempt at a vain resistance, he made himself the tyrant’s -right-hand man, doing an incredible amount of teaching -and performing the most varied official duties. Amidst -all his activities he found time to make important contributions -to science. His theory of parallels is most<span class="pagenum" id="Page_42">[Pg 42]</span> -closely connected with his name, but a study of his -writings shows that he was a man capable of carrying -on mathematics in its main lines of advance, and of a -judgment equal to discerning what these lines were. -Appointed rector of his University, he died at an -advanced age, surrounded by friends, honoured, with the -results of his beneficent activity all around him. To him -no subject came amiss, from the foundations of geometry -to the improvement of the stoves by which the peasants -warmed their houses.</p> - -<p>He was born in 1793. His scientific work was -unnoticed till, in 1867, Houel, the French mathematician, -drew attention to its importance.</p> - -<p>Johann Bolyai de Bolyai was born in Klausenburg, -a town in Transylvania, December 15th, 1802.</p> - -<p>His father, Wolfgang Bolyai, a professor in the -Reformed College of Maros Vasarhely, retained the ardour -in mathematical studies which had made him a chosen -companion of Gauss in their early student days at -Göttingen.</p> - -<p>He found an eager pupil in Johann. He relates that -the boy sprang before him like a devil. As soon as he -had enunciated a problem the child would give the -solution and command him to go on further. As a -thirteen-year-old boy his father sometimes sent him to fill -his place when incapacitated from taking his classes. -The pupils listened to him with more attention than to -his father for they found him clearer to understand.</p> - -<p>In a letter to Gauss Wolfgang Bolyai writes:—</p> - -<p>“My boy is strongly built. He has learned to recognise -many constellations, and the ordinary figures of geometry. -He makes apt applications of his notions, drawing for -instance the positions of the stars with their constellations. -Last winter in the country, seeing Jupiter he asked: -‘How is it that we can see him from here as well as from<span class="pagenum" id="Page_43">[Pg 43]</span> -the town? He must be far off.’ And as to three -different places to which he had been he asked me to tell -him about them in one word. I did not know what he -meant, and then he asked me if one was in a line with -the other and all in a row, or if they were in a triangle.</p> - -<p>“He enjoys cutting paper figures with a pair of scissors, -and without my ever having told him about triangles -remarked that a right-angled triangle which he had cut -out was half of an oblong. I exercise his body with care, -he can dig well in the earth with his little hands. The -blossom can fall and no fruit left. When he is fifteen -I want to send him to you to be your pupil.”</p> - -<p>In Johann’s autobiography he says:—</p> - -<p>“My father called my attention to the imperfections -and gaps in the theory of parallels. He told me he had -gained more satisfactory results than his predecessors, -but had obtained no perfect and satisfying conclusion. -None of his assumptions had the necessary degree of -geometrical certainty, although they sufficed to prove the -eleventh axiom and appeared acceptable on first sight.</p> - -<p>“He begged of me, anxious not without a reason, to -hold myself aloof and to shun all investigation on this -subject, if I did not wish to live all my life in vain.”</p> - -<p>Johann, in the failure of his father to obtain any -response from Gauss, in answer to a letter in which he -asked the great mathematician to make of his son “an -apostle of truth in a far land,” entered the Engineering -School at Vienna. He writes from Temesvar, where he -was appointed sub-lieutenant September, 1823:—</p> - -<div class="blockquote"> -<p class="psig"> -“Temesvar, November 3rd, 1823.</p> - -<p>“<span class="smcap">Dear Good Father</span>, -</p> - -<p>“I have so overwhelmingly much to write -about my discovery that I know no other way of checking -myself than taking a quarter of a sheet only to write on. -I want an answer to my four-sheet letter.</p> - -<p><span class="pagenum" id="Page_44">[Pg 44]</span></p> - -<p>“I am unbroken in my determination to publish a -work on Parallels, as soon as I have put my material in -order and have the means.</p> - -<p>“At present I have not made any discovery, but -the way I have followed almost certainly promises me -the attainment of my object if any possibility of it -exists.</p> - -<p>“I have not got my object yet, but I have produced -such stupendous things that I was overwhelmed myself, -and it would be an eternal shame if they were lost. -When you see them you will find that it is so. Now -I can only say that I have made a new world out of -nothing. Everything that I have sent you before is a -house of cards in comparison with a tower. I am convinced -that it will be no less to my honour than if I had -already discovered it.”</p> -</div> - -<p>The discovery of which Johann here speaks was -published as an appendix to Wolfgang Bolyai’s <i>Tentamen</i>.</p> - -<p>Sending the book to Gauss, Wolfgang writes, after an -interruption of eighteen years in his correspondence:—</p> - -<div class="blockquote"> - -<p>“My son is first lieutenant of Engineers and will soon -be captain. He is a fine youth, a good violin player, -a skilful fencer, and brave, but has had many duels, and -is wild even for a soldier. Yet he is distinguished—light -in darkness and darkness in light. He is an impassioned -mathematician with extraordinary capacities.... He -will think more of your judgment on his work than that -of all Europe.”</p> -</div> - -<p>Wolfgang received no answer from Gauss to this letter, -but sending a second copy of the book received the -following reply:—</p> - -<div class="blockquote"> -<p>“You have rejoiced me, my unforgotten friend, by your -letters. I delayed answering the first because I wanted -to wait for the arrival of the promised little book.</p> - -<p>“Now something about your son’s work.</p> - -<p><span class="pagenum" id="Page_45">[Pg 45]</span></p> - -<p>“If I begin with saying that ‘I ought not to praise it,’ -you will be staggered for a moment. But I cannot say -anything else. To praise it is to praise myself, for the -path your son has broken in upon and the results to which -he has been led are almost exactly the same as my own -reflections, some of which date from thirty to thirty-five -years ago.</p> - -<p>“In fact I am astonished to the uttermost. My intention -was to let nothing be known in my lifetime about -my own work, of which, for the rest, but little is committed -to writing. Most people have but little perception -of the problem, and I have found very few who took any -interest in the views I expressed to them. To be able to -do that one must first of all have had a real live feeling -of what is wanting, and as to that most men are completely -in the dark.</p> - -<p>“Still it was my intention to commit everything to -writing in the course of time, so that at least it should -not perish with me.</p> - -<p>“I am deeply surprised that this task can be spared -me, and I am most of all pleased in this that it is the son -of my old friend who has in so remarkable a manner -preceded me.”</p> -</div> - -<p>The impression which we receive from Gauss’s inexplicable -silence towards his old friend is swept away -by this letter. Hence we breathe the clear air of the -mountain tops. Gauss would not have failed to perceive -the vast significance of his thoughts, sure to be all the -greater in their effect on future ages from the want of -comprehension of the present. Yet there is not a word -or a sign in his writing to claim the thought for himself. -He published no single line on the subject. By the -measure of what he thus silently relinquishes, by such a -measure of a world-transforming thought, we can appreciate -his greatness.</p> - -<p><span class="pagenum" id="Page_46">[Pg 46]</span></p> - -<p>It is a long step from Gauss’s serenity to the disturbed -and passionate life of Johann Bolyai—he and Galois, -the two most interesting figures in the history of mathematics. -For Bolyai, the wild soldier, the duellist, fell -at odds with the world. It is related of him that he was -challenged by thirteen officers of his garrison, a thing not -unlikely to happen considering how differently he thought -from every one else. He fought them all in succession—making -it his only condition that he should be allowed -to play on his violin for an interval between meeting each -opponent. He disarmed or wounded all his antagonists. -It can be easily imagined that a temperament such as -his was one not congenial to his military superiors. He -was retired in 1833.</p> - -<p>His epoch-making discovery awoke no attention. He -seems to have conceived the idea that his father had -betrayed him in some inexplicable way by his communications -with Gauss, and he challenged the excellent -Wolfgang to a duel. He passed his life in poverty, -many a time, says his biographer, seeking to snatch -himself from dissipation and apply himself again to -mathematics. But his efforts had no result. He died -January 27th, 1860, fallen out with the world and with -himself.</p> - - -<h3><span class="smcap">Metageometry</span></h3> - -<p>The theories which are generally connected with the -names of Lobatchewsky and Bolyai bear a singular and -curious relation to the subject of higher space.</p> - -<p>In order to show what this relation is, I must ask the -reader to be at the pains to count carefully the sets of -points by which I shall estimate the volumes of certain -figures.</p> - -<p><span class="pagenum" id="Page_47">[Pg 47]</span></p> - -<p>No mathematical processes beyond this simple one of -counting will be necessary.</p> - -<div class="figleft illowp25" id="fig_19" style="max-width: 12.5em;"> - <img src="images/fig_19.png" alt="" /> - <div class="caption">Fig. 19.</div> -</div> - -<p>Let us suppose we have before us in -<a href="#fig_19">fig. 19</a> a plane covered with points at regular -intervals, so placed that every four determine -a square.</p> - -<p>Now it is evident that as four points -determine a square, so four squares meet in a point.</p> - -<div class="figleft illowp25" id="fig_20" style="max-width: 12.5em;"> - <img src="images/fig_20.png" alt="" /> - <div class="caption">Fig. 20.</div> -</div> - -<p>Thus, considering a point inside a square as -belonging to it, we may say that a point on -the corner of a square belongs to it and to -three others equally: belongs a quarter of it -to each square.</p> - -<p>Thus the square <span class="allsmcap">ACDE</span> (<a href="#fig_21">fig. 21</a>) contains one point, and -has four points at the four corners. Since one-fourth of -each of these four belongs to the square, the four together -count as one point, and the point value of the square is -two points—the one inside and the four at the corner -make two points belonging to it exclusively.</p> - -<div class="figleft illowp25" id="fig_21" style="max-width: 12.5em;"> - <img src="images/fig_21.png" alt="" /> - <div class="caption">Fig. 21.</div> -</div> - -<div class="figright illowp25" id="fig_22" style="max-width: 12.8125em;"> - <img src="images/fig_22.png" alt="" /> - <div class="caption">Fig. 22.</div> -</div> - -<p>Now the area of this square is two unit squares, as can -be seen by drawing two diagonals in <a href="#fig_22">fig. 22</a>.</p> - -<p>We also notice that the square in question is equal to -the sum of the squares on the sides <span class="allsmcap">AB</span>, <span class="allsmcap">BC</span>, of the right-angled -triangle <span class="allsmcap">ABC</span>. Thus we recognise the proposition -that the square on the hypothenuse is equal to the sum -of the squares on the two sides of a right-angled triangle.</p> - -<p>Now suppose we set ourselves the question of determining -the whereabouts in the ordered system of points,<span class="pagenum" id="Page_48">[Pg 48]</span> -the end of a line would come when it turned about a -point keeping one extremity fixed at the point.</p> - -<p>We can solve this problem in a particular case. If we -can find a square lying slantwise amongst the dots which is -equal to one which goes regularly, we shall know that the -two sides are equal, and that the slanting side is equal to the -straight-way side. Thus the volume and shape of a figure -remaining unchanged will be the test of its having rotated -about the point, so that we can say that its side in its first -position would turn into its side in the second position.</p> - -<p>Now, such a square can be found in the one whose side -is five units in length.</p> - -<div class="figcenter illowp66" id="fig_23" style="max-width: 25em;"> - <img src="images/fig_23.png" alt="" /> - <div class="caption">Fig. 23.</div> -</div> - -<p>In <a href="#fig_23">fig. 23</a>, in the square on <span class="allsmcap">AB</span>, there are—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdl">9 points interior</td> -<td class="tdr">9</td> -</tr> -<tr> -<td class="tdl">4 at the corners</td> -<td class="tdr">1</td> -</tr> -<tr> -<td class="tdh"> 4 sides with 3 on each side, considered as -1½ on each side, because belonging -equally to two squares</td> -<td class="tdrb">6</td> -</tr> -</table> - -<p>The total is 16. There are 9 points in the square -on <span class="allsmcap">BC</span>.</p> - -<p><span class="pagenum" id="Page_49">[Pg 49]</span></p> - -<p>In the square on <span class="allsmcap">AC</span> there are—</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdl">24 points inside</td> -<td class="tdr">24</td> -</tr> -<tr> -<td class="tdl"> 4 at the corners</td> -<td class="tdr">1</td> -</tr> -</table> - -<p>or 25 altogether.</p> - -<p>Hence we see again that the square on the hypothenuse -is equal to the squares on the sides.</p> - -<p>Now take the square <span class="allsmcap">AFHG</span>, which is larger than the -square on <span class="allsmcap">AB</span>. It contains 25 points.</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdl">16 inside</td> -<td class="tdr">16</td> -</tr> -<tr> -<td class="tdl">16 on the sides, counting as</td> -<td class="tdr">8</td> -</tr> -<tr> -<td class="tdl"> 4 on the corners</td> -<td class="tdr">1</td> -</tr> -</table> - -<p>making 25 altogether.</p> - -<p>If two squares are equal we conclude the sides are -equal. Hence, the line <span class="allsmcap">AF</span> turning round <span class="allsmcap">A</span> would -move so that it would after a certain turning coincide -with <span class="allsmcap">AC</span>.</p> - -<p>This is preliminary, but it involves all the mathematical -difficulties that will present themselves.</p> - -<p>There are two alterations of a body by which its volume -is not changed.</p> - -<p>One is the one we have just considered, rotation, the -other is what is called shear.</p> - -<p>Consider a book, or heap of loose pages. They can be -slid so that each one slips -over the preceding one, -and the whole assumes -the shape <i>b</i> in <a href="#fig_24">fig. 24</a>.</p> - -<div class="figleft illowp50" id="fig_24" style="max-width: 25em;"> - <img src="images/fig_24.png" alt="" /> - <div class="caption">Fig. 24.</div> -</div> - -<p>This deformation is not shear alone, but shear accompanied -by rotation.</p> - -<p>Shear can be considered as produced in another way.</p> - -<p>Take the square <span class="allsmcap">ABCD</span> (<a href="#fig_25">fig. 25</a>), and suppose that it -is pulled out from along one of its diagonals both ways, -and proportionately compressed along the other diagonal. -It will assume the shape in <a href="#fig_26">fig. 26</a>.</p> - -<p><span class="pagenum" id="Page_50">[Pg 50]</span></p> - -<p>This compression and expansion along two lines at right -angles is what is called shear; it is equivalent to the -sliding illustrated above, combined with a turning round.</p> - -<div class="figleft illowp45" id="fig_25" style="max-width: 12.5em;"> - <img src="images/fig_25.png" alt="" /> - <div class="caption">Fig. 25.</div> -</div> - -<div class="figright illowp50" id="fig_26" style="max-width: 18.75em;"> - <img src="images/fig_26.png" alt="" /> - <div class="caption">Fig. 26.</div> -</div> - -<p>In pure shear a body is compressed and extended in -two directions at right angles to each other, so that its -volume remains unchanged.</p> - -<p>Now we know that our material bodies resist shear—shear -does violence to the internal arrangement of their -particles, but they turn as wholes without such internal -resistance.</p> - -<p>But there is an exception. In a liquid shear and -rotation take place equally easily, there is no more -resistance against a shear than there is against a -rotation.</p> - -<p>Now, suppose all bodies were to be reduced to the liquid -state, in which they yield to shear and to rotation equally -easily, and then were to be reconstructed as solids, but in -such a way that shear and rotation had interchanged -places.</p> - -<p>That is to say, let us suppose that when they had -become solids again they would shear without offering -any internal resistance, but a rotation would do violence -to their internal arrangement.</p> - -<p>That is, we should have a world in which shear would -have taken the place of rotation.</p> - -<p><span class="pagenum" id="Page_51">[Pg 51]</span></p> - -<p>A shear does not alter the volume of a body: thus an -inhabitant living in such a world would look on a body -sheared as we look on a body rotated. He would say -that it was of the same shape, but had turned a bit -round.</p> - -<p>Let us imagine a Pythagoras in this world going to -work to investigate, as is his wont.</p> - -<div class="figleft illowp40" id="fig_27" style="max-width: 12.5em;"> - <img src="images/fig_27.png" alt="" /> - <div class="caption">Fig. 27.</div> -</div> -<div class="figright illowp40" id="fig_28" style="max-width: 13.125em;"> - <img src="images/fig_28.png" alt="" /> - <div class="caption">Fig. 28.</div> -</div> - -<p>Fig. 27 represents a square unsheared. Fig. 28 -represents a square sheared. It is not the figure into -which the square in <a href="#fig_27">fig. 27</a> would turn, but the result of -shear on some square not drawn. It is a simple slanting -placed figure, taken now as we took a simple slanting -placed square before. Now, since bodies in this world of -shear offer no internal resistance to shearing, and keep -their volume when sheared, an inhabitant accustomed to -them would not consider that they altered their shape -under shear. He would call <span class="allsmcap">ACDE</span> as much a square as -the square in <a href="#fig_27">fig. 27</a>. We will call such figures shear -squares. Counting the dots in <span class="allsmcap">ACDE</span>, we find—</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdl">2 inside</td> -<td class="tdc">=</td> -<td class="tdc">2</td> -</tr> -<tr> -<td class="tdl">4 at corners</td> -<td class="tdc">=</td> -<td class="tdc">1</td> -</tr> -</table> - -<p>or a total of 3.</p> - -<p>Now, the square on the side <span class="allsmcap">AB</span> has 4 points, that on <span class="allsmcap">BC</span> -has 1 point. Here the shear square on the hypothenuse -has not 5 points but 3; it is not the sum of the squares on -the sides, but the difference.</p> - -<p><span class="pagenum" id="Page_52">[Pg 52]</span></p> - -<div class="figleft illowp25" id="fig_29" style="max-width: 13.75em;"> - <img src="images/fig_29.png" alt="" /> - <div class="caption">Fig. 29.</div> -</div> - -<p>This relation always holds. Look at -<a href="#fig_29">fig. 29</a>.</p> - -<p>Shear square on hypothenuse—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdl">7 internal</td> -<td class="tdr"> 7</td> -</tr> -<tr> -<td class="tdl">4 at corners</td> -<td class="tdr">1</td> -</tr> -<tr> -<td class="tdl"></td> -<td class="tdr_bt">8</td> -</tr> -</table> - - -<div class="figleft illowp50" id="fig_29bis" style="max-width: 25em;"> - <img src="images/fig_29bis.png" alt="" /> - <div class="caption">Fig. 29 <i>bis</i>.</div> -</div> - -<p>Square on one side—which the reader can draw for -himself—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdl">4 internal</td> -<td class="tdr"> 4</td> -</tr> -<tr> -<td class="tdl">8 on sides</td> -<td class="tdr">4</td> -</tr> -<tr> -<td class="tdl">4 at corners</td> -<td class="tdr">1</td> -</tr> -<tr> -<td class="tdl"></td> -<td class="tdr_bt">9</td> -</tr> -</table> - - -<p>and the square on the other -side is 1. Hence in this -case again the difference is -equal to the shear square on -the hypothenuse, 9 - 1 = 8.</p> - -<p>Thus in a world of shear -the square on the hypothenuse -would be equal to the -difference of the squares on -the sides of a right-angled -triangle.</p> - -<p>In <a href="#fig_29">fig. 29</a> <i>bis</i> another shear square is drawn on which -the above relation can be tested.</p> - -<p>What now would be the position a line on turning by -shear would take up?</p> - -<p>We must settle this in the same way as previously with -our turning.</p> - -<p>Since a body sheared remains the same, we must find two -equal bodies, one in the straight way, one in the slanting -way, which have the same volume. Then the side of one -will by turning become the side of the other, for the two -figures are each what the other becomes by a shear turning.</p> - -<p><span class="pagenum" id="Page_53">[Pg 53]</span></p> - -<p>We can solve the problem in a particular case—</p> - -<div class="figleft illowp50" id="fig_30" style="max-width: 25em;"> - <img src="images/fig_30.png" alt="" /> - <div class="caption">Fig. 30.</div> -</div> - -<p>In the figure <span class="allsmcap">ACDE</span> -(<a href="#fig_30">fig. 30</a>) there are—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdr">15 inside</td> -<td class="tdl">15</td> -</tr> -<tr> -<td class="tdl">4 at corners</td> -<td class="tdr"> 1</td> -</tr> -</table> - -<p>a total of 16.</p> - -<p>Now in the square <span class="allsmcap">ABGF</span>, -there are 16—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdl">9 inside</td> -<td class="tdr"> 9</td> -</tr> -<tr> -<td class="tdl">12 on sides</td> -<td class="tdr">6</td> -</tr> -<tr> -<td class="tdl">4 at corners</td> -<td class="tdr">1</td> -</tr> -<tr> -<td></td> -<td class="tdr_bt">16</td> -</tr> -</table> - -<p>Hence the square on <span class="allsmcap">AB</span> -would, by the shear turning, -become the shear square -<span class="allsmcap">ACDE</span>.</p> - -<p>And hence the inhabitant of this world would say that -the line <span class="allsmcap">AB</span> turned into the line <span class="allsmcap">AC</span>. These two lines -would be to him two lines of equal length, one turned -a little way round from the other.</p> - -<p>That is, putting shear in place of rotation, we get a -different kind of figure, as the result of the shear rotation, -from what we got with our ordinary rotation. And as a -consequence we get a position for the end of a line of -invariable length when it turns by the shear rotation, -different from the position which it would assume on -turning by our rotation.</p> - -<p>A real material rod in the shear world would, on turning -about <span class="allsmcap">A</span>, pass from the position <span class="allsmcap">AB</span> to the position <span class="allsmcap">AC</span>. -We say that its length alters when it becomes <span class="allsmcap">AC</span>, but this -transformation of <span class="allsmcap">AB</span> would seem to an inhabitant of the -shear world like a turning of <span class="allsmcap">AB</span> without altering in -length.</p> - -<p>If now we suppose a communication of ideas that takes -place between one of ourselves and an inhabitant of the<span class="pagenum" id="Page_54">[Pg 54]</span> -shear world, there would evidently be a difference between -his views of distance and ours.</p> - -<p>We should say that his line <span class="allsmcap">AB</span> increased in length in -turning to <span class="allsmcap">AC</span>. He would say that our line <span class="allsmcap">AF</span> (<a href="#fig_23">fig. 23</a>) -decreased in length in turning to <span class="allsmcap">AC</span>. He would think -that what we called an equal line was in reality a shorter -one.</p> - -<p>We should say that a rod turning round would have its -extremities in the positions we call at equal distances. -So would he—but the positions would be different. He -could, like us, appeal to the properties of matter. His -rod to him alters as little as ours does to us.</p> - -<p>Now, is there any standard to which we could appeal, to -say which of the two is right in this argument? There -is no standard.</p> - -<p>We should say that, with a change of position, the -configuration and shape of his objects altered. He would -say that the configuration and shape of our objects altered -in what we called merely a change of position. Hence -distance independent of position is inconceivable, or -practically distance is solely a property of matter.</p> - -<p>There is no principle to which either party in this -controversy could appeal. There is nothing to connect -the definition of distance with our ideas rather than with -his, except the behaviour of an actual piece of matter.</p> - -<p>For the study of the processes which go on in our world -the definition of distance given by taking the sum of the -squares is of paramount importance to us. But as a question -of pure space without making any unnecessary -assumptions the shear world is just as possible and just as -interesting as our world.</p> - -<p>It was the geometry of such conceivable worlds that -Lobatchewsky and Bolyai studied.</p> - -<p>This kind of geometry has evidently nothing to do -directly with four-dimensional space.</p> - -<p><span class="pagenum" id="Page_55">[Pg 55]</span></p> - -<p>But a connection arises in this way. It is evident that, -instead of taking a simple shear as I have done, and -defining it as that change of the arrangement of the -particles of a solid which they will undergo without -offering any resistance due to their mutual action, I -might take a complex motion, composed of a shear and -a rotation together, or some other kind of deformation.</p> - -<p>Let us suppose such an alteration picked out and -defined as the one which means simple rotation, then the -type, according to which all bodies will alter by this -rotation, is fixed.</p> - -<p>Looking at the movements of this kind, we should say -that the objects were altering their shape as well as -rotating. But to the inhabitants of that world they -would seem to be unaltered, and our figures in their -motions would seem to them to alter.</p> - -<p>In such a world the features of geometry are different. -We have seen one such difference in the case of our illustration -of the world of shear, where the square on the -hypothenuse was equal to the difference, not the sum, of -the squares on the sides.</p> - -<p>In our illustration we have the same laws of parallel -lines as in our ordinary rotation world, but in general the -laws of parallel lines are different.</p> - -<p>In one of these worlds of a different constitution of -matter through one point there can be two parallels to -a given line, in another of them there can be none, that -is, although a line be drawn parallel to another it will -meet it after a time.</p> - -<p>Now it was precisely in this respect of parallels that -Lobatchewsky and Bolyai discovered these different -worlds. They did not think of them as worlds of matter, -but they discovered that space did not necessarily mean -that our law of parallels is true. They made the -distinction between laws of space and laws of matter,<span class="pagenum" id="Page_56">[Pg 56]</span> -although that is not the form in which they stated their -results.</p> - -<p>The way in which they were led to these results was the -following. Euclid had stated the existence of parallel lines -as a postulate—putting frankly this unproved proposition—that -one line and only one parallel to a given straight -line can be drawn, as a demand, as something that must -be assumed. The words of his ninth postulate are these: -“If a straight line meeting two other straight lines -makes the interior angles on the same side of it equal -to two right angles, the two straight lines will never -meet.”</p> - -<p>The mathematicians of later ages did not like this bald -assumption, and not being able to prove the proposition -they called it an axiom—the eleventh axiom.</p> - -<p>Many attempts were made to prove the axiom; no one -doubted of its truth, but no means could be found to -demonstrate it. At last an Italian, Sacchieri, unable to -find a proof, said: “Let us suppose it not true.” He deduced -the results of there being possibly two parallels to one -given line through a given point, but feeling the waters -too deep for the human reason, he devoted the latter half -of his book to disproving what he had assumed in the first -part.</p> - -<p>Then Bolyai and Lobatchewsky with firm step entered -on the forbidden path. There can be no greater evidence -of the indomitable nature of the human spirit, or of its -manifest destiny to conquer all those limitations which -bind it down within the sphere of sense than this grand -assertion of Bolyai and Lobatchewsky.</p> - -<div class="figleft illowp25" id="fig_31" style="max-width: 12.5em;"> - <img src="images/fig_31.png" alt="" /> - <div class="caption">Fig. 31.</div> -</div> - -<p>Take a line <span class="allsmcap">AB</span> and a point <span class="allsmcap">C</span>. We -say and see and know that through <span class="allsmcap">C</span> -can only be drawn one line parallel -to <span class="allsmcap">AB</span>.</p> - -<p>But Bolyai said: “I will draw two.” Let <span class="allsmcap">CD</span> be parallel<span class="pagenum" id="Page_57">[Pg 57]</span> -to <span class="allsmcap">AB</span>, that is, not meet <span class="allsmcap">AB</span> however far produced, and let -lines beyond <span class="allsmcap">CD</span> also not meet -<span class="allsmcap">AB</span>; let there be a certain -region between <span class="allsmcap">CD</span> and <span class="allsmcap">CE</span>, -in which no line drawn meets -<span class="allsmcap">AB</span>. <span class="allsmcap">CE</span> and <span class="allsmcap">CD</span> produced -backwards through <span class="allsmcap">C</span> will give a similar region on the -other side of <span class="allsmcap">C</span>.</p> - -<div class="figleft illowp40" id="fig_32" style="max-width: 21.875em;"> - <img src="images/fig_32.png" alt="" /> - <div class="caption">Fig. 32.</div> -</div> - -<p>Nothing so triumphantly, one may almost say so -insolently, ignoring of sense had ever been written before. -Men had struggled against the limitations of the body, -fought them, despised them, conquered them. But no -one had ever thought simply as if the body, the bodily -eyes, the organs of vision, all this vast experience of space, -had never existed. The age-long contest of the soul with -the body, the struggle for mastery, had come to a culmination. -Bolyai and Lobatchewsky simply thought as -if the body was not. The struggle for dominion, the strife -and combat of the soul were over; they had mastered, -and the Hungarian drew his line.</p> - -<p>Can we point out any connection, as in the case of -Parmenides, between these speculations and higher -space? Can we suppose it was any inner perception by -the soul of a motion not known to the senses, which resulted -in this theory so free from the bonds of sense? No -such supposition appears to be possible.</p> - -<p>Practically, however, metageometry had a great influence -in bringing the higher space to the front as a -working hypothesis. This can be traced to the tendency -the mind has to move in the direction of least resistance. -The results of the new geometry could not be neglected, -the problem of parallels had occupied a place too prominent -in the development of mathematical thought for its final -solution to be neglected. But this utter independence of -all mechanical considerations, this perfect cutting loose<span class="pagenum" id="Page_58">[Pg 58]</span> -from the familiar intuitions, was so difficult that almost -any other hypothesis was more easy of acceptance, and -when Beltrami showed that the geometry of Lobatchewsky -and Bolyai was the geometry of shortest lines drawn on -certain curved surfaces, the ordinary definitions of measurement -being retained, attention was drawn to the theory of -a higher space. An illustration of Beltrami’s theory is -furnished by the simple consideration of hypothetical -beings living on a spherical surface.</p> - -<div class="figleft illowp35" id="fig_33" style="max-width: 15.625em;"> - <img src="images/fig_33.png" alt="" /> - <div class="caption">Fig. 33.</div> -</div> - -<p>Let <span class="allsmcap">ABCD</span> be the equator of a globe, and <span class="allsmcap">AP</span>, <span class="allsmcap">BP</span>, -meridian lines drawn to the pole, <span class="allsmcap">P</span>. -The lines <span class="allsmcap">AB</span>, <span class="allsmcap">AP</span>, <span class="allsmcap">BP</span> would seem to be -perfectly straight to a person moving -on the surface of the sphere, and -unconscious of its curvature. Now -<span class="allsmcap">AP</span> and <span class="allsmcap">BP</span> both make right angles -with <span class="allsmcap">AB</span>. Hence they satisfy the -definition of parallels. Yet they -meet in <span class="allsmcap">P</span>. Hence a being living on a spherical surface, -and unconscious of its curvature, would find that parallel -lines would meet. He would also find that the angles -in a triangle were greater than two right angles. In -the triangle <span class="allsmcap">PAB</span>, for instance, the angles at <span class="allsmcap">A</span> and <span class="allsmcap">B</span> -are right angles, so the three angles of the triangle -<span class="allsmcap">PAB</span> are greater than two right angles.</p> - -<p>Now in one of the systems of metageometry (for after -Lobatchewsky had shown the way it was found that other -systems were possible besides his) the angles of a triangle -are greater than two right angles.</p> - -<p>Thus a being on a sphere would form conclusions about -his space which are the same as he would form if he lived -on a plane, the matter in which had such properties as -are presupposed by one of these systems of geometry. -Beltrami also discovered a certain surface on which there -could be drawn more than one “straight” line through a<span class="pagenum" id="Page_59">[Pg 59]</span> -point which would not meet another given line. I use -the word straight as equivalent to the line having the -property of giving the shortest path between any two -points on it. Hence, without giving up the ordinary -methods of measurement, it was possible to find conditions -in which a plane being would necessarily have an experience -corresponding to Lobatchewsky’s geometry. -And by the consideration of a higher space, and a solid -curved in such a higher space, it was possible to account -for a similar experience in a space of three dimensions.</p> - -<p>Now, it is far more easy to conceive of a higher dimensionality -to space than to imagine that a rod in rotating -does not move so that its end describes a circle. Hence, -a logical conception having been found harder than that -of a four dimensional space, thought turned to the latter -as a simple explanation of the possibilities to which -Lobatchewsky had awakened it. Thinkers became accustomed -to deal with the geometry of higher space—it was -Kant, says Veronese, who first used the expression of -“different spaces”—and with familiarity the inevitableness -of the conception made itself felt.</p> - -<p>From this point it is but a small step to adapt the -ordinary mechanical conceptions to a higher spatial -existence, and then the recognition of its objective -existence could be delayed no longer. Here, too, as in so -many cases, it turns out that the order and connection of -our ideas is the order and connection of things.</p> - -<p>What is the significance of Lobatchewsky’s and Bolyai’s -work?</p> - -<p>It must be recognised as something totally different -from the conception of a higher space; it is applicable to -spaces of any number of dimensions. By immersing the -conception of distance in matter to which it properly -belongs, it promises to be of the greatest aid in analysis -for the effective distance of any two particles is the<span class="pagenum" id="Page_60">[Pg 60]</span> -product of complex material conditions and cannot be -measured by hard and fast rules. Its ultimate significance -is altogether unknown. It is a cutting loose -from the bonds of sense, not coincident with the recognition -of a higher dimensionality, but indirectly contributory -thereto.</p> - -<p>Thus, finally, we have come to accept what Plato held -in the hollow of his hand; what Aristotle’s doctrine of -the relativity of substance implies. The vast universe, too, -has its higher, and in recognising it we find that the -directing being within us no longer stands inevitably -outside our systematic knowledge.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_61">[Pg 61]</span></p> - -<h2 class="nobreak" id="CHAPTER_VI">CHAPTER VI<br /> - -<small><i>THE HIGHER WORLD</i></small></h2></div> - - -<p>It is indeed strange, the manner in which we must begin -to think about the higher world.</p> - -<p>Those simplest objects analogous to those which are -about us on every side in our daily experience such as a -door, a table, a wheel are remote and uncognisable in the -world of four dimensions, while the abstract ideas of -rotation, stress and strain, elasticity into which analysis -resolves the familiar elements of our daily experience are -transferable and applicable with no difficulty whatever. -Thus we are in the unwonted position of being obliged -to construct the daily and habitual experience of a four-dimensional -being, from a knowledge of the abstract -theories of the space, the matter, the motion of it; -instead of, as in our case, passing to the abstract theories -from the richness of sensible things.</p> - -<p>What would a wheel be in four dimensions? What -the shafting for the transmission of power which a -four-dimensional being would use.</p> - -<p>The four-dimensional wheel, and the four-dimensional -shafting are what will occupy us for these few pages. And -it is no futile or insignificant enquiry. For in the attempt -to penetrate into the nature of the higher, to grasp within -our ken that which transcends all analogies, because what -we know are merely partial views of it, the purely -material and physical path affords a means of approach<span class="pagenum" id="Page_62">[Pg 62]</span> -pursuing which we are in less likelihood of error than if -we use the more frequently trodden path of framing -conceptions which in their elevation and beauty seem to -us ideally perfect.</p> - -<p>For where we are concerned with our own thoughts, the -development of our own ideals, we are as it were on a -curve, moving at any moment in a direction of tangency. -Whither we go, what we set up and exalt as perfect, -represents not the true trend of the curve, but our own -direction at the present—a tendency conditioned by the -past, and by a vital energy of motion essential but -only true when perpetually modified. That eternal corrector -of our aspirations and ideals, the material universe -draws sublimely away from the simplest things we can -touch or handle to the infinite depths of starry space, -in one and all uninfluenced by what we think or feel, -presenting unmoved fact to which, think it good or -think it evil, we can but conform, yet out of all that -impassivity with a reference to something beyond our -individual hopes and fears supporting us and giving us -our being.</p> - -<p>And to this great being we come with the question: -“You, too, what is your higher?”</p> - -<p>Or to put it in a form which will leave our conclusions in -the shape of no barren formula, and attacking the problem -on its most assailable side: “What is the wheel and the -shafting of the four-dimensional mechanic?”</p> - -<p>In entering on this enquiry we must make a plan of -procedure. The method which I shall adopt is to trace -out the steps of reasoning by which a being confined -to movement in a two-dimensional world could arrive at a -conception of our turning and rotation, and then to apply -an analogous process to the consideration of the higher -movements. The plane being must be imagined as no -abstract figure, but as a real body possessing all three<span class="pagenum" id="Page_63">[Pg 63]</span> -dimensions. His limitation to a plane must be the result -of physical conditions.</p> - -<p>We will therefore think of him as of a figure cut out of -paper placed on a smooth plane. Sliding over this plane, -and coming into contact with other figures equally thin -as he in the third dimension, he will apprehend them only -by their edges. To him they will be completely bounded -by lines. A “solid” body will be to him a two-dimensional -extent, the interior of which can only be reached by -penetrating through the bounding lines.</p> - -<p>Now such a plane being can think of our three-dimensional -existence in two ways.</p> - -<p>First, he can think of it as a series of sections, each like -the solid he knows of extending in a direction unknown -to him, which stretches transverse to his tangible -universe, which lies in a direction at right angles to every -motion which he made.</p> - -<p>Secondly, relinquishing the attempt to think of the -three-dimensional solid body in its entirety he can regard -it as consisting of a number of plane sections, each of them -in itself exactly like the two-dimensional bodies he knows, -but extending away from his two-dimensional space.</p> - -<p>A square lying in his space he regards as a solid -bounded by four lines, each of which lies in his space.</p> - -<p>A square standing at right angles to his plane appears -to him as simply a line in his plane, for all of it except -the line stretches in the third dimension.</p> - -<p>He can think of a three-dimensional body as consisting -of a number of such sections, each of which starts from a -line in his space.</p> - -<p>Now, since in his world he can make any drawing or -model which involves only two dimensions, he can represent -each such upright section as it actually is, and can represent -a turning from a known into the unknown dimension -as a turning from one to another of his known dimensions.</p> - -<p><span class="pagenum" id="Page_64">[Pg 64]</span></p> - -<p>To see the whole he must relinquish part of that which -he has, and take the whole portion by portion.</p> - -<div class= "figleft illowp30" id="fig_34" style="max-width: 15.625em;"> - <img src="images/fig_34.png" alt="" /> - <div class="caption">Fig. 34.</div> -</div> - -<p>Consider now a plane being in front of a square, <a href="#fig_34">fig. 34</a>. -The square can turn about any point -in the plane—say the point <span class="allsmcap">A</span>. But it -cannot turn about a line, as <span class="allsmcap">AB</span>. For, -in order to turn about the line <span class="allsmcap">AB</span>, -the square must leave the plane and -move in the third dimension. This -motion is out of his range of observation, -and is therefore, except for a -process of reasoning, inconceivable to him.</p> - -<p>Rotation will therefore be to him rotation about a point. -Rotation about a line will be inconceivable to him.</p> - -<p>The result of rotation about a line he can apprehend. -He can see the first and last positions occupied in a half-revolution -about the line <span class="allsmcap">AC</span>. The result of such a half revolution -is to place the square <span class="allsmcap">ABCD</span> on the left hand instead -of on the right hand of the line <span class="allsmcap">AC</span>. It would correspond -to a pulling of the whole body <span class="allsmcap">ABCD</span> through the line <span class="allsmcap">AC</span>, -or to the production of a solid body which was the exact -reflection of it in the line <span class="allsmcap">AC</span>. It would be as if the square -<span class="allsmcap">ABCD</span> turned into its image, the line <span class="allsmcap">AB</span> acting as a mirror. -Such a reversal of the positions of the parts of the square -would be impossible in his space. The occurrence of it -would be a proof of the existence of a higher dimensionality.</p> - -<div class="figleft illowp30" id="fig_35" style="max-width: 18.75em;"> - <img src="images/fig_35.png" alt="" /> - <div class="caption">Fig. 35.</div> -</div> - -<p>Let him now, adopting the conception of a three-dimensional -body as a series of -sections lying, each removed a little -farther than the preceding one, in -direction at right angles to his -plane, regard a cube, <a href="#fig_36">fig. 36</a>, as a -series of sections, each like the -square which forms its base, all -rigidly connected together.</p> - -<p><span class="pagenum" id="Page_65">[Pg 65]</span></p> - -<p>If now he turns the square about the point <span class="allsmcap">A</span> in the -plane of <i>xy</i>, each parallel section turns with the square -he moves. In each of the sections there is a point at -rest, that vertically over <span class="allsmcap">A</span>. Hence he would conclude -that in the turning of a three-dimensional body there is -one line which is at rest. That is a three-dimensional -turning in a turning about a line.</p> - -<hr class="tb" /> - -<p>In a similar way let us regard ourselves as limited to a -three-dimensional world by a physical condition. Let us -imagine that there is a direction at right angles to every -direction in which we can move, and that we are prevented -from passing in this direction by a vast solid, that -against which in every movement we make we slip as -the plane being slips against his plane sheet.</p> - -<p>We can then consider a four-dimensional body as consisting -of a series of sections, each parallel to our space, -and each a little farther off than the preceding on the -unknown dimension.</p> - -<div class="figleft illowp35" id="fig_36" style="max-width: 18.75em;"> - <img src="images/fig_36.png" alt="" /> - <div class="caption">Fig. 36.</div> -</div> - -<p>Take the simplest four-dimensional body—one which -begins as a cube, <a href="#fig_36">fig. 36</a>, in our -space, and consists of sections, each -a cube like <a href="#fig_36">fig. 36</a>, lying away from -our space. If we turn the cube -which is its base in our space -about a line, if, <i>e.g.</i>, in <a href="#fig_36">fig. 36</a> we -turn the cube about the line <span class="allsmcap">AB</span>, -not only it but each of the parallel -cubes moves about a line. The -cube we see moves about the line <span class="allsmcap">AB</span>, the cube beyond it -about a line parallel to <span class="allsmcap">AB</span> and so on. Hence the whole -four-dimensional body moves about a plane, for the -assemblage of these lines is our way of thinking about the -plane which, starting from the line <span class="allsmcap">AB</span> in our space, runs -off in the unknown direction.</p> - -<p><span class="pagenum" id="Page_66">[Pg 66]</span></p> - -<p>In this case all that we see of the plane about which -the turning takes place is the line <span class="allsmcap">AB</span>.</p> - -<p>But it is obvious that the axis plane may lie in our -space. A point near the plane determines with it a three-dimensional -space. When it begins to rotate round the -plane it does not move anywhere in this three-dimensional -space, but moves out of it. A point can no more rotate -round a plane in three-dimensional space than a point -can move round a line in two-dimensional space.</p> - -<p>We will now apply the second of the modes of representation -to this case of turning about a plane, building -up our analogy step by step from the turning in a plane -about a point and that in space about a line, and so on.</p> - -<p>In order to reduce our considerations to those of the -greatest simplicity possible, let us realise how the plane -being would think of the motion by which a square is -turned round a line.</p> - -<p>Let, <a href="#fig_34">fig. 34</a>, <span class="allsmcap">ABCD</span> be a square on his plane, and represent -the two dimensions of his space by the axes <span class="allsmcap">A</span><i>x</i> <span class="allsmcap">A</span><i>y</i>.</p> - -<p>Now the motion by which the square is turned over -about the line <span class="allsmcap">AC</span> involves the third dimension.</p> - -<p>He cannot represent the motion of the whole square in -its turning, but he can represent the motions of parts of -it. Let the third axis perpendicular to the plane of the -paper be called the axis of <i>z</i>. Of the three axes <i>x</i>, <i>y</i>, <i>z</i>, -the plane being can represent any two in his space. Let -him then draw, in <a href="#fig_35">fig. 35</a>, two axes, <i>x</i> and <i>z</i>. Here he has -in his plane a representation of what exists in the plane -which goes off perpendicularly to his space.</p> - -<p>In this representation the square would not be shown, -for in the plane of <i>xz</i> simply the line <span class="allsmcap">AB</span> of the square is -contained.</p> - -<p>The plane being then would have before him, in <a href="#fig_35">fig. 35</a>, -the representation of one line <span class="allsmcap">AB</span> of his square and two -axes, <i>x</i> and <i>z</i>, at right angles. Now it would be obvious<span class="pagenum" id="Page_67">[Pg 67]</span> -to him that, by a turning such as he knows, by a rotation -about a point, the line <span class="allsmcap">AB</span> can turn round <span class="allsmcap">A</span>, and occupying -all the intermediate positions, such as <span class="allsmcap">AB</span><sub>1</sub>, come -after half a revolution to lie as <span class="allsmcap">A</span><i>x</i> produced through <span class="allsmcap">A</span>.</p> - -<p>Again, just as he can represent the vertical plane -through <span class="allsmcap">AB</span>, so he can represent the vertical plane -through <span class="allsmcap">A´B´</span>, <a href="#fig_34">fig. 34</a>, and in a like manner can see that -the line <span class="allsmcap">A´B´</span> can turn about the point <span class="allsmcap">A´</span> till it lies in the -opposite direction from that which it ran in at first.</p> - -<p>Now these two turnings are not inconsistent. In his -plane, if <span class="allsmcap">AB</span> turned about <span class="allsmcap">A</span>, and <span class="allsmcap">A´B´</span> about <span class="allsmcap">A´</span>, the consistency -of the square would be destroyed, it would be an -impossible motion for a rigid body to perform. But in -the turning which he studies portion by portion there is -nothing inconsistent. Each line in the square can turn -in this way, hence he would realise the turning of the -whole square as the sum of a number of turnings of -isolated parts. Such turnings, if they took place in his -plane, would be inconsistent, but by virtue of a third -dimension they are consistent, and the result of them all -is that the square turns about the line <span class="allsmcap">AC</span> and lies in a -position in which it is the mirror image of what it was in -its first position. Thus he can realise a turning about a -line by relinquishing one of his axes, and representing his -body part by part.</p> - -<p>Let us apply this method to the turning of a cube so as -to become the mirror image of itself. In our space we can -construct three independent axes, <i>x</i>, <i>y</i>, <i>z</i>, shown in <a href="#fig_36">fig. 36</a>. -Suppose that there is a fourth axis, <i>w</i>, at right angles to -each and every one of them. We cannot, keeping all -three axes, <i>x</i>, <i>y</i>, <i>z</i>, represent <i>w</i> in our space; but if we -relinquish one of our three axes we can let the fourth axis -take its place, and we can represent what lies in the -space, determined by the two axes we retain and the -fourth axis.</p> - -<p><span class="pagenum" id="Page_68">[Pg 68]</span></p> - -<div class="figleft illowp35" id="fig_37" style="max-width: 18.75em;"> - <img src="images/fig_37.png" alt="" /> - <div class="caption">Fig. 37.</div> -</div> - -<p>Let us suppose that we let the <i>y</i> axis drop, and that -we represent the <i>w</i> axis as occupying -its direction. We have in fig. -37 a drawing of what we should -then see of the cube. The square -<span class="allsmcap">ABCD</span>, remains unchanged, for that -is in the plane of <i>xz</i>, and we -still have that plane. But from -this plane the cube stretches out -in the direction of the <i>y</i> axis. Now the <i>y</i> axis is gone, -and so we have no more of the cube than the face <span class="allsmcap">ABCD</span>. -Considering now this face <span class="allsmcap">ABCD</span>, we -see that it is free to turn about the -line <span class="allsmcap">AB</span>. It can rotate in the <i>x</i> to <i>w</i> -direction about this line. In <a href="#fig_38">fig. 38</a> -it is shown on its way, and it can -evidently continue this rotation till -it lies on the other side of the <i>z</i> -axis in the plane of <i>xz</i>.</p> - -<div class="figleft illowp35" id="fig_38" style="max-width: 18.75em;"> - <img src="images/fig_38.png" alt="" /> - <div class="caption">Fig. 38.</div> -</div> - -<p>We can also take a section parallel to the face <span class="allsmcap">ABCD</span>, -and then letting drop all of our space except the plane of -that section, introduce the <i>w</i> axis, running in the old <i>y</i> -direction. This section can be represented by the same -drawing, <a href="#fig_38">fig. 38</a>, and we see that it can rotate about the -line on its left until it swings half way round and runs in -the opposite direction to that which it ran in before. -These turnings of the different sections are not inconsistent, -and taken all together they will bring the cube -from the position shown in <a href="#fig_36">fig. 36</a> to that shown in -<a href="#fig_41">fig. 41</a>.</p> - -<p>Since we have three axes at our disposal in our space, -we are not obliged to represent the <i>w</i> axis by any particular -one. We may let any axis we like disappear, and let the -fourth axis take its place.</p> - -<div class="figleft illowp40" id="fig_39" style="max-width: 18.75em;"> - <img src="images/fig_39.png" alt="" /> - <div class="caption">Fig. 39.</div> -</div> -<div class="figleft illowp40" id="fig_40" style="max-width: 18.75em;"> - <img src="images/fig_40.png" alt="" /> - <div class="caption">Fig. 40.</div> -</div> - -<div class="figleft illowp40" id="fig_41" style="max-width: 21.875em;"> - <img src="images/fig_41.png" alt="" /> - <div class="caption">Fig. 41.</div> -</div> - -<p>In <a href="#fig_36">fig. 36</a> suppose the <i>z</i> axis to go. We have then<span class="pagenum" id="Page_69">[Pg 69]</span> -simply the plane of <i>xy</i> and the square base of the -cube <span class="allsmcap">ACEG</span>, <a href="#fig_39">fig. 39</a>, is all that could -be seen of it. Let now the <i>w</i> axis -take the place of the <i>z</i> axis and -we have, in <a href="#fig_39">fig. 39</a> again, a representation -of the space of <i>xyw</i>, in -which all that exists of the cube is -its square base. Now, by a turning -of <i>x</i> to <i>w</i>, this base can rotate around the line <span class="allsmcap">AE</span>, it is -shown on its way in <a href="#fig_40">fig. 40</a>, and -finally it will, after half a revolution, -lie on the other side of the <i>y</i> axis. -In a similar way we may rotate -sections parallel to the base of the -<i>xw</i> rotation, and each of them comes -to run in the opposite direction from -that which they occupied at first.</p> - -<p>Thus again the cube comes from the position of <a href="#fig_36">fig. 36</a>. -to that of <a href="#fig_41">fig. 41</a>. In this <i>x</i> -to <i>w</i> turning, we see that it -takes place by the rotations of -sections parallel to the front -face about lines parallel to <span class="allsmcap">AB</span>, -or else we may consider it as -consisting of the rotation of -sections parallel to the base -about lines parallel to <span class="allsmcap">AE</span>. It -is a rotation of the whole cube about the plane <span class="allsmcap">ABEF</span>. -Two separate sections could not rotate about two separate -lines in our space without conflicting, but their motion is -consistent when we consider another dimension. Just, -then, as a plane being can think of rotation about a line as -a rotation about a number of points, these rotations not -interfering as they would if they took place in his two-dimensional -space, so we can think of a rotation about a<span class="pagenum" id="Page_70">[Pg 70]</span> -plane as the rotation of a number of sections of a body -about a number of lines in a plane, these rotations not -being inconsistent in a four-dimensional space as they are -in three-dimensional space.</p> - -<p>We are not limited to any particular direction for the -lines in the plane about which we suppose the rotation -of the particular sections to take place. Let us draw -the section of the cube, <a href="#fig_36">fig. 36</a>, through <span class="allsmcap">A</span>, <span class="allsmcap">F</span>, <span class="allsmcap">C</span>, <span class="allsmcap">H</span>, forming a -sloping plane. Now since the fourth dimension is at -right angles to every line in our space it is at right -angles to this section also. We can represent our space -by drawing an axis at right angles to the plane <span class="allsmcap">ACEG</span>, our -space is then determined by the plane <span class="allsmcap">ACEG</span>, and the perpendicular -axis. If we let this axis drop and suppose the -fourth axis, <i>w</i>, to take its place, we have a representation of -the space which runs off in the fourth dimension from the -plane <span class="allsmcap">ACEG</span>. In this space we shall see simply the section -<span class="allsmcap">ACEG</span> of the cube, and nothing else, for one cube does not -extend to any distance in the fourth dimension.</p> - -<div class="figleft illowp40" id="fig_42" style="max-width: 25em;"> - <img src="images/fig_42.png" alt="" /> - <div class="caption">Fig. 42.</div> -</div> - -<p>If, keeping this plane, we bring in the fourth dimension, -we shall have a space in which simply this section of -the cube exists and nothing else. The section can turn -about the line <span class="allsmcap">AF</span>, and parallel sections can turn about -parallel lines. Thus in considering -the rotation about -a plane we can draw any -lines we like and consider -the rotation as taking place -in sections about them.</p> - -<p>To bring out this point -more clearly let us take two -parallel lines, <span class="allsmcap">A</span> and <span class="allsmcap">B</span>, in -the space of <i>xyz</i>, and let <span class="allsmcap">CD</span> -and <span class="allsmcap">EF</span> be two rods running -above and below the plane of <i>xy</i>, from these lines. If we<span class="pagenum" id="Page_71">[Pg 71]</span> -turn these rods in our space about the lines <span class="allsmcap">A</span> and <span class="allsmcap">B</span>, as -the upper end of one, <span class="allsmcap">F</span>, is going down, the lower end of -the other, <span class="allsmcap">C</span>, will be coming up. They will meet and -conflict. But it is quite possible for these two rods -each of them to turn about the two lines without altering -their relative distances.</p> - -<p>To see this suppose the <i>y</i> axis to go, and let the <i>w</i> axis -take its place. We shall see the lines <span class="allsmcap">A</span> and <span class="allsmcap">B</span> no longer, -for they run in the <i>y</i> direction from the points <span class="allsmcap">G</span> and <span class="allsmcap">H</span>.</p> - -<div class="figleft illowp40" id="fig_43" style="max-width: 21.875em;"> - <img src="images/fig_43.png" alt="" /> - <div class="caption">Fig. 43.</div> -</div> - -<p>Fig. 43 is a picture of the two rods seen in the space -of <i>xzw</i>. If they rotate in the -direction shown by the arrows—in -the <i>z</i> to <i>w</i> direction—they -move parallel to one another, -keeping their relative distances. -Each will rotate about its own -line, but their rotation will not -be inconsistent with their forming -part of a rigid body.</p> - -<p>Now we have but to suppose -a central plane with rods crossing -it at every point, like <span class="allsmcap">CD</span> and <span class="allsmcap">EF</span> cross the plane of <i>xy</i>, -to have an image of a mass of matter extending equal -distances on each side of a diametral plane. As two of -these rods can rotate round, so can all, and the whole -mass of matter can rotate round its diametral plane.</p> - -<p>This rotation round a plane corresponds, in four -dimensions, to the rotation round an axis in three -dimensions. Rotation of a body round a plane is the -analogue of rotation of a rod round an axis.</p> - -<p>In a plane we have rotation round a point, in three-space -rotation round an axis line, in four-space rotation -round an axis plane.</p> - -<p>The four-dimensional being’s shaft by which he transmits -power is a disk rotating round its central<span class="pagenum" id="Page_72">[Pg 72]</span> -plane—the whole contour corresponds to the ends of an axis -of rotation in our space. He can impart the rotation at -any point and take it off at any other point on the contour, -just as rotation round a line can in three-space be imparted -at one end of a rod and taken off at the other end.</p> - -<p>A four-dimensional wheel can easily be described from -the analogy of the representation which a plane being -would form for himself of one of our wheels.</p> - -<p>Suppose a wheel to move transverse to a plane, so that -the whole disk, which I will consider to be solid and -without spokes, came at the same time into contact with -the plane. It would appear as a circular portion of plane -matter completely enclosing another and smaller portion—the -axle.</p> - -<p>This appearance would last, supposing the motion of -the wheel to continue until it had traversed the plane by -the extent of its thickness, when there would remain in -the plane only the small disk which is the section of the -axle. There would be no means obvious in the plane -at first by which the axle could be reached, except by -going through the substance of the wheel. But the -possibility of reaching it without destroying the substance -of the wheel would be shown by the continued existence -of the axle section after that of the wheel had disappeared.</p> - -<p>In a similar way a four-dimensional wheel moving -transverse to our space would appear first as a solid sphere, -completely surrounding a smaller solid sphere. The -outer sphere would represent the wheel, and would last -until the wheel has traversed our space by a distance -equal to its thickness. Then the small sphere alone -would remain, representing the section of the axle. The -large sphere could move round the small one quite freely. -Any line in space could be taken as an axis, and round -this line the outer sphere could rotate, while the inner -sphere remained still. But in all these directions of<span class="pagenum" id="Page_73">[Pg 73]</span> -revolution there would be in reality one line which -remained unaltered, that is the line which stretches away -in the fourth direction, forming the axis of the axle. The -four-dimensional wheel can rotate in any number of planes, -but all these planes are such that there is a line at right -angles to them all unaffected by rotation in them.</p> - -<p>An objection is sometimes experienced as to this mode -of reasoning from a plane world to a higher dimensionality. -How artificial, it is argued, this conception of a plane -world is. If any real existence confined to a superficies -could be shown to exist, there would be an argument for -one relative to which our three-dimensional existence is -superficial. But, both on the one side and the other of -the space we are familiar with, spaces either with less -or more than three dimensions are merely arbitrary -conceptions.</p> - -<p>In reply to this I would remark that a plane being -having one less dimension than our three would have one-third -of our possibilities of motion, while we have only -one-fourth less than those of the higher space. It may -very well be that there may be a certain amount of -freedom of motion which is demanded as a condition of an -organised existence, and that no material existence is -possible with a more limited dimensionality than ours. -This is well seen if we try to construct the mechanics of a -two-dimensional world. No tube could exist, for unless -joined together completely at one end two parallel lines -would be completely separate. The possibility of an -organic structure, subject to conditions such as this, is -highly problematical; yet, possibly in the convolutions -of the brain there may be a mode of existence to be -described as two-dimensional.</p> - -<p>We have but to suppose the increase in surface and -the diminution in mass carried on to a certain extent -to find a region which, though without mobility of the<span class="pagenum" id="Page_74">[Pg 74]</span> -constituents, would have to be described as two-dimensional.</p> - -<p>But, however artificial the conception of a plane being -may be, it is none the less to be used in passing to the -conception of a greater dimensionality than ours, and -hence the validity of the first part of this objection -altogether disappears directly we find evidence for such a -state of being.</p> - -<p>The second part of the objection has more weight. -How is it possible to conceive that in a four-dimensional -space any creatures should be confined to a three-dimensional -existence?</p> - -<p>In reply I would say that we know as a matter of fact -that life is essentially a phenomenon of surface. The -amplitude of the movements which we can make is much -greater along the surface of the earth than it is up -or down.</p> - -<p>Now we have but to conceive the extent of a solid -surface increased, while the motions possible tranverse to -it are diminished in the same proportion, to obtain the -image of a three-dimensional world in four-dimensional -space.</p> - -<p>And as our habitat is the meeting of air and earth on -the world, so we must think of the meeting place of two -as affording the condition for our universe. The meeting -of what two? What can that vastness be in the higher -space which stretches in such a perfect level that our -astronomical observations fail to detect the slightest -curvature?</p> - -<p>The perfection of the level suggests a liquid—a lake -amidst what vast scenery!—whereon the matter of the -universe floats speck-like.</p> - -<p>But this aspect of the problem is like what are called -in mathematics boundary conditions.</p> - -<p>We can trace out all the consequences of four-dimensional -movements down to their last detail. Then, knowing<span class="pagenum" id="Page_75">[Pg 75]</span> -the mode of action which would be characteristic of the -minutest particles, if they were free, we can draw conclusions -from what they actually do of what the constraint -on them is. Of the two things, the material conditions and -the motion, one is known, and the other can be inferred. -If the place of this universe is a meeting of two, there -would be a one-sideness to space. If it lies so that what -stretches away in one direction in the unknown is unlike -what stretches away in the other, then, as far as the -movements which participate in that dimension are concerned, -there would be a difference as to which way the -motion took place. This would be shown in the dissimilarity -of phenomena, which, so far as all three-space -movements are concerned, were perfectly symmetrical. -To take an instance, merely, for the sake of precising -our ideas, not for any inherent probability in it; if it could -be shown that the electric current in the positive direction -were exactly like the electric current in the negative -direction, except for a reversal of the components of the -motion in three-dimensional space, then the dissimilarity -of the discharge from the positive and negative poles -would be an indication of a one-sideness to our space. -The only cause of difference in the two discharges would -be due to a component in the fourth dimension, which -directed in one direction transverse to our space, met with -a different resistance to that which it met when directed -in the opposite direction.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_76">[Pg 76]</span></p> - -<h2 class="nobreak" id="CHAPTER_VII">CHAPTER VII<br /> - -<small><i>THE EVIDENCES FOR A FOURTH DIMENSION</i></small></h2></div> - - -<p>The method necessarily to be employed in the search for -the evidences of a fourth dimension, consists primarily in -the formation of the conceptions of four-dimensional -shapes and motions. When we are in possession of these -it is possible to call in the aid of observation, without -them we may have been all our lives in the familiar -presence of a four-dimensional phenomenon without ever -recognising its nature.</p> - -<p>To take one of the conceptions we have already formed, -the turning of a real thing into its mirror image would be -an occurrence which it would be hard to explain, except on -the assumption of a fourth dimension.</p> - -<p>We know of no such turning. But there exist a multitude -of forms which show a certain relation to a plane, -a relation of symmetry, which indicates more than an accidental -juxtaposition of parts. In organic life the universal -type is of right- and left-handed symmetry, there is a plane -on each side of which the parts correspond. Now we have -seen that in four dimensions a plane takes the place of a -line in three dimensions. In our space, rotation about an -axis is the type of rotation, and the origin of bodies symmetrical -about a line as the earth is symmetrical about an -axis can easily be explained. But where there is symmetry -about a plane no simple physical motion, such as we<span class="pagenum" id="Page_77">[Pg 77]</span> -are accustomed to, suffices to explain it. In our space a -symmetrical object must be built up by equal additions -on each side of a central plane. Such additions about -such a plane are as little likely as any other increments. -The probability against the existence of symmetrical -form in inorganic nature is overwhelming in our space, -and in organic forms they would be as difficult of production -as any other variety of configuration. To illustrate -this point we may take the child’s amusement of making -from dots of ink on a piece of paper a lifelike representation -of an insect by simply folding the paper -over. The dots spread out on a symmetrical line, and -give the impression of a segmented form with antennæ -and legs.</p> - -<p>Now seeing a number of such figures we should -naturally infer a folding over. Can, then, a folding over -in four-dimensional space account for the symmetry of -organic forms? The folding cannot of course be of the -bodies we see, but it may be of those minute constituents, -the ultimate elements of living matter which, turned in one -way or the other, become right- or left-handed, and so -produce a corresponding structure.</p> - -<p>There is something in life not included in our conceptions -of mechanical movement. Is this something a four-dimensional -movement?</p> - -<p>If we look at it from the broadest point of view, there is -something striking in the fact that where life comes in -there arises an entirely different set of phenomena to -those of the inorganic world.</p> - -<p>The interest and values of life as we know it in ourselves, -as we know it existing around us in subordinate -forms, is entirely and completely different to anything -which inorganic nature shows. And in living beings we -have a kind of form, a disposition of matter which is -entirely different from that shown in inorganic matter.<span class="pagenum" id="Page_78">[Pg 78]</span> -Right- and left-handed symmetry does not occur in the -configurations of dead matter. We have instances of -symmetry about an axis, but not about a plane. It can -be argued that the occurrence of symmetry in two dimensions -involves the existence of a three-dimensional process, -as when a stone falls into water and makes rings of ripples, -or as when a mass of soft material rotates about an axis. -It can be argued that symmetry in any number of dimensions -is the evidence of an action in a higher dimensionality. -Thus considering living beings, there is an evidence both -in their structure, and their different mode of activity, of a -something coming in from without into the inorganic -world.</p> - -<p>And the objections which will readily occur, such as -those derived from the forms of twin crystals and the -theoretical structure of chemical molecules, do not invalidate -the argument; for in these forms too the -presumable seat of the activity producing them lies in that -very minute region in which we necessarily place the seat -of a four-dimensional mobility.</p> - -<p>In another respect also the existence of symmetrical forms -is noteworthy. It is puzzling to conceive how two shapes -exactly equal can exist which are not superposible. Such -a pair of symmetrical figures as the two hands, right and -left, show either a limitation in our power of movement, -by which we cannot superpose the one on the other, or a -definite influence and compulsion of space on matter, -inflicting limitations which are additional to those of the -proportions of the parts.</p> - -<p>We will, however, put aside the arguments to be drawn -from the consideration of symmetry as inconclusive, -retaining one valuable indication which they afford. If -it is in virtue of a four-dimensional motion that symmetry -exists, it is only in the very minute particles -of bodies that that motion is to be found, for there is<span class="pagenum" id="Page_79">[Pg 79]</span> -no such thing as a bending over in four dimensions of -any object of a size which we can observe. The region -of the extremely minute is the one, then, which we -shall have to investigate. We must look for some -phenomenon which, occasioning movements of the kind -we know, still is itself inexplicable as any form of motion -which we know.</p> - -<p>Now in the theories of the actions of the minute -particles of bodies on one another, and in the motions of -the ether, mathematicians have tacitly assumed that the -mechanical principles are the same as those which prevail -in the case of bodies which can be observed, it has been -assumed without proof that the conception of motion being -three-dimensional, holds beyond the region from observations -in which it was formed.</p> - -<p>Hence it is not from any phenomenon explained by -mathematics that we can derive a proof of four dimensions. -Every phenomenon that has been explained is explained -as three-dimensional. And, moreover, since in the region -of the very minute we do not find rigid bodies acting -on each other at a distance, but elastic substances and -continuous fluids such as ether, we shall have a double -task.</p> - -<p>We must form the conceptions of the possible movements -of elastic and liquid four-dimensional matter, before -we can begin to observe. Let us, therefore, take the four-dimensional -rotation about a plane, and enquire what it -becomes in the case of extensible fluid substances. If -four-dimensional movements exist, this kind of rotation -must exist, and the finer portions of matter must exhibit -it.</p> - -<p>Consider for a moment a rod of flexible and extensible -material. It can turn about an axis, even if not straight; -a ring of india rubber can turn inside out.</p> - -<p>What would this be in the case of four dimensions?</p> - -<p><span class="pagenum" id="Page_80">[Pg 80]</span></p> -<div class="figleft illowp50" id="fig_44" style="max-width: 25em;"> - <img src="images/fig_44.png" alt="" /> - <div class="caption">Fig. 44.<br /> -<i>Axis of x running towards -the observer.</i></div> -</div> - -<p>Let us consider a sphere of our three-dimensional -matter having a definite -thickness. To represent -this thickness let us suppose -that from every point -of the sphere in <a href="#fig_44">fig. 44</a> rods -project both ways, in and -out, like <span class="allsmcap">D</span> and <span class="allsmcap">F</span>. We can -only see the external portion, -because the internal -parts are hidden by the -sphere.</p> - -<p>In this sphere the axis -of <i>x</i> is supposed to come -towards the observer, the -axis of <i>z</i> to run up, the axis of <i>y</i> to go to the right.</p> - -<div class="figleft illowp50" id="fig_45" style="max-width: 25em;"> - <img src="images/fig_45.png" alt="" /> - <div class="caption">Fig. 45.</div> -</div> - -<p>Now take the section determined by the <i>zy</i> plane. -This will be a circle as -shown in <a href="#fig_45">fig. 45</a>. If we -let drop the <i>x</i> axis, this -circle is all we have of -the sphere. Letting the -<i>w</i> axis now run in the -place of the old <i>x</i> axis -we have the space <i>yzw</i>, -and in this space all that -we have of the sphere is -the circle. Fig. 45 then -represents all that there -is of the sphere in the -space of <i>yzw</i>. In this space it is evident that the rods -<span class="allsmcap">CD</span> and <span class="allsmcap">EF</span> can turn round the circumference as an axis. -If the matter of the spherical shell is sufficiently extensible -to allow the particles <span class="allsmcap">C</span> and <span class="allsmcap">E</span> to become as widely -separated as they would be in the positions <span class="allsmcap">D</span> and <span class="allsmcap">F</span>, then<span class="pagenum" id="Page_81">[Pg 81]</span> -the strip of matter represented by <span class="allsmcap">CD</span> and <span class="allsmcap">EF</span> and a -multitude of rods like them can turn round the circular -circumference.</p> - -<p>Thus this particular section of the sphere can turn -inside out, and what holds for any one section holds for -all. Hence in four dimensions the whole sphere can, if -extensible turn inside out. Moreover, any part of it—a -bowl-shaped portion, for instance—can turn inside out, -and so on round and round.</p> - -<p>This is really no more than we had before in the -rotation about a plane, except that we see that the plane -can, in the case of extensible matter, be curved, and still -play the part of an axis.</p> - -<p>If we suppose the spherical shell to be of four-dimensional -matter, our representation will be a little different. -Let us suppose there to be a small thickness to the matter -in the fourth dimension. This would make no difference -in <a href="#fig_44">fig. 44</a>, for that merely shows the view in the <i>xyz</i> -space. But when the <i>x</i> axis is let drop, and the <i>w</i> axis -comes in, then the rods <span class="allsmcap">CD</span> and <span class="allsmcap">EF</span> which represent the -matter of the shell, will have a certain thickness perpendicular -to the plane of the paper on which they are drawn. -If they have a thickness in the fourth dimension they will -show this thickness when looked at from the direction of -the <i>w</i> axis.</p> - -<p>Supposing these rods, then, to be small slabs strung on -the circumference of the circle in <a href="#fig_45">fig. 45</a>, we see that -there will not be in this case either any obstacle to their -turning round the circumference. We can have a shell -of extensible material or of fluid material turning inside -out in four dimensions.</p> - -<p>And we must remember that in four dimensions there -is no such thing as rotation round an axis. If we want to -investigate the motion of fluids in four dimensions we -must take a movement about an axis in our space, and<span class="pagenum" id="Page_82">[Pg 82]</span> -find the corresponding movement about a plane in -four space.</p> - -<p>Now, of all the movements which take place in fluids, -the most important from a physical point of view is -vortex motion.</p> - -<p>A vortex is a whirl or eddy—it is shown in the gyrating -wreaths of dust seen on a summer day; it is exhibited on -a larger scale in the destructive march of a cyclone.</p> - -<p>A wheel whirling round will throw off the water on it. -But when this circling motion takes place in a liquid -itself it is strangely persistent. There is, of course, a -certain cohesion between the particles of water by which -they mutually impede their motions. But in a liquid -devoid of friction, such that every particle is free from -lateral cohesion on its path of motion, it can be shown -that a vortex or eddy separates from the mass of the -fluid a certain portion, which always remain in that -vortex.</p> - -<p>The shape of the vortex may alter, but it always consists -of the same particles of the fluid.</p> - -<p>Now, a very remarkable fact about such a vortex is that -the ends of the vortex cannot remain suspended and -isolated in the fluid. They must always run to the -boundary of the fluid. An eddy in water that remains -half way down without coming to the top is impossible.</p> - -<p>The ends of a vortex must reach the boundary of a -fluid—the boundary may be external or internal—a vortex -may exist between two objects in the fluid, terminating -one end on each object, the objects being internal -boundaries of the fluid. Again, a vortex may have its -ends linked together, so that it forms a ring. Circular -vortex rings of this description are often seen in puffs of -smoke, and that the smoke travels on in the ring is a -proof that the vortex always consists of the same particles -of air.</p> - -<p><span class="pagenum" id="Page_83">[Pg 83]</span></p> - -<p>Let us now enquire what a vortex would be in a four-dimensional -fluid.</p> - -<p>We must replace the line axis by a plane axis. We -should have therefore a portion of fluid rotating round -a plane.</p> - -<p>We have seen that the contour of this plane corresponds -with the ends of the axis line. Hence such a four-dimensional -vortex must have its rim on a boundary of -the fluid. There would be a region of vorticity with a -contour. If such a rotation were started at one part of a -circular boundary, its edges would run round the boundary -in both directions till the whole interior region was filled -with the vortex sheet.</p> - -<p>A vortex in a three-dimensional liquid may consist of a -number of vortex filaments lying together producing a -tube, or rod of vorticity.</p> - -<p>In the same way we can have in four dimensions a -number of vortex sheets alongside each other, each of which -can be thought of as a bowl-shaped portion of a spherical -shell turning inside out. The rotation takes place at any -point not in the space occupied by the shell, but from -that space to the fourth dimension and round back again.</p> - -<p>Is there anything analogous to this within the range -of our observation?</p> - -<p>An electric current answers this description in every -respect. Electricity does not flow through a wire. Its effect -travels both ways from the starting point along the wire. -The spark which shows its passing midway in its circuit -is later than that which occurs at points near its starting -point on either side of it.</p> - -<p>Moreover, it is known that the action of the current -is not in the wire. It is in the region enclosed by the -wire, this is the field of force, the locus of the exhibition -of the effects of the current.</p> - -<p>And the necessity of a conducting circuit for a current is<span class="pagenum" id="Page_84">[Pg 84]</span> -exactly that which we should expect if it were a four-dimensional -vortex. According to Maxwell every current forms -a closed circuit, and this, from the four-dimensional point -of view, is the same as saying a vortex must have its ends -on a boundary of the fluid.</p> - -<p>Thus, on the hypothesis of a fourth dimension, the rotation -of the fluid ether would give the phenomenon of an -electric current. We must suppose the ether to be full of -movement, for the more we examine into the conditions -which prevail in the obscurity of the minute, the more we -find that an unceasing and perpetual motion reigns. Thus -we may say that the conception of the fourth dimension -means that there must be a phenomenon which presents -the characteristics of electricity.</p> - -<p>We know now that light is an electro-magnetic action, -and that so far from being a special and isolated phenomenon -this electric action is universal in the realm of the -minute. Hence, may we not conclude that, so far from -the fourth dimension being remote and far away, being a -thing of symbolic import, a term for the explanation of -dubious facts by a more obscure theory, it is really the -most important fact within our knowledge. Our three-dimensional -world is superficial. These processes, which -really lie at the basis of all phenomena of matter, -escape our observation by their minuteness, but reveal -to our intellect an amplitude of motion surpassing any -that we can see. In such shapes and motions there is a -realm of the utmost intellectual beauty, and one to -which our symbolic methods apply with a better grace -than they do to those of three dimensions.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_85">[Pg 85]</span></p> - -<h2 class="nobreak" id="CHAPTER_VIII">CHAPTER VIII<br /> - -<small><i>THE USE OF FOUR DIMENSIONS IN -THOUGHT</i></small></h2></div> - - -<p>Having held before ourselves this outline of a conjecture -of the world as four-dimensional, having roughly thrown -together those facts of movement which we can see apply -to our actual experience, let us pass to another branch -of our subject.</p> - -<p>The engineer uses drawings, graphical constructions, -in a variety of manners. He has, for instance, diagrams -which represent the expansion of steam, the efficiency -of his valves. These exist alongside the actual plans of -his machines. They are not the pictures of anything -really existing, but enable him to think about the relations -which exist in his mechanisms.</p> - -<p>And so, besides showing us the actual existence of that -world which lies beneath the one of visible movements, -four-dimensional space enables us to make ideal constructions -which serve to represent the relations of things, -and throw what would otherwise be obscure into a definite -and suggestive form.</p> - -<p>From amidst the great variety of instances which lies -before me I will select two, one dealing with a subject -of slight intrinsic interest, which however gives within -a limited field a striking example of the method<span class="pagenum" id="Page_86">[Pg 86]</span> -of drawing conclusions and the use of higher space -figures.<a id="FNanchor_1" href="#Footnote_1" class="fnanchor">[1]</a></p> - -<div class="footnotes"> -<div class="footnote"> - -<p><a id="Footnote_1" href="#FNanchor_1" class="label">[1]</a> It is suggestive also in another respect, because it shows very -clearly that in our processes of thought there are in play faculties other -than logical; in it the origin of the idea which proves to be justified is -drawn from the consideration of symmetry, a branch of the beautiful.</p> - -</div></div> - -<p>The other instance is chosen on account of the bearing -it has on our fundamental conceptions. In it I try to -discover the real meaning of Kant’s theory of experience.</p> - -<p>The investigation of the properties of numbers is much -facilitated by the fact that relations between numbers are -themselves able to be represented as numbers—<i>e.g.</i>, 12, -and 3 are both numbers, and the relation between them -is 4, another number. The way is thus opened for a -process of constructive theory, without there being any -necessity for a recourse to another class of concepts -besides that which is given in the phenomena to be -studied.</p> - -<p>The discipline of number thus created is of great and -varied applicability, but it is not solely as quantitative -that we learn to understand the phenomena of nature. -It is not possible to explain the properties of matter -by number simply, but all the activities of matter are -energies in space. They are numerically definite and also, -we may say, directedly definite, <i>i.e.</i> definite in direction.</p> - -<p>Is there, then, a body of doctrine about space which, like -that of number, is available in science? It is needless -to answer: Yes; geometry. But there is a method -lying alongside the ordinary methods of geometry, which -tacitly used and presenting an analogy to the method -of numerical thought deserves to be brought into greater -prominence than it usually occupies.</p> - -<p>The relation of numbers is a number.</p> - -<p>Can we say in the same way that the relation of -shapes is a shape?</p> - -<p>We can.</p> - -<p><span class="pagenum" id="Page_87">[Pg 87]</span></p> -<div class="figleft illowp50" id="fig_46" style="max-width: 25em;"> - <img src="images/fig_46.png" alt="" /> - <div class="caption">Fig. 46.</div> -</div> - -<p>To take an instance chosen on account of its ready -availability. Let us take -two right-angled triangles of -a given hypothenuse, but -having sides of different -lengths (<a href="#fig_46">fig. 46</a>). These -triangles are shapes which have a certain relation to each -other. Let us exhibit their relation as a figure.</p> - -<div class="figleft illowp40" id="fig_47" style="max-width: 18.75em;"> - <img src="images/fig_47.png" alt="" /> - <div class="caption">Fig. 47.</div> -</div> - -<p>Draw two straight lines at right angles to each other, -the one <span class="allsmcap">HL</span> a horizontal level, the -other <span class="allsmcap">VL</span> a vertical level (<a href="#fig_47">fig. 47</a>). -By means of these two co-ordinating -lines we can represent a -double set of magnitudes; one set -as distances to the right of the vertical -level, the other as distances -above the horizontal level, a suitable unit being chosen.</p> - -<p>Thus the line marked 7 will pick out the assemblage -of points whose distance from the vertical level is 7, -and the line marked 1 will pick out the points whose -distance above the horizontal level is 1. The meeting -point of these two lines, 7 and 1, will define a point -which with regard to the one set of magnitudes is 7, -with regard to the other is 1. Let us take the sides of -our triangles as the two sets of magnitudes in question.</p> - -<div class="figleft illowp40" id="fig_48" style="max-width: 18.75em;"> - <img src="images/fig_48.png" alt="" /> - <div class="caption">Fig. 48.</div> -</div> - -<p>Then the point 7, 1, will represent the triangle whose -sides are 7 and 1. Similarly the point 5, 5—5, that -is, to the right of the vertical level and 5 above the -horizontal level—will represent the -triangle whose sides are 5 and 5 -(<a href="#fig_48">fig. 48</a>).</p> - -<p>Thus we have obtained a figure -consisting of the two points 7, 1, -and 5, 5, representative of our two -triangles. But we can go further, and, drawing an arc<span class="pagenum" id="Page_88">[Pg 88]</span> -of a circle about <span class="allsmcap">O</span>, the meeting point of the horizontal -and vertical levels, which passes through 7, 1, and 5, 5, -assert that all the triangles which are right-angled and -have a hypothenuse whose square is 50 are represented -by the points on this arc.</p> - -<p>Thus, each individual of a class being represented by a -point, the whole class is represented by an assemblage of -points forming a figure. Accepting this representation -we can attach a definite and calculable significance to the -expression, resemblance, or similarity between two individuals -of the class represented, the difference being -measured by the length of the line between two representative -points. It is needless to multiply examples, or -to show how, corresponding to different classes of triangles, -we obtain different curves.</p> - -<p>A representation of this kind in which an object, a -thing in space, is represented as a point, and all its properties -are left out, their effect remaining only in the -relative position which the representative point bears -to the representative points of the other objects, may be -called, after the analogy of Sir William R. Hamilton’s -hodograph, a “Poiograph.”</p> - -<p>Representations thus made have the character of -natural objects; they have a determinate and definite -character of their own. Any lack of completeness in them -is probably due to a failure in point of completeness -of those observations which form the ground of their -construction.</p> - -<p>Every system of classification is a poiograph. In -Mendeléeff’s scheme of the elements, for instance, each -element is represented by a point, and the relations -between the elements are represented by the relations -between the points.</p> - -<p>So far I have simply brought into prominence processes -and considerations with which we are all familiar. But<span class="pagenum" id="Page_89">[Pg 89]</span> -it is worth while to bring into the full light of our attention -our habitual assumptions and processes. It often -happens that we find there are two of them which have -a bearing on each other, which, without this dragging into -the light, we should have allowed to remain without -mutual influence.</p> - -<p>There is a fact which it concerns us to take into account -in discussing the theory of the poiograph.</p> - -<p>With respect to our knowledge of the world we are -far from that condition which Laplace imagined when he -asserted that an all-knowing mind could determine the -future condition of every object, if he knew the co-ordinates -of its particles in space, and their velocity at any -particular moment.</p> - -<p>On the contrary, in the presence of any natural object, -we have a great complexity of conditions before us, -which we cannot reduce to position in space and date -in time.</p> - -<p>There is mass, attraction apparently spontaneous, electrical -and magnetic properties which must be superadded -to spatial configuration. To cut the list short we must -say that practically the phenomena of the world present -us problems involving many variables, which we must -take as independent.</p> - -<p>From this it follows that in making poiographs we -must be prepared to use space of more than three dimensions. -If the symmetry and completeness of our representation -is to be of use to us we must be prepared to -appreciate and criticise figures of a complexity greater -than of those in three dimensions. It is impossible to give -an example of such a poiograph which will not be merely -trivial, without going into details of some kind irrelevant -to our subject. I prefer to introduce the irrelevant details -rather than treat this part of the subject perfunctorily.</p> - -<p>To take an instance of a poiograph which does not lead<span class="pagenum" id="Page_90">[Pg 90]</span> -us into the complexities incident on its application in -classificatory science, let us follow Mrs. Alicia Boole Stott -in her representation of the syllogism by its means. She -will be interested to find that the curious gap she detected -has a significance.</p> - -<div class= "figleft illowp40" id="fig_49" style="max-width: 13.75em;"> - <img src="images/fig_49.png" alt="" /> - <div class="caption">Fig. 49.</div> -</div> - -<p>A syllogism consists of two statements, the major and -the minor premiss, with the conclusion that can be drawn -from them. Thus, to take an instance, <a href="#fig_49">fig. 49</a>. It is -evident, from looking at the successive figures that, if we -know that the region <span class="allsmcap">M</span> lies altogether within the region -<span class="allsmcap">P</span>, and also know that the region <span class="allsmcap">S</span> lies altogether within -the region <span class="allsmcap">M</span>, we can conclude that the region <span class="allsmcap">S</span> lies -altogether within the region <span class="allsmcap">P</span>. <span class="allsmcap">M</span> is <span class="allsmcap">P</span>, -major premiss; <span class="allsmcap">S</span> is <span class="allsmcap">M</span>, minor premiss; <span class="allsmcap">S</span> -is <span class="allsmcap">P</span>, conclusion. Given the first two data -we must conclude that <span class="allsmcap">S</span> lies in <span class="allsmcap">P</span>. The -conclusion <span class="allsmcap">S</span> is <span class="allsmcap">P</span> involves two terms, <span class="allsmcap">S</span> and -<span class="allsmcap">P</span>, which are respectively called the subject -and the predicate, the letters <span class="allsmcap">S</span> and <span class="allsmcap">P</span> -being chosen with reference to the parts -the notions they designate play in the -conclusion. <span class="allsmcap">S</span> is the subject of the conclusion, -<span class="allsmcap">P</span> is the predicate of the conclusion. -The major premiss we take to be, that -which does not involve <span class="allsmcap">S</span>, and here we -always write it first.</p> - -<p>There are several varieties of statement -possessing different degrees of universality and manners of -assertiveness. These different forms of statement are -called the moods.</p> - -<p>We will take the major premiss as one variable, as a -thing capable of different modifications of the same kind, -the minor premiss as another, and the different moods we -will consider as defining the variations which these -variables undergo.</p> - -<p><span class="pagenum" id="Page_91">[Pg 91]</span></p> - -<p>There are four moods:—</p> - -<p>1. The universal affirmative; all <span class="allsmcap">M</span> is <span class="allsmcap">P</span>, called mood <span class="allsmcap">A</span>.</p> - -<p>2. The universal negative; no <span class="allsmcap">M</span> is <span class="allsmcap">P</span>, mood <span class="allsmcap">E</span>.</p> - -<p>3. The particular affirmative; some <span class="allsmcap">M</span> is <span class="allsmcap">P</span>, mood <span class="allsmcap">I</span>.</p> - -<p>4. The particular negative; some <span class="allsmcap">M</span> is not <span class="allsmcap">P</span>, mood <span class="allsmcap">O</span>.</p> - -<div class="figcenter illowp100" id="fig_50" style="max-width: 62.5em;"> - <img src="images/fig_50.png" alt="" /> - <div class="caption">Figure 50. -</div></div> - - -<p>The dotted lines in 3 and 4, <a href="#fig_50">fig. 50</a>, denote that it is -not known whether or no any objects exist, corresponding -to the space of which the dotted line forms one delimiting -boundary; thus, in mood <span class="allsmcap">I</span> we do not know if there are -any <span class="smcap">M’s</span> which are not <span class="allsmcap">P</span>, we only know some <span class="smcap">M’s</span> are <span class="allsmcap">P</span>.</p> - -<div class="figleft illowp30" id="fig_51" style="max-width: 15.625em;"> - <img src="images/fig_51.png" alt="" /> - <div class="caption">Fig. 51.</div> -</div> - -<p>Representing the first premiss in its various moods by -regions marked by vertical lines to -the right of <span class="allsmcap">PQ</span>, we have in <a href="#fig_51">fig. 51</a>, -running up from the four letters <span class="allsmcap">AEIO</span>, -four columns, each of which indicates -that the major premiss is in the mood -denoted by the respective letter. In -the first column to the right of <span class="allsmcap">PQ</span> is -the mood <span class="allsmcap">A</span>. Now above the line <span class="allsmcap">RS</span> let there be marked -off four regions corresponding to the four moods of the -minor premiss. Thus, in the first row above <span class="allsmcap">RS</span> all the -region between <span class="allsmcap">RS</span> and the first horizontal line above it -denotes that the minor premiss is in the mood <span class="allsmcap">A</span>. The<span class="pagenum" id="Page_92">[Pg 92]</span> -letters <span class="allsmcap">E</span>, <span class="allsmcap">I</span>, <span class="allsmcap">O</span>, in the same way show the mood characterising -the minor premiss in the rows opposite these letters.</p> - -<p>We have still to exhibit the conclusion. To do this we -must consider the conclusion as a third variable, characterised -in its different varieties by four moods—this being -the syllogistic classification. The introduction of a third -variable involves a change in our system of representation.</p> - -<div class="figleft illowp25" id="fig_52" style="max-width: 12.5em;"> - <img src="images/fig_52.png" alt="" /> - <div class="caption">Fig. 52.</div> -</div> - -<p>Before we started with the regions to the right of a -certain line as representing successively the major premiss -in its moods; now we must start with the regions to the -right of a certain plane. Let <span class="allsmcap">LMNR</span> -be the plane face of a cube, <a href="#fig_52">fig. 52</a>, and -let the cube be divided into four parts -by vertical sections parallel to <span class="allsmcap">LMNR</span>. -The variable, the major premiss, is represented -by the successive regions -which occur to the right of the plane -<span class="allsmcap">LMNR</span>—that region to which <span class="allsmcap">A</span> stands opposite, that -slice of the cube, is significative of the mood <span class="allsmcap">A</span>. This -whole quarter-part of the cube represents that for every -part of it the major premiss is in the mood <span class="allsmcap">A</span>.</p> - -<p>In a similar manner the next section, the second with -the letter <span class="allsmcap">E</span> opposite it, represents that for every one of -the sixteen small cubic spaces in it, the major premiss is -in the mood <span class="allsmcap">E</span>. The third and fourth compartments made -by the vertical sections denote the major premiss in the -moods <span class="allsmcap">I</span> and <span class="allsmcap">O</span>. But the cube can be divided in other -ways by other planes. Let the divisions, of which four -stretch from the front face, correspond to the minor -premiss. The first wall of sixteen cubes, facing the -observer, has as its characteristic that in each of the small -cubes, whatever else may be the case, the minor premiss is -in the mood <span class="allsmcap">A</span>. The variable—the minor premiss—varies -through the phases <span class="allsmcap">A</span>, <span class="allsmcap">E</span>, <span class="allsmcap">I</span>, <span class="allsmcap">O</span>, away from the front face of the -cube, or the front plane of which the front face is a part.</p> - -<p><span class="pagenum" id="Page_93">[Pg 93]</span></p> - -<p>And now we can represent the third variable in a precisely -similar way. We can take the conclusion as the third -variable, going through its four phases from the ground -plane upwards. Each of the small cubes at the base of -the whole cube has this true about it, whatever else may -be the case, that the conclusion is, in it, in the mood <span class="allsmcap">A</span>. -Thus, to recapitulate, the first wall of sixteen small cubes, -the first of the four walls which, proceeding from left to -right, build up the whole cube, is characterised in each -part of it by this, that the major premiss is in the mood <span class="allsmcap">A</span>.</p> - -<p>The next wall denotes that the major premiss is in the -mood <span class="allsmcap">E</span>, and so on. Proceeding from the front to the -back the first wall presents a region in every part of -which the minor premiss is in the mood <span class="allsmcap">A</span>. The second -wall is a region throughout which the minor premiss is in -the mood <span class="allsmcap">E</span>, and so on. In the layers, from the bottom -upwards, the conclusion goes through its various moods -beginning with <span class="allsmcap">A</span> in the lowest, <span class="allsmcap">E</span> in the second, <span class="allsmcap">I</span> in the -third, <span class="allsmcap">O</span> in the fourth.</p> - -<p>In the general case, in which the variables represented -in the poiograph pass through a wide range of values, the -planes from which we measure their degrees of variation -in our representation are taken to be indefinitely extended. -In this case, however, all we are concerned with is the -finite region.</p> - -<p>We have now to represent, by some limitation of the -complex we have obtained, the fact that not every combination -of premisses justifies any kind of conclusion. -This can be simply effected by marking the regions in -which the premisses, being such as are defined by the -positions, a conclusion which is valid is found.</p> - -<p>Taking the conjunction of the major premiss, all <span class="allsmcap">M</span> is -<span class="allsmcap">P</span>, and the minor, all <span class="allsmcap">S</span> is <span class="allsmcap">M</span>, we conclude that all <span class="allsmcap">S</span> is <span class="allsmcap">P</span>. -Hence, that region must be marked in which we have the -conjunction of major premiss in mood <span class="allsmcap">A</span>; minor premiss,<span class="pagenum" id="Page_94">[Pg 94]</span> -mood <span class="allsmcap">A</span>; conclusion, mood <span class="allsmcap">A</span>. This is the cube occupying -the lowest left-hand corner of the large cube.</p> - -<div class="figleft illowp25" id="fig_53" style="max-width: 12.5em;"> - <img src="images/fig_53.png" alt="" /> - <div class="caption">Fig. 53.</div> -</div> - - -<p>Proceeding in this way, we find that the regions which -must be marked are those shown in <a href="#fig_53">fig. 53</a>. -To discuss the case shown in the marked -cube which appears at the top of <a href="#fig_53">fig. 53</a>. -Here the major premiss is in the second -wall to the right—it is in the mood <span class="allsmcap">E</span> and -is of the type no <span class="allsmcap">M</span> is <span class="allsmcap">P</span>. The minor -premiss is in the mood characterised by -the third wall from the front. It is of -the type some <span class="allsmcap">S</span> is <span class="allsmcap">M</span>. From these premisses we draw -the conclusion that some <span class="allsmcap">S</span> is not <span class="allsmcap">P</span>, a conclusion in the -mood <span class="allsmcap">O</span>. Now the mood <span class="allsmcap">O</span> of the conclusion is represented -in the top layer. Hence we see that the marking is -correct in this respect.</p> - -<div class="figleft illowp50" id="fig_54" style="max-width: 25em;"> - <img src="images/fig_54.png" alt="" /> - <div class="caption">Fig. 54.</div> -</div> - -<p>It would, of course, be possible to represent the cube on -a plane by means of four -squares, as in <a href="#fig_54">fig. 54</a>, if we -consider each square to represent -merely the beginning -of the region it stands for. -Thus the whole cube can be -represented by four vertical -squares, each standing for a -kind of vertical tray, and the -markings would be as shown. In No. 1 the major premiss -is in mood <span class="allsmcap">A</span> for the whole of the region indicated by the -vertical square of sixteen divisions; in No. 2 it is in the -mood <span class="allsmcap">E</span>, and so on.</p> - -<p>A creature confined to a plane would have to adopt some -such disjunctive way of representing the whole cube. He -would be obliged to represent that which we see as a -whole in separate parts, and each part would merely -represent, would not be, that solid content which we see.</p> - -<p><span class="pagenum" id="Page_95">[Pg 95]</span></p> - -<p>The view of these four squares which the plane creature -would have would not be such as ours. He would not -see the interior of the four squares represented above, but -each would be entirely contained within its outline, the -internal boundaries of the separate small squares he could -not see except by removing the outer squares.</p> - -<p>We are now ready to introduce the fourth variable -involved in the syllogism.</p> - -<p>In assigning letters to denote the terms of the syllogism -we have taken <span class="allsmcap">S</span> and <span class="allsmcap">P</span> to represent the subject and -predicate in the conclusion, and thus in the conclusion -their order is invariable. But in the premisses we have -taken arbitrarily the order all <span class="allsmcap">M</span> is <span class="allsmcap">P</span>, and all <span class="allsmcap">S</span> is <span class="allsmcap">M</span>. -There is no reason why <span class="allsmcap">M</span> instead of <span class="allsmcap">P</span> should not be the -predicate of the major premiss, and so on.</p> - -<p>Accordingly we take the order of the terms in the premisses -as the fourth variable. Of this order there are four -varieties, and these varieties are called figures.</p> - -<p>Using the order in which the letters are written to -denote that the letter first written is subject, the one -written second is predicate, we have the following possibilities:—</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdc"></td> -<td class="tdc">1st Figure.</td> -<td class="tdc">2nd Figure.</td> -<td class="tdc">3rd Figure.</td> -<td class="tdc">4th Figure.</td> -</tr> -<tr> -<td class="tdc">Major</td> -<td class="tdc"><span class="allsmcap">M P</span></td> -<td class="tdc"><span class="allsmcap">P M</span></td> -<td class="tdc"><span class="allsmcap">M P</span></td> -<td class="tdc"><span class="allsmcap">P M</span></td> -</tr> -<tr> -<td class="tdc">Minor</td> -<td class="tdc"><span class="allsmcap">S M</span></td> -<td class="tdc"><span class="allsmcap">S M</span></td> -<td class="tdc"><span class="allsmcap">M S</span></td> -<td class="tdc"><span class="allsmcap">M S</span></td> -</tr> -</table> - -<p>There are therefore four possibilities with regard to -this fourth variable as with regard to the premisses.</p> - -<p>We have used up our dimensions of space in representing -the phases of the premisses and the conclusion in -respect of mood, and to represent in an analogous manner -the variations in figure we require a fourth dimension.</p> - -<p>Now in bringing in this fourth dimension we must -make a change in our origins of measurement analogous -to that which we made in passing from the plane to the -solid.</p> - -<p><span class="pagenum" id="Page_96">[Pg 96]</span></p> - -<p>This fourth dimension is supposed to run at right -angles to any of the three space dimensions, as the third -space dimension runs at right angles to the two dimensions -of a plane, and thus it gives us the opportunity of -generating a new kind of volume. If the whole cube -moves in this dimension, the solid itself traces out a path, -each section of which, made at right angles to the -direction in which it moves, is a solid, an exact repetition -of the cube itself.</p> - -<p>The cube as we see it is the beginning of a solid of such -a kind. It represents a kind of tray, as the square face of -the cube is a kind of tray against which the cube rests.</p> - -<p>Suppose the cube to move in this fourth dimension in -four stages, and let the hyper-solid region traced out in -the first stage of its progress be characterised by this, that -the terms of the syllogism are in the first figure, then we -can represent in each of the three subsequent stages the -remaining three figures. Thus the whole cube forms -the basis from which we measure the variation in figure. -The first figure holds good for the cube as we see it, and -for that hyper-solid which lies within the first stage; -the second figure holds good in the second stage, and -so on.</p> - -<p>Thus we measure from the whole cube as far as figures -are concerned.</p> - -<p>But we saw that when we measured in the cube itself -having three variables, namely, the two premisses and -the conclusion, we measured from three planes. The base -from which we measured was in every case the same.</p> - -<p>Hence, in measuring in this higher space we should -have bases of the same kind to measure from, we should -have solid bases.</p> - -<p>The first solid base is easily seen, it is the cube itself. -The other can be found from this consideration.</p> - -<p>That solid from which we measure figure is that in<span class="pagenum" id="Page_97">[Pg 97]</span> -which the remaining variables run through their full -range of varieties.</p> - -<p>Now, if we want to measure in respect of the moods of -the major premiss, we must let the minor premiss, the -conclusion, run through their range, and also the order -of the terms. That is we must take as basis of measurement -in respect to the moods of the major that which -represents the variation of the moods of the minor, the -conclusion and the variation of the figures.</p> - -<p>Now the variation of the moods of the minor and of the -conclusion are represented in the square face on the left -of the cube. Here are all varieties of the minor premiss -and the conclusion. The varieties of the figures are -represented by stages in a motion proceeding at right -angles to all space directions, at right angles consequently -to the face in question, the left-hand face of the cube.</p> - -<p>Consequently letting the left-hand face move in this -direction we get a cube, and in this cube all the varieties -of the minor premiss, the conclusion, and the figure are -represented.</p> - -<p>Thus another cubic base of measurement is given to -the cube, generated by movement of the left-hand square -in the fourth dimension.</p> - -<p>We find the other bases in a similar manner, one is the -cube generated by the front square moved in the fourth -dimension so as to generate a cube. From this cube -variations in the mood of the minor are measured. The -fourth base is that found by moving the bottom square of -the cube in the fourth dimension. In this cube the -variations of the major, the minor, and the figure are given. -Considering this as a basis in the four stages proceeding -from it, the variation in the moods of the conclusion are -given.</p> - -<p>Any one of these cubic bases can be represented in space, -and then the higher solid generated from them lies out of<span class="pagenum" id="Page_98">[Pg 98]</span> -our space. It can only be represented by a device analogous -to that by which the plane being represents a cube.</p> - -<p>He represents the cube shown above, by taking four -square sections and placing them arbitrarily at convenient -distances the one from the other.</p> - -<p>So we must represent this higher solid by four cubes: -each cube represents only the beginning of the corresponding -higher volume.</p> - -<p>It is sufficient for us, then, if we draw four cubes, the -first representing that region in which the figure is of the -first kind, the second that region in which the figure is -of the second kind, and so on. These cubes are the -beginnings merely of the respective regions—they are -the trays, as it were, against which the real solids must -be conceived as resting, from which they start. The first -one, as it is the beginning of the region of the first figure, -is characterised by the order of the terms in the premisses -being that of the first figure. The second similarly has -the terms of the premisses in the order of the second -figure, and so on.</p> - -<p>These cubes are shown below.</p> - -<p>For the sake of showing the properties of the method -of representation, not for the logical problem, I will make -a digression. I will represent in space the moods of the -minor and of the conclusion and the different figures, -keeping the major always in mood <span class="allsmcap">A</span>. Here we have -three variables in different stages, the minor, the conclusion, -and the figure. Let the square of the left-hand -side of the original cube be imagined to be standing by -itself, without the solid part of the cube, represented by -(2) <a href="#fig_55">fig. 55</a>. The <span class="allsmcap">A</span>, <span class="allsmcap">E</span>, <span class="allsmcap">I</span>, <span class="allsmcap">O</span>, which run away represent the -moods of the minor, the <span class="allsmcap">A</span>, <span class="allsmcap">E</span>, <span class="allsmcap">I</span>, <span class="allsmcap">O</span>, which run up represent -the moods of the conclusion. The whole square, since it -is the beginning of the region in the major premiss, mood -<span class="allsmcap">A</span>, is to be considered as in major premiss, mood <span class="allsmcap">A</span>.</p> - -<p><span class="pagenum" id="Page_99">[Pg 99]</span></p> - -<p>From this square, let it be supposed that that direction -in which the figures are represented runs to the -left hand. Thus we have a cube (1) running from the -square above, in which the square itself is hidden, but -the letters <span class="allsmcap">A</span>, <span class="allsmcap">E</span>, <span class="allsmcap">I</span>, <span class="allsmcap">O</span>, of the conclusion are seen. In this -cube we have the minor premiss and the conclusion in all -their moods, and all the figures represented. With regard -to the major premiss, since the face (2) belongs to the first -wall from the left in the original arrangement, and in this -arrangement was characterised by the major premiss in the -mood <span class="allsmcap">A</span>, we may say that the whole of the cube we now -have put up represents the mood <span class="allsmcap">A</span> of the major premiss.</p> - -<div class="figcenter illowp100" id="fig_55" style="max-width: 50em;"> - <img src="images/fig_55.png" alt="" /> - <div class="caption">Fig. 55.</div> -</div> - -<p>Hence the small cube at the bottom to the right in 1, -nearest to the spectator, is major premiss, mood <span class="allsmcap">A</span>; minor -premiss, mood <span class="allsmcap">A</span>; conclusion, mood <span class="allsmcap">A</span>; and figure the first. -The cube next to it, running to the left, is major premiss, -mood <span class="allsmcap">A</span>; minor premiss, mood <span class="allsmcap">A</span>; conclusion, mood <span class="allsmcap">A</span>; -figure 2.</p> - -<p>So in this cube we have the representations of all the -combinations which can occur when the major premiss, -remaining in the mood <span class="allsmcap">A</span>, the minor premiss, the conclusion, -and the figures pass through their varieties.</p> - -<p>In this case there is no room in space for a natural -representation of the moods of the major premiss. To -represent them we must suppose as before that there is a -fourth dimension, and starting from this cube as base in -the fourth direction in four equal stages, all the first volume -corresponds to major premiss <span class="allsmcap">A</span>, the second to major<span class="pagenum" id="Page_100">[Pg 100]</span> -premiss, mood <span class="allsmcap">E</span>, the next to the mood <span class="allsmcap">I</span>, and the last -to mood <span class="allsmcap">O</span>.</p> - -<p>The cube we see is as it were merely a tray against -which the four-dimensional figure rests. Its section at -any stage is a cube. But a transition in this direction -being transverse to the whole of our space is represented -by no space motion. We can exhibit successive stages of -the result of transference of the cube in that direction, -but cannot exhibit the product of a transference, however -small, in that direction.</p> - -<div class="figcenter illowp100" id="fig_56" style="max-width: 62.5em;"> - <img src="images/fig_56.png" alt="" /> - <div class="caption">Fig. 56.</div> -</div> - -<p>To return to the original method of representing our -variables, consider <a href="#fig_56">fig. 56</a>. These four cubes represent -four sections of the figure derived from the first of them -by moving it in the fourth dimension. The first portion -of the motion, which begins with 1, traces out a -more than solid body, which is all in the first figure. -The beginning of this body is shown in 1. The next -portion of the motion traces out a more than solid body, -all of which is in the second figure; the beginning of -this body is shown in 2; 3 and 4 follow on in like -manner. Here, then, in one four-dimensional figure we -have all the combinations of the four variables, major -premiss, minor premiss, figure, conclusion, represented, -each variable going through its four varieties. The disconnected -cubes drawn are our representation in space by -means of disconnected sections of this higher body.</p> - -<p><span class="pagenum" id="Page_101">[Pg 101]</span></p> - -<p>Now it is only a limited number of conclusions which -are true—their truth depends on the particular combinations -of the premisses and figures which they accompany. -The total figure thus represented may be called the -universe of thought in respect to these four constituents, -and out of the universe of possibly existing combinations -it is the province of logic to select those which correspond -to the results of our reasoning faculties.</p> - -<p>We can go over each of the premisses in each of the -moods, and find out what conclusion logically follows. -But this is done in the works on logic; most simply and -clearly I believe in “Jevon’s Logic.” As we are only concerned -with a formal presentation of the results we will -make use of the mnemonic lines printed below, in which -the words enclosed in brackets refer to the figures, and -are not significative:—</p> - -<ul> -<li>Barbara celarent Darii ferio<i>que</i> [prioris].</li> -<li>Caesare Camestris Festino Baroko [secundae].</li> -<li>[Tertia] darapti disamis datisi felapton.</li> -<li>Bokardo ferisson <i>habet</i> [Quarta insuper addit].</li> -<li>Bramantip camenes dimaris ferapton fresison.</li> -</ul> - -<p>In these lines each significative word has three vowels, -the first vowel refers to the major premiss, and gives the -mood of that premiss, “a” signifying, for instance, that -the major mood is in mood <i>a</i>. The second vowel refers -to the minor premiss, and gives its mood. The third -vowel refers to the conclusion, and gives its mood. Thus -(prioris)—of the first figure—the first mnemonic word is -“barbara,” and this gives major premiss, mood <span class="allsmcap">A</span>; minor -premiss, mood <span class="allsmcap">A</span>; conclusion, mood <span class="allsmcap">A</span>. Accordingly in the -first of our four cubes we mark the lowest left-hand front -cube. To take another instance in the third figure “Tertia,” -the word “ferisson” gives us major premiss mood <span class="allsmcap">E</span>—<i>e.g.</i>, -no <span class="allsmcap">M</span> is <span class="allsmcap">P</span>, minor premiss mood <span class="allsmcap">I</span>; some <span class="allsmcap">M</span> is <span class="allsmcap">S</span>, conclusion, -mood <span class="allsmcap">O</span>; some <span class="allsmcap">S</span> is not <span class="allsmcap">P</span>. The region to be marked then<span class="pagenum" id="Page_102">[Pg 102]</span> -in the third representative cube is the one in the second -wall to the right for the major premiss, the third wall -from the front for the minor premiss, and the top layer -for the conclusion.</p> - -<p>It is easily seen that in the diagram this cube is -marked, and so with all the valid conclusions. The -regions marked in the total region show which combinations -of the four variables, major premiss, minor -premiss, figure, and conclusion exist.</p> - -<p>That is to say, we objectify all possible conclusions, and -build up an ideal manifold, containing all possible combinations -of them with the premisses, and then out of -this we eliminate all that do not satisfy the laws of logic. -The residue is the syllogism, considered as a canon of -reasoning.</p> - -<p>Looking at the shape which represents the totality -of the valid conclusions, it does not present any obvious -symmetry, or easily characterisable nature. A striking -configuration, however, is obtained, if we project the four-dimensional -figure obtained into a three-dimensional one; -that is, if we take in the base cube all those cubes which -have a marked space anywhere in the series of four -regions which start from that cube.</p> - -<p>This corresponds to making abstraction of the figures, -giving all the conclusions which are valid whatever the -figure may be.</p> - -<div class="figcenter illowp25" id="fig_57" style="max-width: 12.5em;"> - <img src="images/fig_57.png" alt="" /> - <div class="caption">Fig. 57.</div> -</div> - -<p>Proceeding in this way we obtain the arrangement of -marked cubes shown in <a href="#fig_57">fig. 57</a>. We see -that the valid conclusions are arranged -almost symmetrically round one cube—the -one on the top of the column starting from -<span class="allsmcap">AAA</span>. There is one breach of continuity -however in this scheme. One cube is -unmarked, which if marked would give -symmetry. It is the one which would be denoted by the<span class="pagenum" id="Page_103">[Pg 103]</span> -letters <span class="allsmcap">I</span>, <span class="allsmcap">E</span>, <span class="allsmcap">O</span>, in the third wall to the right, the second -wall away, the topmost layer. Now this combination of -premisses in the mood <span class="allsmcap">IE</span>, with a conclusion in the mood -<span class="allsmcap">O</span>, is not noticed in any book on logic with which I am -familiar. Let us look at it for ourselves, as it seems -that there must be something curious in connection with -this break of continuity in the poiograph.</p> - -<div class="figcenter illowp100" id="fig_58" style="max-width: 62.5em;"> - <img src="images/fig_58.png" alt="" /> - <div class="caption">Fig. 58.</div> -</div> - -<p>The propositions <span class="allsmcap">I</span>, <span class="allsmcap">E</span>, in the various figures are the -following, as shown in the accompanying scheme, <a href="#fig_58">fig. 58</a>:—First -figure: some <span class="allsmcap">M</span> is <span class="allsmcap">P</span>; no <span class="allsmcap">S</span> is <span class="allsmcap">M</span>. Second figure: -some <span class="allsmcap">P</span> is <span class="allsmcap">M</span>; no <span class="allsmcap">S</span> is <span class="allsmcap">M</span>. Third figure: some <span class="allsmcap">M</span> is <span class="allsmcap">P</span>; no -<span class="allsmcap">M</span> is <span class="allsmcap">S</span>. Fourth figure: some <span class="allsmcap">P</span> is <span class="allsmcap">M</span>; no <span class="allsmcap">M</span> is <span class="allsmcap">S</span>.</p> - -<p>Examining these figures, we see, taking the first, that -if some <span class="allsmcap">M</span> is <span class="allsmcap">P</span> and no <span class="allsmcap">S</span> is <span class="allsmcap">M</span>, we have no conclusion of<span class="pagenum" id="Page_104">[Pg 104]</span> -the form <span class="allsmcap">S</span> is <span class="allsmcap">P</span> in the various moods. It is quite indeterminate -how the circle representing <span class="allsmcap">S</span> lies with regard -to the circle representing <span class="allsmcap">P</span>. It may lie inside, outside, -or partly inside <span class="allsmcap">P</span>. The same is true in the other figures -2 and 3. But when we come to the fourth figure, since -<span class="allsmcap">M</span> and <span class="allsmcap">S</span> lie completely outside each other, there cannot -lie inside <span class="allsmcap">S</span> that part of <span class="allsmcap">P</span> which lies inside <span class="allsmcap">M</span>. Now -we know by the major premiss that some of <span class="allsmcap">P</span> does lie -in <span class="allsmcap">M</span>. Hence <span class="allsmcap">S</span> cannot contain the whole of <span class="allsmcap">P</span>. In -words, some <span class="allsmcap">P</span> is <span class="allsmcap">M</span>, no <span class="allsmcap">M</span> is <span class="allsmcap">S</span>, therefore <span class="allsmcap">S</span> does not contain -the whole of <span class="allsmcap">P</span>. If we take <span class="allsmcap">P</span> as the subject, this gives -us a conclusion in the mood <span class="allsmcap">O</span> about <span class="allsmcap">P</span>. Some <span class="allsmcap">P</span> is not <span class="allsmcap">S</span>. -But it does not give us conclusion about <span class="allsmcap">S</span> in any one -of the four forms recognised in the syllogism and called -its moods. Hence the breach of the continuity in the -poiograph has enabled us to detect a lack of completeness -in the relations which are considered in the syllogism.</p> - -<p>To take an instance:—Some Americans (<span class="allsmcap">P</span>) are of -African stock (<span class="allsmcap">M</span>); No Aryans (<span class="allsmcap">S</span>) are of African stock -(<span class="allsmcap">M</span>); Aryans (<span class="allsmcap">S</span>) do not include all of Americans (<span class="allsmcap">P</span>).</p> - -<p>In order to draw a conclusion about <span class="allsmcap">S</span> we have to admit -the statement, “<span class="allsmcap">S</span> does not contain the whole of <span class="allsmcap">P</span>,” as -a valid logical form—it is a statement about <span class="allsmcap">S</span> which can -be made. The logic which gives us the form, “some <span class="allsmcap">P</span> -is not <span class="allsmcap">S</span>,” and which does not allow us to give the exactly -equivalent and equally primary form, “<span class="allsmcap">S</span> does not contain -the whole of <span class="allsmcap">P</span>,” is artificial.</p> - -<p>And I wish to point out that this artificiality leads -to an error.</p> - -<p>If one trusted to the mnemonic lines given above, one -would conclude that no logical conclusion about <span class="allsmcap">S</span> can -be drawn from the statement, “some <span class="allsmcap">P</span> are <span class="allsmcap">M</span>, no <span class="allsmcap">M</span> are <span class="allsmcap">S</span>.”</p> - -<p>But a conclusion can be drawn: <span class="allsmcap">S</span> does not contain -the whole of <span class="allsmcap">P</span>.</p> - -<p>It is not that the result is given expressed in another<span class="pagenum" id="Page_105">[Pg 105]</span> -form. The mnemonic lines deny that any conclusion -can be drawn from premisses in the moods <span class="allsmcap">I</span>, <span class="allsmcap">E</span>, respectively.</p> - -<p>Thus a simple four-dimensional poiograph has enabled -us to detect a mistake in the mnemonic lines which have -been handed down unchallenged from mediæval times. -To discuss the subject of these lines more fully a logician -defending them would probably say that a particular -statement cannot be a major premiss; and so deny the -existence of the fourth figure in the combination of moods.</p> - -<p>To take our instance: some Americans are of African -stock; no Aryans are of African stock. He would say -that the conclusion is some Americans are not Aryans; -and that the second statement is the major. He would -refuse to say anything about Aryans, condemning us to -an eternal silence about them, as far as these premisses -are concerned! But, if there is a statement involving -the relation of two classes, it must be expressible as a -statement about either of them.</p> - -<p>To bar the conclusion, “Aryans do not include the -whole of Americans,” is purely a makeshift in favour of -a false classification.</p> - -<p>And the argument drawn from the universality of the -major premiss cannot be consistently maintained. It -would preclude such combinations as major <span class="allsmcap">O</span>, minor <span class="allsmcap">A</span>, -conclusion <span class="allsmcap">O</span>—<i>i.e.</i>, such as some mountains (<span class="allsmcap">M</span>) are not -permanent (<span class="allsmcap">P</span>); all mountains (<span class="allsmcap">M</span>) are scenery (<span class="allsmcap">S</span>); some -scenery (<span class="allsmcap">S</span>) is not permanent (<span class="allsmcap">P</span>).</p> - -<p>This is allowed in “Jevon’s Logic,” and his omission to -discuss <span class="allsmcap">I</span>, <span class="allsmcap">E</span>, <span class="allsmcap">O</span>, in the fourth figure, is inexplicable. A -satisfactory poiograph of the logical scheme can be made -by admitting the use of the words some, none, or all, -about the predicate as well as about the subject. Then -we can express the statement, “Aryans do not include the -whole of Americans,” clumsily, but, when its obscurity -is fathomed, correctly, as “Some Aryans are not all<span class="pagenum" id="Page_106">[Pg 106]</span> -Americans.” And this method is what is called the -“quantification of the predicate.”</p> - -<p>The laws of formal logic are coincident with the conclusions -which can be drawn about regions of space, which -overlap one another in the various possible ways. It is -not difficult so to state the relations or to obtain a -symmetrical poiograph. But to enter into this branch of -geometry is beside our present purpose, which is to show -the application of the poiograph in a finite and limited -region, without any of those complexities which attend its -use in regard to natural objects.</p> - -<p>If we take the latter—plants, for instance—and, without -assuming fixed directions in space as representative of -definite variations, arrange the representative points in -such a manner as to correspond to the similarities of the -objects, we obtain configuration of singular interest; and -perhaps in this way, in the making of shapes of shapes, -bodies with bodies omitted, some insight into the structure -of the species and genera might be obtained.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_107">[Pg 107]</span></p> - -<h2 class="nobreak" id="CHAPTER_IX">CHAPTER IX<br /> - -<small><i>APPLICATION TO KANT’S THEORY OF -EXPERIENCE</i></small></h2></div> - - -<p>When we observe the heavenly bodies we become aware -that they all participate in one universal motion—a -diurnal revolution round the polar axis.</p> - -<p>In the case of fixed stars this is most unqualifiedly true, -but in the case of the sun, and the planets also, the single -motion of revolution can be discerned, modified, and -slightly altered by other and secondary motions.</p> - -<p>Hence the universal characteristic of the celestial bodies -is that they move in a diurnal circle.</p> - -<p>But we know that this one great fact which is true of -them all has in reality nothing to do with them. The -diurnal revolution which they visibly perform is the result -of the condition of the observer. It is because the -observer is on a rotating earth that a universal statement -can be made about all the celestial bodies.</p> - -<p>The universal statement which is valid about every one -of the celestial bodies is that which does not concern -them at all, and is but a statement of the condition of -the observer.</p> - -<p>Now there are universal statements of other kinds -which we can make. We can say that all objects of -experience are in space and subject to the laws of -geometry.</p> - -<p><span class="pagenum" id="Page_108">[Pg 108]</span></p> - -<p>Does this mean that space and all that it means is due -to a condition of the observer?</p> - -<p>If a universal law in one case means nothing affecting -the objects themselves, but only a condition of observation, -is this true in every case? There is shown us in -astronomy a <i>vera causa</i> for the assertion of a universal. -Is the same cause to be traced everywhere?</p> - -<p>Such is a first approximation to the doctrine of Kant’s -critique.</p> - -<p>It is the apprehension of a relation into which, on the -one side and the other, perfectly definite constituents -enter—the human observer and the stars—and a transference -of this relation to a region in which the constituents -on either side are perfectly unknown.</p> - -<p>If spatiality is due to a condition of the observer, the -observer cannot be this bodily self of ours—the body, like -the objects around it, are equally in space.</p> - -<p>This conception Kant applied, not only to the intuitions -of sense, but to the concepts of reason—wherever a universal -statement is made there is afforded him an opportunity -for the application of his principle. He constructed a -system in which one hardly knows which the most to -admire, the architectonic skill, or the reticence with regard -to things in themselves, and the observer in himself.</p> - -<p>His system can be compared to a garden, somewhat -formal perhaps, but with the charm of a quality more -than intellectual, a <i>besonnenheit</i>, an exquisite moderation -over all. And from the ground he so carefully prepared -with that buried in obscurity, which it is fitting should -be obscure, science blossoms and the tree of real knowledge -grows.</p> - -<p>The critique is a storehouse of ideas of profound interest. -The one of which I have given a partial statement leads, -as we shall see on studying it in detail, to a theory of -mathematics suggestive of enquiries in many directions.</p> - -<p><span class="pagenum" id="Page_109">[Pg 109]</span></p> - -<p>The justification for my treatment will be found -amongst other passages in that part of the transcendental -analytic, in which Kant speaks of objects of experience -subject to the forms of sensibility, not subject to the -concepts of reason.</p> - -<p>Kant asserts that whenever we think we think of -objects in space and time, but he denies that the space -and time exist as independent entities. He goes about -to explain them, and their universality, not by assuming -them, as most other philosophers do, but by postulating -their absence. How then does it come to pass that the -world is in space and time to us?</p> - -<p>Kant takes the same position with regard to what we -call nature—a great system subject to law and order. -“How do you explain the law and order in nature?” we -ask the philosophers. All except Kant reply by assuming -law and order somewhere, and then showing how we can -recognise it.</p> - -<p>In explaining our notions, philosophers from other than -the Kantian standpoint, assume the notions as existing -outside us, and then it is no difficult task to show how -they come to us, either by inspiration or by observation.</p> - -<p>We ask “Why do we have an idea of law in nature?” -“Because natural processes go according to law,” we are -answered, “and experience inherited or acquired, gives us -this notion.”</p> - -<p>But when we speak about the law in nature we are -speaking about a notion of our own. So all that these -expositors do is to explain our notion by an assumption -of it.</p> - -<p>Kant is very different. He supposes nothing. An experience -such as ours is very different from experience -in the abstract. Imagine just simply experience, succession -of states, of consciousness! Why, there would -be no connecting any two together, there would be no<span class="pagenum" id="Page_110">[Pg 110]</span> -personal identity, no memory. It is out of a general -experience such as this, which, in respect to anything we -call real, is less than a dream, that Kant shows the -genesis of an experience such as ours.</p> - -<p>Kant takes up the problem of the explanation of space, -time, order, and so quite logically does not presuppose -them.</p> - -<p>But how, when every act of thought is of things in -space, and time, and ordered, shall we represent to ourselves -that perfectly indefinite somewhat which is Kant’s -necessary hypothesis—that which is not in space or time -and is not ordered. That is our problem, to represent -that which Kant assumes not subject to any of our forms -of thought, and then show some function which working -on that makes it into a “nature” subject to law and -order, in space and time. Such a function Kant calls the -“Unity of Apperception”; <i>i.e.</i>, that which makes our state -of consciousness capable of being woven into a system -with a self, an outer world, memory, law, cause, and order.</p> - -<p>The difficulty that meets us in discussing Kant’s -hypothesis is that everything we think of is in space -and time—how then shall we represent in space an existence -not in space, and in time an existence not in time? -This difficulty is still more evident when we come to -construct a poiograph, for a poiograph is essentially a -space structure. But because more evident the difficulty -is nearer a solution. If we always think in space, <i>i.e.</i> -using space concepts, the first condition requisite for -adapting them to the representation of non-spatial existence, -is to be aware of the limitation of our thought, -and so be able to take the proper steps to overcome it. -The problem before us, then, is to represent in space an -existence not in space.</p> - -<p>The solution is an easy one. It is provided by the -conception of alternativity.</p> - -<p><span class="pagenum" id="Page_111">[Pg 111]</span></p> - -<p>To get our ideas clear let us go right back behind the -distinctions of an inner and an outer world. Both of -these, Kant says, are products. Let us take merely states -of consciousness, and not ask the question whether they are -produced or superinduced—to ask such a question is to -have got too far on, to have assumed something of which -we have not traced the origin. Of these states let us -simply say that they occur. Let us now use the word -a “posit” for a phase of consciousness reduced to its -last possible stage of evanescence; let a posit be that -phase of consciousness of which all that can be said is -that it occurs.</p> - -<p>Let <i>a</i>, <i>b</i>, <i>c</i>, be three such posits. We cannot represent -them in space without placing them in a certain order, -as <i>a</i>, <i>b</i>, <i>c</i>. But Kant distinguishes between the forms -of sensibility and the concepts of reason. A dream in -which everything happens at haphazard would be an -experience subject to the form of sensibility and only -partially subject to the concepts of reason. It is partially -subject to the concepts of reason because, although -there is no order of sequence, still at any given time -there is order. Perception of a thing as in space is a -form of sensibility, the perception of an order is a concept -of reason.</p> - -<p>We must, therefore, in order to get at that process -which Kant supposes to be constitutive of an ordered -experience imagine the posits as in space without -order.</p> - -<p>As we know them they must be in some order, <i>abc</i>, -<i>bca</i>, <i>cab</i>, <i>acb</i>, <i>cba</i>, <i>bac</i>, one or another.</p> - -<p>To represent them as having no order conceive all -these different orders as equally existing. Introduce the -conception of alternativity—let us suppose that the order -<i>abc</i>, and <i>bac</i>, for example, exist equally, so that we -cannot say about <i>a</i> that it comes before or after <i>b</i>. This<span class="pagenum" id="Page_112">[Pg 112]</span> -would correspond to a sudden and arbitrary change of <i>a</i> -into <i>b</i> and <i>b</i> into <i>a</i>, so that, to use Kant’s words, it would -be possible to call one thing by one name at one time -and at another time by another name.</p> - -<p>In an experience of this kind we have a kind of chaos, -in which no order exists; it is a manifold not subject to -the concepts of reason.</p> - -<p>Now is there any process by which order can be introduced -into such a manifold—is there any function of -consciousness in virtue of which an ordered experience -could arise?</p> - -<p>In the precise condition in which the posits are, as -described above, it does not seem to be possible. But -if we imagine a duality to exist in the manifold, a -function of consciousness can be easily discovered which -will produce order out of no order.</p> - -<p>Let us imagine each posit, then, as having, a dual aspect. -Let <i>a</i> be 1<i>a</i> in which the dual aspect is represented by the -combination of symbols. And similarly let <i>b</i> be 2<i>b</i>, -<i>c</i> be 3<i>c</i>, in which 2 and <i>b</i> represent the dual aspects -of <i>b</i>, 3 and <i>c</i> those of <i>c</i>.</p> - -<p>Since <i>a</i> can arbitrarily change into <i>b</i>, or into <i>c</i>, and -so on, the particular combinations written above cannot -be kept. We have to assume the equally possible occurrence -of form such as 2<i>a</i>, 2<i>b</i>, and so on; and in order -to get a representation of all those combinations out of -which any set is alternatively possible, we must take -every aspect with every aspect. We must, that is, have -every letter with every number.</p> - -<p>Let us now apply the method of space representation.</p> - -<div class="blockquote"> - -<p><i>Note.</i>—At the beginning of the next chapter the same -structures as those which follow are exhibited in -more detail and a reference to them will remove -any obscurity which may be found in the immediately -following passages. They are there carried</p> - -<p><span class="pagenum" id="Page_113">[Pg 113]</span></p> - -<p>on to a greater multiplicity of dimensions, and the -significance of the process here briefly explained -becomes more apparent.</p> -</div> -<div class="figleft illowp25" id="fig_59" style="max-width: 12.5em;"> - <img src="images/fig_59.png" alt="" /> - <div class="caption">Fig. 59.</div> -</div> - -<p>Take three mutually rectangular axes in space 1, 2, 3 -(<a href="#fig_59">fig. 59</a>), and on each mark three points, -the common meeting point being the -first on each axis. Then by means of -these three points on each axis we -define 27 positions, 27 points in a -cubical cluster, shown in <a href="#fig_60">fig. 60</a>, the -same method of co-ordination being -used as has been described before. -Each of these positions can be named by means of the -axes and the points combined.</p> - -<div class="figleft illowp30" id="fig_60" style="max-width: 18.75em;"> - <img src="images/fig_60.png" alt="" /> - <div class="caption">Fig. 60.</div> -</div> - - -<p>Thus, for instance, the one marked by an asterisk can -be called 1<i>c</i>, 2<i>b</i>, 3<i>c</i>, because it is -opposite to <i>c</i> on 1, to <i>b</i> on 2, to -<i>c</i> on 3.</p> - -<p>Let us now treat of the states of -consciousness corresponding to these -positions. Each point represents a -composite of posits, and the manifold -of consciousness corresponding -to them is of a certain complexity.</p> - -<p>Suppose now the constituents, the points on the axes, -to interchange arbitrarily, any one to become any other, -and also the axes 1, 2, and 3, to interchange amongst -themselves, any one to become any other, and to be subject -to no system or law, that is to say, that order does -not exist, and that the points which run <i>abc</i> on each axis -may run <i>bac</i>, and so on.</p> - -<p>Then any one of the states of consciousness represented -by the points in the cluster can become any other. We -have a representation of a random consciousness of a -certain degree of complexity.</p> - -<p><span class="pagenum" id="Page_114">[Pg 114]</span></p> - -<p>Now let us examine carefully one particular case of -arbitrary interchange of the points, <i>a</i>, <i>b</i>, <i>c</i>; as one such -case, carefully considered, makes the whole clear.</p> - -<div class="figleft illowp40" id="fig_61" style="max-width: 15.625em;"> - <img src="images/fig_61.png" alt="" /> - <div class="caption">Fig. 61.</div> -</div> - -<p>Consider the points named in the figure 1<i>c</i>, 2<i>a</i>, 3<i>c</i>; -1<i>c</i>, 2<i>c</i>, 3<i>a</i>; 1<i>a</i>, 2<i>c</i>, 3<i>c</i>, and -examine the effect on them -when a change of order takes -place. Let us suppose, for -instance, that <i>a</i> changes into <i>b</i>, -and let us call the two sets of -points we get, the one before -and the one after, their change -conjugates.</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdl">Before the change</td> - -<td class="tdl">1<i>c</i> 2<i>a</i> 3<i>c</i></td> -<td class="tdlp">1<i>c</i> 2<i>c</i> 3<i>a</i></td> -<td class="tdlp">1<i>a</i> 2<i>c</i> 3<i>c</i></td> -<td class="tdl" rowspan="2">} Conjugates.</td> -</tr> -<tr> -<td class="tdl">After the change</td> -<td class="tdl">1<i>c</i> 2<i>b</i> 3<i>c</i></td> -<td class="tdlp">1<i>c</i> 2<i>c</i> 3<i>b</i></td> -<td class="tdlp">1<i>b</i> 2<i>c</i> 3<i>c</i></td> -</tr> -</table> - -<p>The points surrounded by rings represent the conjugate -points.</p> - -<p>It is evident that as consciousness, represented first by -the first set of points and afterwards by the second set of -points, would have nothing in common in its two phases. -It would not be capable of giving an account of itself. -There would be no identity.</p> - -<div class="figleft illowp35" id="fig_62" style="max-width: 18.75em;"> - <img src="images/fig_62.png" alt="" /> - <div class="caption">Fig. 62.</div> -</div> - -<p>If, however, we can find any set of points in the -cubical cluster, which, when any arbitrary change takes -place in the points on the axes, or in the axes themselves, -repeats itself, is reproduced, then a consciousness represented -by those points would have a permanence. It -would have a principle of identity. Despite the no law, -the no order, of the ultimate constituents, it would have -an order, it would form a system, the condition of a -personal identity would be fulfilled.</p> - -<p>The question comes to this, then. Can we find a -system of points which is self-conjugate which is such -that when any posit on the axes becomes any other, or<span class="pagenum" id="Page_115">[Pg 115]</span> -when any axis becomes any other, such a set is transformed -into itself, its identity -is not submerged, but rises -superior to the chaos of its -constituents?</p> - -<p>Such a set can be found. -Consider the set represented -in <a href="#fig_62">fig. 62</a>, and written down in -the first of the two lines—</p> - - -<table class="standard" summary=""> -<tr> -<td class="tdl" rowspan="2">Self-<br />conjugate</td> -<td class="tdl" rowspan="2">{</td> -<td class="tdl">1<i>a</i> 2<i>b</i> 3<i>c</i></td> -<td class="tdlp">1<i>b</i> 2<i>a</i> 3<i>c</i></td> -<td class="tdlp">1<i>c</i> 2<i>a</i> 3<i>b</i></td> -<td class="tdlp">1<i>c</i> 2<i>b</i> 3<i>a</i></td> -<td class="tdlp">1<i>b</i> 2<i>c</i> 3<i>a</i></td> -<td class="tdlp">1<i>a</i> 2<i>c</i> 3<i>b</i></td> -</tr> -<tr> -<td class="tdl">1<i>c</i> 2<i>b</i> 3<i>a</i></td> -<td class="tdlp">1<i>b</i> 2<i>c</i> 3<i>a</i></td> -<td class="tdlp">1<i>a</i> 2<i>c</i> 3<i>b</i></td> -<td class="tdlp">1<i>a</i> 2<i>b</i> 3<i>c</i></td> -<td class="tdlp">1<i>b</i> 2<i>a</i> 3<i>c</i></td> -<td class="tdlp">1<i>c</i> 2<i>a</i> 3<i>b</i></td> -</tr> -</table> - -<p>If now <i>a</i> change into <i>c</i> and <i>c</i> into <i>a</i>, we get the set in -the second line, which has the same members as are in the -upper line. Looking at the diagram we see that it would -correspond simply to the turning of the figures as a -whole.<a id="FNanchor_2" href="#Footnote_2" class="fnanchor">[2]</a> Any arbitrary change of the points on the axes, -or of the axes themselves, reproduces the same set.</p> - -<div class="footnotes"> -<div class="footnote"> - -<p><a id="Footnote_2" href="#FNanchor_2" class="label">[2]</a> These figures are described more fully, and extended, in the next -chapter.</p> - -</div></div> - -<p>Thus, a function, by which a random, an unordered, consciousness -could give an ordered and systematic one, can -be represented. It is noteworthy that it is a system of -selection. If out of all the alternative forms that only is -attended to which is self-conjugate, an ordered consciousness -is formed. A selection gives a feature of permanence.</p> - -<p>Can we say that the permanent consciousness is this -selection?</p> - -<p>An analogy between Kant and Darwin comes into light. -That which is swings clear of the fleeting, in virtue of its -presenting a feature of permanence. There is no need -to suppose any function of “attending to.” A consciousness -capable of giving an account of itself is one -which is characterised by this combination. All combinations -exist—of this kind is the consciousness which -can give an account of itself. And the very duality which<span class="pagenum" id="Page_116">[Pg 116]</span> -we have presupposed may be regarded as originated by -a process of selection.</p> - -<p>Darwin set himself to explain the origin of the fauna -and flora of the world. He denied specific tendencies. -He assumed an indefinite variability—that is, chance—but -a chance confined within narrow limits as regards the -magnitude of any consecutive variations. He showed that -organisms possessing features of permanence, if they -occurred would be preserved. So his account of any -structure or organised being was that it possessed features -of permanence.</p> - -<p>Kant, undertaking not the explanation of any particular -phenomena but of that which we call nature as a whole, -had an origin of species of his own, an account of the -flora and fauna of consciousness. He denied any specific -tendency of the elements of consciousness, but taking our -own consciousness, pointed out that in which it resembled -any consciousness which could survive, which could give -an account of itself.</p> - -<p>He assumes a chance or random world, and as great -and small were not to him any given notions of which he -could make use, he did not limit the chance, the randomness, -in any way. But any consciousness which is permanent -must possess certain features—those attributes -namely which give it permanence. Any consciousness -like our own is simply a consciousness which possesses -those attributes. The main thing is that which he calls -the unity of apperception, which we have seen above is -simply the statement that a particular set of phases of -consciousness on the basis of complete randomness will be -self-conjugate, and so permanent.</p> - -<p>As with Darwin so with Kant, the reason for existence -of any feature comes to this—show that it tends to the -permanence of that which possesses it.</p> - -<p>We can thus regard Kant as the creator of the first of<span class="pagenum" id="Page_117">[Pg 117]</span> -the modern evolution theories. And, as is so often the -case, the first effort was the most stupendous in its scope. -Kant does not investigate the origin of any special part -of the world, such as its organisms, its chemical elements, -its social communities of men. He simply investigates -the origin of the whole—of all that is included in consciousness, -the origin of that “thought thing” whose -progressive realisation is the knowable universe.</p> - -<p>This point of view is very different from the ordinary -one, in which a man is supposed to be placed in a world -like that which he has come to think of it, and then to -learn what he has found out from this model which he -himself has placed on the scene.</p> - -<p>We all know that there are a number of questions in -attempting an answer to which such an assumption is not -allowable.</p> - -<p>Mill, for instance, explains our notion of “law” by an -invariable sequence in nature. But what we call nature -is something given in thought. So he explains a thought -of law and order by a thought of an invariable sequence. -He leaves the problem where he found it.</p> - -<p>Kant’s theory is not unique and alone. It is one of -a number of evolution theories. A notion of its import -and significance can be obtained by a comparison of it -with other theories.</p> - -<p>Thus in Darwin’s theoretical world of natural selection -a certain assumption is made, the assumption of indefinite -variability—slight variability it is true, over any appreciable -lapse of time, but indefinite in the postulated -epochs of transformation—and a whole chain of results -is shown to follow.</p> - -<p>This element of chance variation is not, however, an -ultimate resting place. It is a preliminary stage. This -supposing the all is a preliminary step towards finding -out what is. If every kind of organism can come into<span class="pagenum" id="Page_118">[Pg 118]</span> -being, those that do survive will present such and such -characteristics. This is the necessary beginning for ascertaining -what kinds of organisms do come into existence. -And so Kant’s hypothesis of a random consciousness is -the necessary beginning for the rational investigation -of consciousness as it is. His assumption supplies, as -it were, the space in which we can observe the phenomena. -It gives the general laws constitutive of any -experience. If, on the assumption of absolute randomness -in the constituents, such and such would be -characteristic of the experience, then, whatever the constituents, -these characteristics must be universally valid.</p> - -<p>We will now proceed to examine more carefully the -poiograph, constructed for the purpose of exhibiting an -illustration of Kant’s unity of apperception.</p> - -<p>In order to show the derivation order out of non-order -it has been necessary to assume a principle of duality—we -have had the axes and the posits on the axes—there -are two sets of elements, each non-ordered, and it is in -the reciprocal relation of them that the order, the definite -system, originates.</p> - -<p>Is there anything in our experience of the nature of a -duality?</p> - -<p>There certainly are objects in our experience which -have order and those which are incapable of order. The -two roots of a quadratic equation have no order. No one -can tell which comes first. If a body rises vertically and -then goes at right angles to its former course, no one can -assign any priority to the direction of the north or to the -east. There is no priority in directions of turning. We -associate turnings with no order progressions in a line -with order. But in the axes and points we have assumed -above there is no such distinction. It is the same, whether -we assume an order among the turnings, and no order -among the points on the axes, or, <i>vice versa</i>, an order in<span class="pagenum" id="Page_119">[Pg 119]</span> -the points and no order in the turnings. A being with -an infinite number of axes mutually at right angles, -with a definite sequence between them and no sequence -between the points on the axes, would be in a condition -formally indistinguishable from that of a creature who, -according to an assumption more natural to us, had on -each axis an infinite number of ordered points and no -order of priority amongst the axes. A being in such -a constituted world would not be able to tell which -was turning and which was length along an axis, in -order to distinguish between them. Thus to take a pertinent -illustration, we may be in a world of an infinite -number of dimensions, with three arbitrary points on -each—three points whose order is indifferent, or in a -world of three axes of arbitrary sequence with an infinite -number of ordered points on each. We can’t tell which -is which, to distinguish it from the other.</p> - -<p>Thus it appears the mode of illustration which we -have used is not an artificial one. There really exists -in nature a duality of the kind which is necessary to -explain the origin of order out of no order—the duality, -namely, of dimension and position. Let us use the term -group for that system of points which remains unchanged, -whatever arbitrary change of its constituents takes place. -We notice that a group involves a duality, is inconceivable -without a duality.</p> - -<p>Thus, according to Kant, the primary element of experience -is the group, and the theory of groups would be -the most fundamental branch of science. Owing to an -expression in the critique the authority of Kant is sometimes -adduced against the assumption of more than three -dimensions to space. It seems to me, however, that the -whole tendency of his theory lies in the opposite direction, -and points to a perfect duality between dimension and -position in a dimension.</p> - -<p><span class="pagenum" id="Page_120">[Pg 120]</span></p> - -<p>If the order and the law we see is due to the conditions -of conscious experience, we must conceive nature as -spontaneous, free, subject to no predication that we can -devise, but, however apprehended, subject to our logic.</p> - -<p>And our logic is simply spatiality in the general sense—that -resultant of a selection of the permanent from the -unpermanent, the ordered from the unordered, by the -means of the group and its underlying duality.</p> - -<p>We can predicate nothing about nature, only about the -way in which we can apprehend nature. All that we can -say is that all that which experience gives us will be conditioned -as spatial, subject to our logic. Thus, in exploring -the facts of geometry from the simplest logical relations -to the properties of space of any number of dimensions, -we are merely observing ourselves, becoming aware of -the conditions under which we must perceive. Do any -phenomena present themselves incapable of explanation -under the assumption of the space we are dealing with, -then we must habituate ourselves to the conception of a -higher space, in order that our logic may be equal to the -task before us.</p> - -<p>We gain a repetition of the thought that came before, -experimentally suggested. If the laws of the intellectual -comprehension of nature are those derived from considering -her as absolute chance, subject to no law save -that derived from a process of selection, then, perhaps, the -order of nature requires different faculties from the intellectual -to apprehend it. The source and origin of -ideas may have to be sought elsewhere than in reasoning.</p> - -<p>The total outcome of the critique is to leave the -ordinary man just where he is, justified in his practical -attitude towards nature, liberated from the fetters of his -own mental representations.</p> - -<p>The truth of a picture lies in its total effect. It is vain -to seek information about the landscape from an examina<span class="pagenum" id="Page_121">[Pg 121]</span>tion -of the pigments. And in any method of thought it -is the complexity of the whole that brings us to a knowledge -of nature. Dimensions are artificial enough, but in -the multiplicity of them we catch some breath of nature.</p> - -<p>We must therefore, and this seems to me the practical -conclusion of the whole matter, proceed to form means of -intellectual apprehension of a greater and greater degree -of complexity, both dimensionally and in extent in any -dimension. Such means of representation must always -be artificial, but in the multiplicity of the elements with -which we deal, however incipiently arbitrary, lies our -chance of apprehending nature.</p> - -<p>And as a concluding chapter to this part of the book, -I will extend the figures, which have been used to represent -Kant’s theory, two steps, so that the reader may -have the opportunity of looking at a four-dimensional -figure which can be delineated without any of the special -apparatus, to the consideration of which I shall subsequently -pass on.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_122">[Pg 122]</span></p> - -<h2 class="nobreak" id="CHAPTER_X">CHAPTER X<br /> - -<small><i>A FOUR-DIMENSIONAL FIGURE</i></small></h2></div> - - -<p>The method used in the preceding chapter to illustrate -the problem of Kant’s critique, gives a singularly easy -and direct mode of constructing a series of important -figures in any number of dimensions.</p> - -<p>We have seen that to represent our space a plane being -must give up one of his axes, and similarly to represent -the higher shapes we must give up one amongst our -three axes.</p> - -<p>But there is another kind of giving up which reduces -the construction of higher shapes to a matter of the -utmost simplicity.</p> - -<p>Ordinarily we have on a straight line any number of -positions. The wealth of space in position is illimitable, -while there are only three dimensions.</p> - -<p>I propose to give up this wealth of positions, and to -consider the figures obtained by taking just as many -positions as dimensions.</p> - -<p>In this way I consider dimensions and positions as two -“kinds,” and applying the simple rule of selecting every -one of one kind with every other of every other kind, -get a series of figures which are noteworthy because -they exactly fill space of any number of dimensions -(as the hexagon fills a plane) by equal repetitions of -themselves.</p> - -<p><span class="pagenum" id="Page_123">[Pg 123]</span></p> - -<p>The rule will be made more evident by a simple -application.</p> - -<p>Let us consider one dimension and one position. I will -call the axis <i>i</i>, and the position <i>o</i>.</p> - -<p class="center"> -———————————————-<i>i</i><br /> -<span style="margin-left: 3.5em;"><i>o</i></span> -</p> - -<p>Here the figure is the position <i>o</i> on the line <i>i</i>. Take -now two dimensions and two positions on each.</p> - -<div class="figleft illowp30" id="fig_63" style="max-width: 12.125em;"> - <img src="images/fig_63.png" alt="" /> - <div class="caption">Fig. 63.</div> -</div> - -<p>We have the two positions <i>o</i>; 1 on <i>i</i>, and the two -positions <i>o</i>, 1 on <i>j</i>, <a href="#fig_63">fig. 63</a>. These give -rise to a certain complexity. I will -let the two lines <i>i</i> and <i>j</i> meet in the -position I call <i>o</i> on each, and I will -consider <i>i</i> as a direction starting equally -from every position on <i>j</i>, and <i>j</i> as -starting equally from every position on <i>i</i>. We thus -obtain the following figure:—<span class="allsmcap">A</span> is both <i>oi</i> and <i>oj</i>, <span class="allsmcap">B</span> is 1<i>i</i> -and <i>oj</i>, and so on as shown in <a href="#fig_63">fig. 63</a><i>b</i>. -The positions on <span class="allsmcap">AC</span> are all <i>oi</i> positions. -They are, if we like to consider it in -that way, points at no distance in the <i>i</i> -direction from the line <span class="allsmcap">AC</span>. We can -call the line <span class="allsmcap">AC</span> the <i>oi</i> line. Similarly -the points on <span class="allsmcap">AB</span> are those no distance -from <span class="allsmcap">AB</span> in the <i>j</i> direction, and we can -call them <i>oj</i> points and the line <span class="allsmcap">AB</span> the <i>oj</i> line. Again, -the line <span class="allsmcap">CD</span> can be called the 1<i>j</i> line because the points -on it are at a distance, 1 in the <i>j</i> direction.</p> - -<div class="figleft illowp30" id="fig_63b" style="max-width: 12.5em;"> - <img src="images/fig_63b.png" alt="" /> - <div class="caption">Fig. 63<i>b</i>.</div> -</div> - -<p>We have then four positions or points named as shown, -and, considering directions and positions as “kinds,” we -have the combination of two kinds with two kinds. Now, -selecting every one of one kind with every other of every -other kind will mean that we take 1 of the kind <i>i</i> and<span class="pagenum" id="Page_124">[Pg 124]</span> -with it <i>o</i> of the kind <i>j</i>; and then, that we take <i>o</i> of the -kind <i>i</i> and with it 1 of the kind <i>j</i>.</p> - -<div class="figleft illowp25" id="fig_64" style="max-width: 12.5em;"> - <img src="images/fig_64.png" alt="" /> - <div class="caption">Fig. 64.</div> -</div> - -<p>Thus we get a pair of positions lying in the straight -line <span class="allsmcap">BC</span>, <a href="#fig_64">fig. 64</a>. We can call this pair 10 -and 01 if we adopt the plan of mentally, -adding an <i>i</i> to the first and a <i>j</i> to the -second of the symbols written thus—01 -is a short expression for O<i>i</i>, 1<i>j</i>.</p> - -<div class="figcenter illowp80" id="fig_65" style="max-width: 62.5em;"> - <img src="images/fig_65.png" alt="" /> - <div class="caption">Fig. 65.</div> -</div> - -<p>Coming now to our space, we have three -dimensions, so we take three positions on each. These -positions I will suppose to be at equal distances along each -axis. The three axes and the three positions on each are -shown in the accompanying diagrams, <a href="#fig_65">fig. 65</a>, of which -the first represents a cube with the front faces visible, the -second the rear faces of the same cube; the positions I -will call 0, 1, 2; the axes, <i>i</i>, <i>j</i>, <i>k</i>. I take the base <span class="allsmcap">ABC</span> as -the starting place, from which to determine distances in -the <i>k</i> direction, and hence every point in the base <span class="allsmcap">ABC</span> -will be an <i>ok</i> position, and the base <span class="allsmcap">ABC</span> can be called an -<i>ok</i> plane.</p> - -<p>In the same way, measuring the distances from the face -<span class="allsmcap">ADC</span>, we see that every position in the face <span class="allsmcap">ADC</span> is an <i>oi</i> -position, and the whole plane of the face may be called an -<i>oi</i> plane. Thus we see that with the introduction of a<span class="pagenum" id="Page_125">[Pg 125]</span> -new dimension the signification of a compound symbol, -such as “<i>oi</i>,” alters. In the plane it meant the line <span class="allsmcap">AC</span>. -In space it means the whole plane <span class="allsmcap">ACD</span>.</p> - -<p>Now, it is evident that we have twenty-seven positions, -each of them named. If the reader will follow this -nomenclature in respect of the positions marked in the -figures he will have no difficulty in assigning names to -each one of the twenty-seven positions. <span class="allsmcap">A</span> is <i>oi</i>, <i>oj</i>, <i>ok</i>. -It is at the distance 0 along <i>i</i>, 0 along <i>j</i>, 0 along <i>k</i>, and -<i>io</i> can be written in short 000, where the <i>ijk</i> symbols -are omitted.</p> - -<p>The point immediately above is 001, for it is no distance -in the <i>i</i> direction, and a distance of 1 in the <i>k</i> -direction. Again, looking at <span class="allsmcap">B</span>, it is at a distance of 2 -from <span class="allsmcap">A</span>, or from the plane <span class="allsmcap">ADC</span>, in the <i>i</i> direction, 0 in the -<i>j</i> direction from the plane <span class="allsmcap">ABD</span>, and 0 in the <i>k</i> direction, -measured from the plane <span class="allsmcap">ABC</span>. Hence it is 200 written -for 2<i>i</i>, 0<i>j</i>, 0<i>k</i>.</p> - -<p>Now, out of these twenty-seven “things” or compounds -of position and dimension, select those which are given by -the rule, every one of one kind with every other of every -other kind.</p> - -<div class="figleft illowp30" id="fig_66" style="max-width: 15.625em;"> - <img src="images/fig_66.png" alt="" /> - <div class="caption">Fig. 66.</div> -</div> - -<p>Take 2 of the <i>i</i> kind. With this -we must have a 1 of the <i>j</i> kind, -and then by the rule we can only -have a 0 of the <i>k</i> kind, for if we -had any other of the <i>k</i> kind we -should repeat one of the kinds we -already had. In 2<i>i</i>, 1<i>j</i>, 1<i>k</i>, for -instance, 1 is repeated. The point -we obtain is that marked 210, <a href="#fig_66">fig. 66</a>.</p> - -<div class="figleft illowp30" id="fig_67" style="max-width: 15.625em;"> - <img src="images/fig_67.png" alt="" /> - <div class="caption">Fig. 67.</div> -</div> - -<p>Proceeding in this way, we pick out the following -cluster of points, <a href="#fig_67">fig. 67</a>. They are joined by lines, -dotted where they are hidden by the body of the cube, -and we see that they form a figure—a hexagon which<span class="pagenum" id="Page_126">[Pg 126]</span> -could be taken out of the cube and placed on a plane. -It is a figure which will fill a -plane by equal repetitions of itself. -The plane being representing this -construction in his plane would -take three squares to represent the -cube. Let us suppose that he -takes the <i>ij</i> axes in his space and -<i>k</i> represents the axis running out -of his space, <a href="#fig_68">fig. 68</a>. In each of -the three squares shown here as drawn separately he -could select the points given by the rule, and he would -then have to try to discover the figure determined by -the three lines drawn. The line from 210 to 120 is -given in the figure, but the line from 201 to 102 or <span class="allsmcap">GK</span> -is not given. He can determine <span class="allsmcap">GK</span> by making another -set of drawings and discovering in them what the relation -between these two extremities is.</p> - -<div class="figcenter illowp100" id="fig_68" style="max-width: 62.5em;"> - <img src="images/fig_68.png" alt="" /> - <div class="caption">Fig. 68.</div> -</div> - -<div class="figcenter illowp80" id="fig_69" style="max-width: 50em;"> - <img src="images/fig_69.png" alt="" /> - <div class="caption">Fig. 69.</div> -</div> - -<p>Let him draw the <i>i</i> and <i>k</i> axes in his plane, <a href="#fig_69">fig. 69</a>. -The <i>j</i> axis then runs out and he has the accompanying -figure. In the first of these three squares, <a href="#fig_69">fig. 69</a>, he can<span class="pagenum" id="Page_127">[Pg 127]</span> -pick out by the rule the two points 201, 102—<span class="allsmcap">G</span>, and <span class="allsmcap">K</span>. -Here they occur in one plane and he can measure the -distance between them. In his first representation they -occur at <span class="allsmcap">G</span> and <span class="allsmcap">K</span> in separate figures.</p> - -<p>Thus the plane being would find that the ends of each -of the lines was distant by the diagonal of a unit square -from the corresponding end of the last and he could then -place the three lines in their right relative position. -Joining them he would have the figure of a hexagon.</p> - -<div class="figleft illowp30" id="fig_70" style="max-width: 15.625em;"> - <img src="images/fig_70.png" alt="" /> - <div class="caption">Fig. 70.</div> -</div> - -<p>We may also notice that the plane being could make -a representation of the whole cube -simultaneously. The three squares, -shown in perspective in <a href="#fig_70">fig. 70</a>, all -lie in one plane, and on these the -plane being could pick out any -selection of points just as well as on -three separate squares. He would -obtain a hexagon by joining the -points marked. This hexagon, as -drawn, is of the right shape, but it would not be so if -actual squares were used instead of perspective, because -the relation between the separate squares as they lie in -the plane figure is not their real relation. The figure, -however, as thus constructed, would give him an idea of -the correct figure, and he could determine it accurately -by remembering that distances in each square were -correct, but in passing from one square to another their -distance in the third dimension had to be taken into -account.</p> - -<p>Coming now to the figure made by selecting according -to our rule from the whole mass of points given by four -axes and four positions in each, we must first draw a -catalogue figure in which the whole assemblage is shown.</p> - -<p>We can represent this assemblage of points by four -solid figures. The first giving all those positions which<span class="pagenum" id="Page_128">[Pg 128]</span> -are at a distance <span class="allsmcap">O</span> from our space in the fourth dimension, -the second showing all those that are at a distance 1, -and so on.</p> - -<p>These figures will each be cubes. The first two are -drawn showing the front faces, the second two the rear -faces. We will mark the points 0, 1, 2, 3, putting points -at those distances along each of these axes, and suppose -all the points thus determined to be contained in solid -models of which our drawings in <a href="#fig_71">fig. 71</a> are representatives. -Here we notice that as on the plane 0<i>i</i> meant -the whole line from which the distances in the <i>i</i> direction -was measured, and as in space 0<i>i</i> means the whole plane -from which distances in the <i>i</i> direction are measured, so -now 0<i>h</i> means the whole space in which the first cube -stands—measuring away from that space by a distance -of one we come to the second cube represented.</p> - -<div class="figcenter illowp80" id="fig_71" style="max-width: 62.5em;"> - <img src="images/fig_71.png" alt="" /> - <div class="caption">Fig. 71.</div> -</div> - -<p><span class="pagenum" id="Page_129">[Pg 129]</span></p> - -<p>Now selecting according to the rule every one of one -kind with every other of every other kind, we must take, -for instance, 3<i>i</i>, 2<i>j</i>, 1<i>k</i>, 0<i>h</i>. This point is marked -3210 at the lower star in the figure. It is 3 in the -<i>i</i> direction, 2 in the <i>j</i> direction, 1 in the <i>k</i> direction, -0 in the <i>h</i> direction.</p> - -<p>With 3<i>i</i> we must also take 1<i>j</i>, 2<i>k</i>, 0<i>h</i>. This point -is shown by the second star in the cube 0<i>h</i>.</p> - -<div class="figcenter illowp80" id="fig_72" style="max-width: 62.5em;"> - <img src="images/fig_72.png" alt="" /> - <div class="caption">Fig. 72.</div> -</div> - -<p>In the first cube, since all the points are 0<i>h</i> points, -we can only have varieties in which <i>i</i>, <i>j</i>, <i>k</i>, are accompanied -by 3, 2, 1.</p> - -<p>The points determined are marked off in the diagram -fig. 72, and lines are drawn joining the adjacent pairs -in each figure, the lines being dotted when they pass -within the substance of the cube in the first two diagrams.</p> - -<p>Opposite each point, on one side or the other of each<span class="pagenum" id="Page_130">[Pg 130]</span> -cube, is written its name. It will be noticed that the -figures are symmetrical right and left; and right and -left the first two numbers are simply interchanged.</p> - -<p>Now this being our selection of points, what figure do -they make when all are put together in their proper -relative positions?</p> - -<p>To determine this we must find the distance between -corresponding corners of the separate hexagons.</p> -<div class="figcenter illowp80" id="fig_73" style="max-width: 62.5em;"> - <img src="images/fig_73.png" alt="" /> - <div class="caption">Fig. 73.</div> -</div> - - -<p>To do this let us keep the axes <i>i</i>, <i>j</i>, in our space, and -draw <i>h</i> instead of <i>k</i>, letting <i>k</i> run out in the fourth -dimension, <a href="#fig_73">fig. 73</a>.</p> - -<div class="figright illowp50" id="fig_74" style="max-width: 37.5em;"> - <img src="images/fig_74.png" alt="" /> - <div class="caption">Fig. 74.</div> -</div> - -<p>Here we have four cubes again, in the first of which all -the points are 0<i>k</i> points; that is, points at a distance zero -in the <i>k</i> direction from the space of the three dimensions -<i>ijh</i>. We have all the points selected before, and some -of the distances, which in the last diagram led from figure -to figure are shown here in the same figure, and so capable<span class="pagenum" id="Page_131">[Pg 131]</span> -of measurement. Take for instance the points 3120 to -3021, which in the first diagram (<a href="#fig_72">fig. 72</a>) lie in the first -and second figures. Their actual relation is shown in -fig. 73 in the cube marked 2<span class="allsmcap">K</span>, where the points in question -are marked with a *. We see that the -distance in question is the diagonal of a unit square. In -like manner we find that the distance between corresponding -points of any two hexagonal figures is the -diagonal of a unit square. The total figure is now easily -constructed. An idea -of it may be gained by -drawing all the four -cubes in the catalogue -figure in one (fig. 74). -These cubes are exact -repetitions of one -another, so one drawing -will serve as a -representation of the -whole series, if we -take care to remember -where we are, whether -in a 0<i>h</i>, a 1<i>h</i>, a 2<i>h</i>, -or a 3<i>h</i> figure, when -we pick out the points required. Fig. 74 is a representation -of all the catalogue cubes put in one. For the -sake of clearness the front faces and the back faces of -this cube are represented separately.</p> - -<p>The figure determined by the selected points is shown -below.</p> - -<p>In putting the sections together some of the outlines -in them disappear. The line <span class="allsmcap">TW</span> for instance is not -wanted.</p> - -<p>We notice that <span class="allsmcap">PQTW</span> and <span class="allsmcap">TWRS</span> are each the half -of a hexagon. Now <span class="allsmcap">QV</span> and <span class="allsmcap">VR</span> lie in one straight line.<span class="pagenum" id="Page_132">[Pg 132]</span> -Hence these two hexagons fit together, forming one -hexagon, and the line <span class="allsmcap">TW</span> is only wanted when we consider -a section of the whole figure, we thus obtain the -solid represented in the lower part of <a href="#fig_74">fig. 74</a>. Equal -repetitions of this figure, called a tetrakaidecagon, will -fill up three-dimensional space.</p> - -<p>To make the corresponding four-dimensional figure we -have to take five axes mutually at right angles with five -points on each. A catalogue of the positions determined -in five-dimensional space can be found thus.</p> -<div class="figleft illowp60" id="fig_75" style="max-width: 37.5em;"> - <img src="images/fig_75.png" alt="" /> - <div class="caption">Fig. 75.</div> -</div> - -<p>Take a cube with five points on each of its axes, the -fifth point is at a distance of four units of length from the -first on any one of the axes. And since the fourth dimension -also stretches to a distance of four we shall need to -represent the successive -sets of points at -distances 0, 1, 2, 3, 4, -in the fourth dimensions, -five cubes. Now -all of these extend to -no distance at all in -the fifth dimension. -To represent what -lies in the fifth dimension -we shall have to -draw, starting from -each of our cubes, five -similar cubes to represent -the four steps -on in the fifth dimension. By this assemblage we get a -catalogue of all the points shown in <a href="#fig_75">fig. 75</a>, in which -<i>L</i> represents the fifth dimension.</p> - -<p>Now, as we saw before, there is nothing to prevent us -from putting all the cubes representing the different -stages in the fourth dimension in one figure, if we take<span class="pagenum" id="Page_133">[Pg 133]</span> -note when we look at it, whether we consider it as a 0<i>h</i>, a -1<i>h</i>, a 2<i>h</i>, etc., cube. Putting then the 0<i>h</i>, 1<i>h</i>, 2<i>h</i>, 3<i>h</i>, 4<i>h</i> -cubes of each row in one, we have five cubes with the sides -of each containing five positions, the first of these five -cubes represents the 0<i>l</i> points, and has in it the <i>i</i> points -from 0 to 4, the <i>j</i> points from 0 to 4, the <i>k</i> points from -0 to 4, while we have to specify with regard to any -selection we make from it, whether we regard it as a 0<i>h</i>, -a 1<i>h</i>, a 2<i>h</i>, a 3<i>h</i>, or a 4<i>h</i> figure. In <a href="#fig_76">fig. 76</a> each cube -is represented by two drawings, one of the front part, the -other of the rear part.</p> - -<p>Let then our five cubes be arranged before us and our -selection be made according to the rule. Take the first -figure in which all points are 0<i>l</i> points. We cannot -have 0 with any other letter. Then, keeping in the first -figure, which is that of the 0<i>l</i> positions, take first of all -that selection which always contains 1<i>h</i>. We suppose, -therefore, that the cube is a 1<i>h</i> cube, and in it we take -<i>i</i>, <i>j</i>, <i>k</i> in combination with 4, 3, 2 according to the rule.</p> - -<p>The figure we obtain is a hexagon, as shown, the one -in front. The points on the right hand have the same -figures as those on the left, with the first two numerals -interchanged. Next keeping still to the 0<i>l</i> figure let -us suppose that the cube before us represents a section -at a distance of 2 in the <i>h</i> direction. Let all the points -in it be considered as 2<i>h</i> points. We then have a 0<i>l</i>, 2<i>h</i> -region, and have the sets <i>ijk</i> and 431 left over. We -must then pick out in accordance with our rule all such -points as 4<i>i</i>, 3<i>j</i>, 1<i>k</i>.</p> - -<p>These are shown in the figure and we find that we can -draw them without confusion, forming the second hexagon -from the front. Going on in this way it will be seen -that in each of the five figures a set of hexagons is picked -out, which put together form a three-space figure something -like the tetrakaidecagon.</p> - -<p><span class="pagenum" id="Page_134">[Pg 134]</span></p> - -<div class="figcenter illowp100" id="fig_76" style="max-width: 93.75em;"> - <img src="images/fig_76.png" alt="" /> - <div class="caption">Fig. 76.</div> -</div> - -<p><span class="pagenum" id="Page_135">[Pg 135]</span></p> - -<p>These separate figures are the successive stages in -which the whole four-dimensional figure in which they -cohere can be apprehended.</p> - -<p>The first figure and the last are tetrakaidecagons. -These are two of the solid boundaries of the figure. The -other solid boundaries can be traced easily. Some of -them are complete from one face in the figure to the -corresponding face in the next, as for instance the solid -which extends from the hexagonal base of the first figure -to the equal hexagonal base of the second figure. This -kind of boundary is a hexagonal prism. The hexagonal -prism also occurs in another sectional series, as for -instance, in the square at the bottom of the first figure, -the oblong at the base of the second and the square at -the base of the third figure.</p> - -<p>Other solid boundaries can be traced through four of -the five sectional figures. Thus taking the hexagon at -the top of the first figure we find in the next a hexagon -also, of which some alternate sides are elongated. The -top of the third figure is also a hexagon with the other -set of alternate rules elongated, and finally we come in -the fourth figure to a regular hexagon.</p> - -<p>These four sections are the sections of a tetrakaidecagon -as can be recognised from the sections of this figure -which we have had previously. Hence the boundaries -are of two kinds, hexagonal prisms and tetrakaidecagons.</p> - -<p>These four-dimensional figures exactly fill four-dimensional -space by equal repetitions of themselves.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_136">[Pg 136]</span></p> - -<h2 class="nobreak" id="CHAPTER_XI">CHAPTER XI<br /> - -<small><i>NOMENCLATURE AND ANALOGIES PRELIMINARY -TO THE STUDY OF FOUR-DIMENSIONAL -FIGURES</i></small></h2></div> - - -<p>In the following pages a method of designating different -regions of space by a systematic colour scheme has been -adopted. The explanations have been given in such a -manner as to involve no reference to models, the diagrams -will be found sufficient. But to facilitate the study a -description of a set of models is given in an appendix -which the reader can either make for himself or obtain. -If models are used the diagrams in Chapters XI. and XII. -will form a guide sufficient to indicate their use. Cubes -of the colours designated by the diagrams should be picked -out and used to reinforce the diagrams. The reader, -in the following description, should -suppose that a board or wall -stretches away from him, against -which the figures are placed.</p> - -<div class="figleft illowp30" id="fig_77" style="max-width: 15.625em;"> - <img src="images/fig_77.png" alt="" /> - <div class="caption">Fig. 77.</div> -</div> - -<p>Take a square, one of those -shown in Fig. 77 and give it a -neutral colour, let this colour be -called “null,” and be such that it -makes no appreciable difference<span class="pagenum" id="Page_137">[Pg 137]</span> -to any colour with which it mixed. If there is no -such real colour let us imagine such a colour, and -assign to it the properties of the number zero, which -makes no difference in any number to which it is -added.</p> - -<p>Above this square place a red square. Thus we symbolise -the going up by adding red to null.</p> - -<p>Away from this null square place a yellow square, and -represent going away by adding yellow to null.</p> - -<div class="figleft illowp40" id="fig_78" style="max-width: 15.625em;"> - <img src="images/fig_78.png" alt="" /> - <div class="caption">Fig. 78.</div> -</div> - -<p>To complete the figure we need a fourth square. -Colour this orange, which is a mixture of red and -yellow, and so appropriately represents a going in a -direction compounded of up and away. We have thus -a colour scheme which will serve to name the set of -squares drawn. We have two axes of colours—red and -yellow—and they may occupy -as in the figure the -direction up and away, or -they may be turned about; -in any case they enable us -to name the four squares -drawn in their relation to -one another.</p> - -<p>Now take, in Fig. 78, -nine squares, and suppose -that at the end of the -going in any direction the -colour started with repeats itself.</p> - -<p>We obtain a square named as shown.</p> - -<p>Let us now, in <a href="#fig_79">fig. 79</a>, suppose the number of squares to -be increased, keeping still to the principle of colouring -already used.</p> - -<p>Here the nulls remain four in number. There -are three reds between the first null and the null -above it, three yellows between the first null and the<span class="pagenum" id="Page_138">[Pg 138]</span> -null beyond it, while the oranges increase in a double -way.</p> - -<div class="figcenter illowp80" id="fig_79" style="max-width: 62.5em;"> - <img src="images/fig_79.png" alt="" /> - <div class="caption">Fig. 79.</div> -</div> - -<p>Suppose this process of enlarging the number of the -squares to be indefinitely pursued and -the total figure obtained to be reduced -in size, we should obtain a square of -which the interior was all orange, -while the lines round it were red and -yellow, and merely the points null -colour, as in <a href="#fig_80">fig. 80</a>. Thus all the points, lines, and the -area would have a colour.</p> - -<div class="figleft illowp25" id="fig_80" style="max-width: 15.625em;"> - <img src="images/fig_80.png" alt="" /> - <div class="caption">Fig. 80.</div> -</div> - - -<p>We can consider this scheme to originate thus:—Let -a null point move in a yellow direction and trace out a -yellow line and end in a null point. Then let the whole -line thus traced move in a red direction. The null points -at the ends of the line will produce red lines, and end in<span class="pagenum" id="Page_139">[Pg 139]</span> -null points. The yellow line will trace out a yellow and -red, or orange square.</p> - -<p>Now, turning back to <a href="#fig_78">fig. 78</a>, we see that these two -ways of naming, the one we started with and the one we -arrived at, can be combined.</p> - -<p>By its position in the group of four squares, in <a href="#fig_77">fig. 77</a>, -the null square has a relation to the yellow and to the red -directions. We can speak therefore of the red line of the -null square without confusion, meaning thereby the line -<span class="allsmcap">AB</span>, <a href="#fig_81">fig. 81</a>, which runs up from the -initial null point <span class="allsmcap">A</span> in the figure as -drawn. The yellow line of the null -square is its lower horizontal line <span class="allsmcap">AC</span> -as it is situated in the figure.</p> - -<div class="figleft illowp30" id="fig_81" style="max-width: 15.625em;"> - <img src="images/fig_81.png" alt="" /> - <div class="caption">Fig. 81.</div> -</div> - -<p>If we wish to denote the upper -yellow line <span class="allsmcap">BD</span>, <a href="#fig_81">fig. 81</a>, we can speak -of it as the yellow γ line, meaning -the yellow line which is separated -from the primary yellow line by the red movement.</p> - -<p>In a similar way each of the other squares has null -points, red and yellow lines. Although the yellow square -is all yellow, its line <span class="allsmcap">CD</span>, for instance, can be referred to as -its red line.</p> - -<p>This nomenclature can be extended.</p> - -<p>If the eight cubes drawn, in <a href="#fig_82">fig. 82</a>, are put close -together, as on the right hand of the diagram, they form -a cube, and in them, as thus arranged, a going up is -represented by adding red to the zero, or null colour, a -going away by adding yellow, a going to the right by -adding white. White is used as a colour, as a pigment, -which produces a colour change in the pigments with which -it is mixed. From whatever cube of the lower set we -start, a motion up brings us to a cube showing a change -to red, thus light yellow becomes light yellow red, or -light orange, which is called ochre. And going to the<span class="pagenum" id="Page_140">[Pg 140]</span> -right from the null on the left we have a change involving -the introduction of white, while the yellow change runs -from front to back. There are three colour axes—the red, -the white, the yellow—and these run in the position the -cubes occupy in the drawing—up, to the right, away—but -they could be turned about to occupy any positions in space.</p> - -<div class="figcenter illowp100" id="fig_82" style="max-width: 62.5em;"> - <img src="images/fig_82.png" alt="" /> - <div class="caption">Fig. 82.</div> -</div> - - -<div class="figcenter illowp100" id="fig_83" style="max-width: 62.5em;"> - <img src="images/fig_83.png" alt="" /> - <div class="caption">Fig. 83.</div> -</div> - -<p>We can conveniently represent a block of cubes by -three sets of squares, representing each the base of a cube.</p> - -<p>Thus the block, <a href="#fig_83">fig. 83</a>, can be represented by the<span class="pagenum" id="Page_141">[Pg 141]</span> -layers on the right. Here, as in the case of the plane, -the initial colours repeat themselves at the end of the -series.</p> - -<div class="figleft illowp50" id="fig_84" style="max-width: 31.25em;"> - <img src="images/fig_84.png" alt="" /> - <div class="caption">Fig. 84.</div> -</div> - -<p>Proceeding now to increase the number of the cubes -we obtain <a href="#fig_84">fig. 84</a>, -in which the initial -letters of the colours -are given instead of -their full names.</p> - -<p>Here we see that -there are four null -cubes as before, but -the series which spring -from the initial corner -will tend to become -lines of cubes, as also -the sets of cubes -parallel to them, starting -from other corners. -Thus, from the initial -null springs a line of -red cubes, a line of -white cubes, and a line -of yellow cubes.</p> - -<p>If the number of the -cubes is largely increased, -and the size -of the whole cube is -diminished, we get -a cube with null -points, and the edges -coloured with these three colours.</p> - -<p>The light yellow cubes increase in two ways, forming -ultimately a sheet of cubes, and the same is true of -the orange and pink sets. Hence, ultimately the cube<span class="pagenum" id="Page_142">[Pg 142]</span> -thus formed would have red, white, and yellow lines -surrounding pink, orange, and light yellow faces. The -ochre cubes increase in three ways, and hence ultimately -the whole interior of the cube would be coloured -ochre.</p> - -<p>We have thus a nomenclature for the points, lines, -faces, and solid content of a cube, and it can be named -as exhibited in <a href="#fig_85">fig. 85</a>.</p> - -<div class="figleft illowp30" id="fig_85" style="max-width: 15.625em;"> - <img src="images/fig_85.png" alt="" /> - <div class="caption">Fig. 85.</div> -</div> - -<p>We can consider the cube to be produced in the -following way. A null point -moves in a direction to which -we attach the colour indication -yellow; it generates a yellow line -and ends in a null point. The -yellow line thus generated moves -in a direction to which we give -the colour indication red. This -lies up in the figure. The yellow -line traces out a yellow, red, or -orange square, and each of its null points trace out a -red line, and ends in a null point.</p> - -<p>This orange square moves in a direction to which -we attribute the colour indication white, in this case -the direction is the right. The square traces out a -cube coloured orange, red, or ochre, the red lines trace -out red to white or pink squares, and the yellow -lines trace out light yellow squares, each line ending -in a line of its own colour. While the points each -trace out a null + white, or white line to end in a null -point.</p> - -<p>Now returning to the first block of eight cubes we can -name each point, line, and square in them by reference to -the colour scheme, which they determine by their relation -to each other.</p> - -<p>Thus, in <a href="#fig_86">fig. 86</a>, the null cube touches the red cube by<span class="pagenum" id="Page_143">[Pg 143]</span> -a light yellow square; it touches the yellow cube by a -pink square, and touches -the white cube by an -orange square.</p> - -<div class="figleft illowp50" id="fig_86" style="max-width: 25em;"> - <img src="images/fig_86.png" alt="" /> - <div class="caption">Fig. 86.</div> -</div> - -<p>There are three axes -to which the colours red, -yellow, and white are -assigned, the faces of -each cube are designated -by taking these colours in pairs. Taking all the colours -together we get a colour name for the solidity of a cube.</p> - - -<p>Let us now ask ourselves how the cube could be presented -to the plane being. Without going into the question -of how he could have a real experience of it, let us see -how, if we could turn it about and show it to him, he, -under his limitations, could get information about it. -If the cube were placed with its red and yellow axes -against a plane, that is resting against it by its orange -face, the plane being would observe a square surrounded -by red and yellow lines, and having null points. See the -dotted square, <a href="#fig_87">fig. 87</a>.</p> - -<div class="figcenter illowp80" id="fig_87" style="max-width: 37.5em;"> - <img src="images/fig_87.png" alt="" /> - <div class="caption">Fig. 87.</div> -</div> - -<p>We could turn the cube about the red line so that -a different face comes into juxtaposition with the plane.</p> - -<p>Suppose the cube turned about the red line. As it<span class="pagenum" id="Page_144">[Pg 144]</span> -is turning from its first position all of it except the red -line leaves the plane—goes absolutely out of the range -of the plane being’s apprehension. But when the yellow -line points straight out from the plane then the pink -face comes into contact with it. Thus the same red line -remaining as he saw it at first, now towards him comes -a face surrounded by white and red lines.</p> - -<div class="figleft illowp35" id="fig_88" style="max-width: 18.75em;"> - <img src="images/fig_88.png" alt="" /> - <div class="caption">Fig. 88.</div> -</div> - -<p>If we call the direction to the right the unknown -direction, then the line he saw before, the yellow line, -goes out into this unknown direction, and the line which -before went into the unknown direction, comes in. It -comes in in the opposite direction to that in which the -yellow line ran before; the interior of the face now -against the plane is pink. It is -a property of two lines at right -angles that, if one turns out of -a given direction and stands at -right angles to it, then the other -of the two lines comes in, but -runs the opposite way in that -given direction, as in <a href="#fig_88">fig. 88</a>.</p> - -<p>Now these two presentations of the cube would seem, -to the plane creature like perfectly different material -bodies, with only that line in common in which they -both meet.</p> - -<p>Again our cube can be turned about the yellow line. -In this case the yellow square would disappear as before, -but a new square would come into the plane after the -cube had rotated by an angle of 90° about this line. -The bottom square of the cube would come in thus -in figure 89. The cube supposed in contact with the -plane is rotated about the lower yellow line and then -the bottom face is in contact with the plane.</p> - -<p>Here, as before, the red line going out into the unknown -dimension, the white line which before ran in the<span class="pagenum" id="Page_145">[Pg 145]</span> -unknown dimension would come in downwards in the -opposite sense to that in which the red line ran before.</p> - -<div class="figcenter illowp80" id="fig_89" style="max-width: 62.5em;"> - <img src="images/fig_89.png" alt="" /> - <div class="caption">Fig. 89.</div> -</div> - -<p>Now if we use <i>i</i>, <i>j</i>, <i>k</i>, for the three space directions, -<i>i</i> left to right, <i>j</i> from near away, <i>k</i> from below up; then, -using the colour names for the axes, we have that first -of all white runs <i>i</i>, yellow runs <i>j</i>, red runs <i>k</i>; then after -the first turning round the <i>k</i> axis, white runs negative <i>j</i>, -yellow runs <i>i</i>, red runs <i>k</i>; thus we have the table:—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdc"></td> -<td class="tdc"><i>i</i></td> -<td class="tdc"><i>j</i></td> -<td class="tdc"><i>k</i></td> -</tr> -<tr> -<td class="tdl">1st position</td> -<td class="tdc">white</td> -<td class="tdc">yellow</td> -<td class="tdc">red</td> -</tr> -<tr> -<td class="tdl">2nd position</td> -<td class="tdc">yellow</td> -<td class="tdc">white—</td> -<td class="tdc">red</td> -</tr> -<tr> -<td class="tdl">3rd position</td> -<td class="tdc">red</td> -<td class="tdc">yellow</td> -<td class="tdc">white—</td> -</tr> -</table> - - -<p>Here white with a negative sign after it in the column -under <i>j</i> means that white runs in the negative sense of -the <i>j</i> direction.</p> - -<p>We may express the fact in the following way:— -In the plane there is room for two axes while the body -has three. Therefore in the plane we can represent any -two. If we want to keep the axis that goes in the -unknown dimension always running in the positive sense, -then the axis which originally ran in the unknown<span class="pagenum" id="Page_146">[Pg 146]</span> -dimension (the white axis) must come in in the negative -sense of that axis which goes out of the plane into the -unknown dimension.</p> - -<p>It is obvious that the unknown direction, the direction -in which the white line runs at first, is quite distinct from -any direction which the plane creature knows. The white -line may come in towards him, or running down. If he -is looking at a square, which is the face of a cube -(looking at it by a line), then any one of the bounding lines -remaining unmoved, another face of the cube may come -in, any one of the faces, namely, which have the white line -in them. And the white line comes sometimes in one -of the space directions he knows, sometimes in another.</p> - -<p>Now this turning which leaves a line unchanged is -something quite unlike any turning he knows in the -plane. In the plane a figure turns round a point. The -square can turn round the null point in his plane, and -the red and yellow lines change places, only of course, as -with every rotation of lines at right angles, if red goes -where yellow went, yellow comes in negative of red’s old -direction.</p> - -<p>This turning, as the plane creature conceives it, we -should call turning about an axis perpendicular to the -plane. What he calls turning about the null point we -call turning about the white line as it stands out from -his plane. There is no such thing as turning about a -point, there is always an axis, and really much more turns -than the plane being is aware of.</p> - -<p>Taking now a different point of view, let us suppose the -cubes to be presented to the plane being by being passed -transverse to his plane. Let us suppose the sheet of -matter over which the plane being and all objects in his -world slide, to be of such a nature that objects can pass -through it without breaking it. Let us suppose it to be -of the same nature as the film of a soap bubble, so that<span class="pagenum" id="Page_147">[Pg 147]</span> -it closes around objects pushed through it, and, however -the object alters its shape as it passes through it, let us -suppose this film to run up to the contour of the object -in every part, maintaining its plane surface unbroken.</p> - -<p>Then we can push a cube or any object through the -film and the plane being who slips about in the film -will know the contour of the cube just and exactly where -the film meets it.</p> - -<div class="figleft illowp40" id="fig_90" style="max-width: 18.75em;"> - <img src="images/fig_90.png" alt="" /> - <div class="caption">Fig. 90.</div> -</div> - -<p>Fig. 90 represents a cube passing through a plane film. -The plane being now comes into -contact with a very thin slice -of the cube somewhere between -the left and right hand faces. -This very thin slice he thinks -of as having no thickness, and -consequently his idea of it is -what we call a section. It is -bounded by him by pink lines -front and back, coming from -the part of the pink face he is -in contact with, and above and below, by light yellow -lines. Its corners are not null-coloured points, but white -points, and its interior is ochre, the colour of the interior -of the cube.</p> - -<p>If now we suppose the cube to be an inch in each -dimension, and to pass across, from right to left, through -the plane, then we should explain the appearances presented -to the plane being by saying: First of all you -have the face of a cube, this lasts only a moment; then -you have a figure of the same shape but differently -coloured. This, which appears not to move to you in any -direction which you know of, is really moving transverse -to your plane world. Its appearance is unaltered, but -each moment it is something different—a section further -on, in the white, the unknown dimension. Finally, at the<span class="pagenum" id="Page_148">[Pg 148]</span> -end of the minute, a face comes in exactly like the face -you first saw. This finishes up the cube—it is the further -face in the unknown dimension.</p> - -<p>The white line, which extends in length just like the -red or the yellow, you do not see as extensive; you apprehend -it simply as an enduring white point. The null -point, under the condition of movement of the cube, -vanishes in a moment, the lasting white point is really -your apprehension of a white line, running in the unknown -dimension. In the same way the red line of the face by -which the cube is first in contact with the plane lasts only -a moment, it is succeeded by the pink line, and this pink -line lasts for the inside of a minute. This lasting pink -line in your apprehension of a surface, which extends in -two dimensions just like the orange surface extends, as you -know it, when the cube is at rest.</p> - -<p>But the plane creature might answer, “This orange -object is substance, solid substance, bounded completely -and on every side.”</p> - -<p>Here, of course, the difficulty comes in. His solid is our -surface—his notion of a solid is our notion of an abstract -surface with no thickness at all.</p> - -<p>We should have to explain to him that, from every point -of what he called a solid, a new dimension runs away. -From every point a line can be drawn in a direction -unknown to him, and there is a solidity of a kind greater -than that which he knows. This solidity can only be -realised by him by his supposing an unknown direction, -by motion in which what he conceives to be solid matter -instantly disappears. The higher solid, however, which -extends in this dimension as well as in those which he -knows, lasts when a motion of that kind takes place, -different sections of it come consecutively in the plane of -his apprehension, and take the place of the solid which he -at first conceives to be all. Thus, the higher solid—our<span class="pagenum" id="Page_149">[Pg 149]</span> -solid in contradistinction to his area solid, his two-dimensional -solid, must be conceived by him as something -which has duration in it, under circumstances in which his -matter disappears out of his world.</p> - -<p>We may put the matter thus, using the conception of -motion.</p> - -<p>A null point moving in a direction away generates a -yellow line, and the yellow line ends in a null point. We -suppose, that is, a point to move and mark out the -products of this motion in such a manner. Now -suppose this whole line as thus produced to move in -an upward direction; it traces out the two-dimensional -solid, and the plane being gets an orange square. The -null point moves in a red line and ends in a null point, -the yellow line moves and generates an orange square and -ends in a yellow line, the farther null point generates -a red line and ends in a null point. Thus, by movement -in two successive directions known to him, he -can imagine his two-dimensional solid produced with all -its boundaries.</p> - -<p>Now we tell him: “This whole two-dimensional solid -can move in a third or unknown dimension to you. The -null point moving in this dimension out of your world -generates a white line and ends in a null point. The -yellow line moving generates a light yellow two-dimensional -solid and ends in a yellow line, and this -two-dimensional solid, lying end on to your plane world, is -bounded on the far side by the other yellow line. In -the same way each of the lines surrounding your square -traces out an area, just like the orange area you know. -But there is something new produced, something which -you had no idea of before; it is that which is produced by -the movement of the orange square. That, than which -you can imagine nothing more solid, itself moves in a -direction open to it and produces a three-dimensional<span class="pagenum" id="Page_150">[Pg 150]</span> -solid. Using the addition of white to symbolise the -products of this motion this new kind of solid will be light -orange or ochre, and it will be bounded on the far side by -the final position of the orange square which traced it -out, and this final position we suppose to be coloured like -the square in its first position, orange with yellow and -red boundaries and null corners.”</p> - -<p>This product of movement, which it is so easy for us to -describe, would be difficult for him to conceive. But this -difficulty is connected rather with its totality than with -any particular part of it.</p> - -<p>Any line, or plane of this, to him higher, solid we could -show to him, and put in his sensible world.</p> - -<p>We have already seen how the pink square could be put -in his world by a turning of the cube about the red line. -And any section which we can conceive made of the cube -could be exhibited to him. You have simply to turn the -cube and push it through, so that the plane of his existence -is the plane which cuts out the given section of the cube, -then the section would appear to him as a solid. In his -world he would see the contour, get to any part of it by -digging down into it.</p> - - -<p><span class="smcap">The Process by which a Plane Being would gain -a Notion of a Solid.</span></p> - -<p>If we suppose the plane being to have a general idea of -the existence of a higher solid—our solid—we must next -trace out in detail the method, the discipline, by which -he would acquire a working familiarity with our space -existence. The process begins with an adequate realisation -of a simple solid figure. For this purpose we will -suppose eight cubes forming a larger cube, and first we -will suppose each cube to be coloured throughout uniformly.<span class="pagenum" id="Page_151">[Pg 151]</span> -Let the cubes in <a href="#fig_91">fig. 91</a> be the eight making a larger -cube.</p> - -<div class="figcenter illowp80" id="fig_91" style="max-width: 62.5em;"> - <img src="images/fig_91.png" alt="" /> - <div class="caption">Fig. 91.</div> -</div> - - -<p>Now, although each cube is supposed to be coloured -entirely through with the colour, the name of which is -written on it, still we can speak of the faces, edges, and -corners of each cube as if the colour scheme we have -investigated held for it. Thus, on the null cube we can -speak of a null point, a red line, a white line, a pink face, and -so on. These colour designations are shown on No. 1 of -the views of the tesseract in the plate. Here these colour -names are used simply in their geometrical significance. -They denote what the particular line, etc., referred to would -have as its colour, if in reference to the particular cube -the colour scheme described previously were carried out.</p> - -<p>If such a block of cubes were put against the plane and -then passed through it from right to left, at the rate of an -inch a minute, each cube being an inch each way, the -plane being would have the following appearances:—</p> - -<p>First of all, four squares null, yellow, red, orange, lasting -each a minute; and secondly, taking the exact places -of these four squares, four others, coloured white, light -yellow, pink, ochre. Thus, to make a catalogue of the -solid body, he would have to put side by side in his world -two sets of four squares each, as in <a href="#fig_92">fig. 92</a>. The first<span class="pagenum" id="Page_152">[Pg 152]</span> -are supposed to last a minute, and then the others to -come in in place of them, -and also last a minute.</p> - -<div class="figleft illowp50" id="fig_92" style="max-width: 25em;"> - <img src="images/fig_92.png" alt="" /> - <div class="caption">Fig. 92.</div> -</div> - -<p>In speaking of them -he would have to denote -what part of the respective -cube each square represents. -Thus, at the beginning -he would have null -cube orange face, and after -the motion had begun he -would have null cube ochre -section. As he could get -the same coloured section whichever way the cube passed -through, it would be best for him to call this section white -section, meaning that it is transverse to the white axis. -These colour-names, of course, are merely used as names, -and do not imply in this case that the object is really -coloured. Finally, after a minute, as the first cube was -passing beyond his plane he would have null cube orange -face again.</p> - -<p>The same names will hold for each of the other cubes, -describing what face or section of them the plane being -has before him; and the second wall of cubes will come -on, continue, and go out in the same manner. In the -area he thus has he can represent any movement which -we carry out in the cubes, as long as it does not involve -a motion in the direction of the white axis. The relation -of parts that succeed one another in the direction of the -white axis is realised by him as a consecution of states.</p> - -<p>Now, his means of developing his space apprehension -lies in this, that that which is represented as a time -sequence in one position of the cubes, can become a real -co-existence, <i>if something that has a real co-existence -becomes a time sequence</i>.</p> - -<p><span class="pagenum" id="Page_153">[Pg 153]</span></p> - -<p>We must suppose the cubes turned round each of the -axes, the red line, and the yellow line, then something, -which was given as time before, will now be given as the -plane creature’s space; something, which was given as space -before, will now be given as a time series as the cube is -passed through the plane.</p> - -<p>The three positions in which the cubes must be studied -are the one given above and the two following ones. In -each case the original null point which was nearest to us -at first is marked by an asterisk. In figs. 93 and 94 the -point marked with a star is the same in the cubes and in -the plane view.</p> - -<div class="figcenter illowp100" id="fig_93" style="max-width: 62.5em;"> - <img src="images/fig_93.png" alt="" /> - <div class="caption">Fig. 93.<br /> -The cube swung round the red line, so that the white line points -towards us.</div> -</div> - -<p>In <a href="#fig_93">fig. 93</a> the cube is swung round the red line so as to -point towards us, and consequently the pink face comes -next to the plane. As it passes through there are two -varieties of appearance designated by the figures 1 and 2 -in the plane. These appearances are named in the figure, -and are determined by the order in which the cubes<span class="pagenum" id="Page_154">[Pg 154]</span> -come in the motion of the whole block through the -plane.</p> - -<p>With regard to these squares severally, however, -different names must be used, determined by their -relations in the block.</p> - -<p>Thus, in <a href="#fig_93">fig. 93</a>, when the cube first rests against the -plane the null cube is in contact by its pink face; as the -block passes through we get an ochre section of the null -cube, but this is better called a yellow section, as it is -made by a plane perpendicular to the yellow line. When -the null cube has passed through the plane, as it is -leaving it, we get again a pink face.</p> - -<div class="figcenter illowp100" id="fig_94" style="max-width: 62.5em;"> - <img src="images/fig_94.png" alt="" /> - <div class="caption">Fig. 94.<br /> -The cube swung round yellow line, with red line running from left -to right, and white line running down.</div> -</div> - -<p>The same series of changes take place with the cube -appearances which follow on those of the null cube. In -this motion the yellow cube follows on the null cube, and -the square marked yellow in 2 in the plane will be first -“yellow pink face,” then “yellow yellow section,” then -“yellow pink face.”</p> - -<p>In <a href="#fig_94">fig. 94</a>, in which the cube is turned about the yellow -line, we have a certain difficulty, for the plane being will<span class="pagenum" id="Page_155">[Pg 155]</span> -find that the position his squares are to be placed in will -lie below that which they first occupied. They will come -where the support was on which he stood his first set of -squares. He will get over this difficulty by moving his -support.</p> - -<p>Then, since the cubes come upon his plane by the light -yellow face, he will have, taking the null cube as before for -an example, null, light yellow face; null, red section, -because the section is perpendicular to the red line; and -finally, as the null cube leaves the plane, null, light yellow -face. Then, in this case red following on null, he will -have the same series of views of the red as he had of the -null cube.</p> - -<div class="figcenter illowp100" id="fig_95" style="max-width: 62.5em;"> - <img src="images/fig_95.png" alt="" /> - <div class="caption">Fig. 95.</div> -</div> - -<p>There is another set of considerations which we will -briefly allude to.</p> - -<p>Suppose there is a hollow cube, and a string is stretched -across it from null to null, <i>r</i>, <i>y</i>, <i>wh</i>, as we may call the -far diagonal point, how will this string appear to the -plane being as the cube moves transverse to his plane?</p> - -<p>Let us represent the cube as a number of sections, say -5, corresponding to 4 equal divisions made along the white -line perpendicular to it.</p> - -<p>We number these sections 0, 1, 2, 3, 4, corresponding -to the distances along the white line at which they are<span class="pagenum" id="Page_156">[Pg 156]</span> -taken, and imagine each section to come in successively, -taking the place of the preceding one.</p> - -<p>These sections appear to the plane being, counting from -the first, to exactly coincide each with the preceding one. -But the section of the string occupies a different place in -each to that which it does in the preceding section. The -section of the string appears in the position marked by -the dots. Hence the slant of the string appears as a -motion in the frame work marked out by the cube sides. -If we suppose the motion of the cube not to be recognised, -then the string appears to the plane being as a moving -point. Hence extension on the unknown dimension -appears as duration. Extension sloping in the unknown -direction appears as continuous movement.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_157">[Pg 157]</span></p> - -<h2 class="nobreak" id="CHAPTER_XII">CHAPTER XII<br /> - -<small><i>THE SIMPLEST FOUR-DIMENSIONAL SOLID</i></small></h2></div> - - -<p>A plane being, in learning to apprehend solid existence, -must first of all realise that there is a sense of direction -altogether wanting to him. That which we call right -and left does not exist in his perception. He must -assume a movement in a direction, and a distinction of -positive and negative in that direction, which has no -reality corresponding to it in the movements he can -make. This direction, this new dimension, he can only -make sensible to himself by bringing in time, and supposing -that changes, which take place in time, are due to -objects of a definite configuration in three dimensions -passing transverse to his plane, and the different sections -of it being apprehended as changes of one and the same -plane figure.</p> - -<p>He must also acquire a distinct notion about his plane -world, he must no longer believe that it is the all of -space, but that space extends on both sides of it. In -order, then, to prevent his moving off in this unknown -direction, he must assume a sheet, an extended solid sheet, -in two dimensions, against which, in contact with which, -all his movements take place.</p> - -<p>When we come to think of a four-dimensional solid, -what are the corresponding assumptions which we must -make?</p> - -<p>We must suppose a sense which we have not, a sense<span class="pagenum" id="Page_158">[Pg 158]</span> -of direction wanting in us, something which a being in -a four-dimensional world has, and which we have not. It -is a sense corresponding to a new space direction, a -direction which extends positively and negatively from -every point of our space, and which goes right away from -any space direction we know of. The perpendicular to a -plane is perpendicular, not only to two lines in it, but to -every line, and so we must conceive this fourth dimension -as running perpendicularly to each and every line we can -draw in our space.</p> - -<p>And as the plane being had to suppose something -which prevented his moving off in the third, the -unknown dimension to him, so we have to suppose -something which prevents us moving off in the direction -unknown to us. This something, since we must be in -contact with it in every one of our movements, must not -be a plane surface, but a solid; it must be a solid, which -in every one of our movements we are against, not in. It -must be supposed as stretching out in every space dimension -that we know; but we are not in it, we are against it, we -are next to it, in the fourth dimension.</p> - -<p>That is, as the plane being conceives himself as having -a very small thickness in the third dimension, of which -he is not aware in his sense experience, so we must -suppose ourselves as having a very small thickness in -the fourth dimension, and, being thus four-dimensional -beings, to be prevented from realising that we are -such beings by a constraint which keeps us always in -contact with a vast solid sheet, which stretches on in -every direction. We are against that sheet, so that, if we -had the power of four-dimensional movement, we should -either go away from it or through it; all our space -movements as we know them being such that, performing -them, we keep in contact with this solid sheet.</p> - -<p>Now consider the exposition a plane being would make<span class="pagenum" id="Page_159">[Pg 159]</span> -for himself as to the question of the enclosure of a square, -and of a cube.</p> - -<p>He would say the square <span class="allsmcap">A</span>, in Fig. 96, is completely -enclosed by the four squares, <span class="allsmcap">A</span> far, -<span class="allsmcap">A</span> near, <span class="allsmcap">A</span> above, <span class="allsmcap">A</span> below, or as they -are written <span class="allsmcap">A</span><i>n</i>, <span class="allsmcap">A</span><i>f</i>, <span class="allsmcap">A</span><i>a</i>, <span class="allsmcap">A</span><i>b</i>.</p> - -<div class="figleft illowp30" id="fig_96" style="max-width: 15.625em;"> - <img src="images/fig_96.png" alt="" /> - <div class="caption">Fig. 96.</div> -</div> - -<p>If now he conceives the square <span class="allsmcap">A</span> -to move in the, to him, unknown -dimension it will trace out a cube, -and the bounding squares will form -cubes. Will these completely surround -the cube generated by <span class="allsmcap">A</span>? No; -there will be two faces of the cube -made by <span class="allsmcap">A</span> left uncovered; the first, -that face which coincides with the -square <span class="allsmcap">A</span> in its first position; the next, that which coincides -with the square <span class="allsmcap">A</span> in its final position. Against these -two faces cubes must be placed in order to completely -enclose the cube <span class="allsmcap">A</span>. These may be called the cubes left -and right or <span class="allsmcap">A</span><i>l</i> and <span class="allsmcap">A</span><i>r</i>. Thus each of the enclosing -squares of the square <span class="allsmcap">A</span> becomes a cube and two more -cubes are wanted to enclose the cube formed by the -movement of <span class="allsmcap">A</span> in the third dimension.</p> - -<div class="figleft illowp30" id="fig_97" style="max-width: 34.6875em;"> - <img src="images/fig_97.png" alt="" /> - <div class="caption">Fig. 97.</div> -</div> - -<p>The plane being could not see the square <span class="allsmcap">A</span> with the -squares <span class="allsmcap">A</span><i>n</i>, <span class="allsmcap">A</span><i>f</i>, etc., placed about it, -because they completely hide it from -view; and so we, in the analogous -case in our three-dimensional world, -cannot see a cube <span class="allsmcap">A</span> surrounded by -six other cubes. These cubes we -will call <span class="allsmcap">A</span> near <span class="allsmcap">A</span><i>n</i>, <span class="allsmcap">A</span> far <span class="allsmcap">A</span><i>f</i>, <span class="allsmcap">A</span> above -<span class="allsmcap">A</span><i>a</i>, <span class="allsmcap">A</span> below <span class="allsmcap">A</span><i>b</i>, <span class="allsmcap">A</span> left <span class="allsmcap">A</span><i>l</i>, <span class="allsmcap">A</span> right <span class="allsmcap">A</span><i>r</i>, -shown in <a href="#fig_97">fig. 97</a>. If now the cube <span class="allsmcap">A</span> -moves in the fourth dimension right out of space, it traces -out a higher cube—a tesseract, as it may be called.<span class="pagenum" id="Page_160">[Pg 160]</span> -Each of the six surrounding cubes carried on in the same -motion will make a tesseract also, and these will be -grouped around the tesseract formed by <span class="allsmcap">A</span>. But will they -enclose it completely?</p> - -<p>All the cubes <span class="allsmcap">A</span><i>n</i>, <span class="allsmcap">A</span><i>f</i>, etc., lie in our space. But there is -nothing between the cube <span class="allsmcap">A</span> and that solid sheet in contact -with which every particle of matter is. When the cube <span class="allsmcap">A</span> -moves in the fourth direction it starts from its position, -say <span class="allsmcap">A</span><i>k</i>, and ends in a final position <span class="allsmcap">A</span><i>n</i> (using the words -“ana” and “kata” for up and down in the fourth dimension). -Now the movement in this fourth dimension is -not bounded by any of the cubes <span class="allsmcap">A</span><i>n</i>, <span class="allsmcap">A</span><i>f</i>, nor by what -they form when thus moved. The tesseract which <span class="allsmcap">A</span> -becomes is bounded in the positive and negative ways in -this new direction by the first position of <span class="allsmcap">A</span> and the last -position of <span class="allsmcap">A</span>. Or, if we ask how many tesseracts lie -around the tesseract which <span class="allsmcap">A</span> forms, there are eight, of -which one meets it by the cube <span class="allsmcap">A</span>, and another meets it -by a cube like <span class="allsmcap">A</span> at the end of its motion.</p> - -<p>We come here to a very curious thing. The whole -solid cube <span class="allsmcap">A</span> is to be looked on merely as a boundary of -the tesseract.</p> - -<p>Yet this is exactly analogous to what the plane being -would come to in his study of the solid world. The -square <span class="allsmcap">A</span> (<a href="#fig_96">fig. 96</a>), which the plane being looks on as a -solid existence in his plane world, is merely the boundary -of the cube which he supposes generated by its motion.</p> - -<p>The fact is that we have to recognise that, if there is -another dimension of space, our present idea of a solid -body, as one which has three dimensions only, does not -correspond to anything real, but is the abstract idea of a -three-dimensional boundary limiting a four-dimensional -solid, which a four-dimensional being would form. The -plane being’s thought of a square is not the thought -of what we should call a possibly existing real square,<span class="pagenum" id="Page_161">[Pg 161]</span> -but the thought of an abstract boundary, the face of -a cube.</p> - -<p>Let us now take our eight coloured cubes, which form -a cube in space, and ask what additions we must make -to them to represent the simplest collection of four-dimensional -bodies—namely, a group of them of the same extent -in every direction. In plane space we have four squares. -In solid space we have eight cubes. So we should expect -in four-dimensional space to have sixteen four-dimensional -bodies-bodies which in four-dimensional space -correspond to cubes in three-dimensional space, and these -bodies we call tesseracts.</p> - -<div class="figleft illowp30" id="fig_98" style="max-width: 15.625em;"> - <img src="images/fig_98.png" alt="" /> - <div class="caption">Fig. 98.</div> -</div> - -<p>Given then the null, white, red, yellow cubes, and -those which make up the block, we -notice that we represent perfectly -well the extension in three directions -(fig. 98). From the null point of -the null cube, travelling one inch, we -come to the white cube; travelling -one inch away we come to the yellow -cube; travelling one inch up we come -to the red cube. Now, if there is -a fourth dimension, then travelling -from the same null point for one -inch in that direction, we must come to the body lying -beyond the null region.</p> - -<p>I say null region, not cube; for with the introduction -of the fourth dimension each of our cubes must become -something different from cubes. If they are to have -existence in the fourth dimension, they must be “filled -up from” in this fourth dimension.</p> - -<p>Now we will assume that as we get a transference from -null to white going in one way, from null to yellow going -in another, so going from null in the fourth direction we -have a transference from null to blue, using thus the<span class="pagenum" id="Page_162">[Pg 162]</span> -colours white, yellow, red, blue, to denote transferences in -each of the four directions—right, away, up, unknown or -fourth dimension.</p> - -<div class="figleft illowp60" id="fig_99" style="max-width: 25em;"> - <img src="images/fig_99.png" alt="" /> - <div class="caption">Fig. 99.<br /> -A plane being’s representation of a block -of eight cubes by two sets of four squares.</div> -</div> - -<p>Hence, as the plane being must represent the solid regions, -he would come to by going right, as four squares lying -in some position in -his plane, arbitrarily -chosen, side by side -with his original four -squares, so we must -represent those eight -four-dimensional regions, -which we -should come to by -going in the fourth -dimension from each -of our eight cubes, by eight cubes placed in some arbitrary -position relative to our first eight cubes.</p> - -<div class="figcenter illowp80" id="fig_100" style="max-width: 50em;"> - <img src="images/fig_100.png" alt="" /> - <div class="caption">Fig. 100.</div> -</div> - -<p>Our representation of a block of sixteen tesseracts by -two blocks of eight cubes.<a id="FNanchor_3" href="#Footnote_3" class="fnanchor">[3]</a></p> - - -<div class="footnotes"><div class="footnote"> - -<p><a id="Footnote_3" href="#FNanchor_3" class="label">[3]</a> The eight cubes used here in 2 can be found in the second of the -model blocks. They can be taken out and used.</p> - -</div></div> - -<p>Hence, of the two sets of eight cubes, each one will serve<span class="pagenum" id="Page_163">[Pg 163]</span> -us as a representation of one of the sixteen tesseracts -which form one single block in four-dimensional space. -Each cube, as we have it, is a tray, as it were, against -which the real four-dimensional figure rests—just as each -of the squares which the plane being has is a tray, so to -speak, against which the cube it represents could rest.</p> - -<p>If we suppose the cubes to be one inch each way, then -the original eight cubes will give eight tesseracts of the -same colours, or the cubes, extending each one inch in the -fourth dimension.</p> - -<p>But after these there come, going on in the fourth dimension, -eight other bodies, eight other tesseracts. These -must be there, if we suppose the four-dimensional body -we make up to have two divisions, one inch each in each -of four directions.</p> - -<p>The colour we choose to designate the transference to -this second region in the fourth dimension is blue. Thus, -starting from the null cube and going in the fourth -dimension, we first go through one inch of the null -tesseract, then we come to a blue cube, which is the -beginning of a blue tesseract. This blue tesseract stretches -one inch farther on in the fourth dimension.</p> - -<p>Thus, beyond each of the eight tesseracts, which are of -the same colour as the cubes which are their bases, lie -eight tesseracts whose colours are derived from the colours -of the first eight by adding blue. Thus—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdl">Null</td> -<td class="tdc">gives</td> -<td class="tdl">blue</td> -</tr> -<tr> -<td class="tdl">Yellow</td> -<td class="tdc">”</td> -<td class="tdl">green</td> -</tr> -<tr> -<td class="tdl">Red</td> -<td class="tdc">”</td> -<td class="tdl">purple</td> -</tr> -<tr> -<td class="tdl">Orange</td> -<td class="tdc">”</td> -<td class="tdl">brown</td> -</tr> -<tr> -<td class="tdl">White</td> -<td class="tdc">”</td> -<td class="tdl">light blue</td> -</tr> -<tr> -<td class="tdl">Pink</td> -<td class="tdc">”</td> -<td class="tdl">light purple</td> -</tr> -<tr> -<td class="tdl">Light yellow</td> -<td class="tdc">”</td> -<td class="tdl">light green</td> -</tr> -<tr> -<td class="tdl">Ochre</td> -<td class="tdc">”</td> -<td class="tdl">light brown</td> -</tr> -</table> - -<p>The addition of blue to yellow gives green—this is a<span class="pagenum" id="Page_164">[Pg 164]</span> -natural supposition to make. It is also natural to suppose -that blue added to red makes purple. Orange and blue -can be made to give a brown, by using certain shades and -proportions. And ochre and blue can be made to give a -light brown.</p> - -<p>But the scheme of colours is merely used for getting -a definite and realisable set of names and distinctions -visible to the eye. Their naturalness is apparent to any -one in the habit of using colours, and may be assumed to -be justifiable, as the sole purpose is to devise a set of -names which are easy to remember, and which will give -us a set of colours by which diagrams may be made easy -of comprehension. No scientific classification of colours -has been attempted.</p> - -<p>Starting, then, with these sixteen colour names, we have -a catalogue of the sixteen tesseracts, which form a four-dimensional -block analogous to the cubic block. But -the cube which we can put in space and look at is not one -of the constituent tesseracts; it is merely the beginning, -the solid face, the side, the aspect, of a tesseract.</p> - -<p>We will now proceed to derive a name for each region, -point, edge, plane face, solid and a face of the tesseract.</p> - -<p>The system will be clear, if we look at a representation -in the plane of a tesseract with three, and one with four -divisions in its side.</p> - -<p>The tesseract made up of three tesseracts each way -corresponds to the cube made up of three cubes each way, -and will give us a complete nomenclature.</p> - -<p>In this diagram, <a href="#fig_101">fig. 101</a>, 1 represents a cube of 27 -cubes, each of which is the beginning of a tesseract. -These cubes are represented simply by their lowest squares, -the solid content must be understood. 2 represents the -27 cubes which are the beginnings of the 27 tesseracts -one inch on in the fourth dimension. These tesseracts -are represented as a block of cubes put side by side with<span class="pagenum" id="Page_165">[Pg 165]</span> -the first block, but in their proper positions they could -not be in space with the first set. 3 represents 27 cubes -(forming a larger cube) which are the beginnings of the -tesseracts, which begin two inches in the fourth direction -from our space and continue another inch.</p> - -<div class="figcenter illowp100" id="fig_101" style="max-width: 62.5em;"> - <img src="images/fig_101.png" alt="" /> - <div class="caption">Fig. 101.<br /> - - -<table class="standard" summary=""> -<col width="30%" /> <col width="30%" /> <col width="30%" /> -<tr> -<td class="tdc">1</td> -<td class="tdc">2</td> -<td class="tdc">3</td> -</tr> -<tr> -<td class="tdc">Each cube is the -beginning of the first -tesseract going in the -fourth dimension. -</td> -<td class="tdc">Each cube is the -beginning of the -second tesseract. -</td> -<td class="tdc">Each cube is the -beginning of the -third tesseract. -</td> -</tr> -</table></div> -</div> - - -<p><span class="pagenum" id="Page_166">[Pg 166]</span></p> - - -<div class="figcenter illowp100" id="fig_102" style="max-width: 62.5em;"> - <img src="images/fig_102.png" alt="" /> - <div class="caption">Fig. 102.<a id="FNanchor_4" href="#Footnote_4" class="fnanchor">[4]</a></div> -<table class="standard" summary=""> -<col width="25%" /> <col width="25%" /> <col width="25%" /> <col width="25%" /> -<tr> -<td class="tdc">1</td> -<td class="tdc">2</td> -<td class="tdc">3</td> -<td class="tdc">4</td> -</tr> -<tr> -<td class="tdl">A cube of 64 cubes -each 1. in × 1 in., the beginning of a tesseract. -</td> -<td class="tdl">A cube of 64 cubes, -each 1 in. × 1 in. × 1 in. the beginning -of tesseracts 1 in. from our space -in the 4th dimension. -</td> -<td class="tdl">A cube of 64 cubes, -each 1 in. × 1 in. × 1 in. the beginning -of tesseracts 2 in. from our space -in the 4th dimension. -</td> -<td class="tdl">A cube of 64 cubes, -each 1 in. × 1 in. × 1 in. the beginning -of tesseracts 3 in. from our space -in the 4th dimension. -</td> -</tr> -</table></div> - - -<div class="footnotes"> -<div class="footnote"> - -<p><a id="Footnote_4" href="#FNanchor_4" class="label">[4]</a> The coloured plate, figs. 1, 2, 3, shows these relations more -conspicuously.</p> - -</div></div> - - -<p>In <a href="#fig_102">fig. 102</a>, we have the representation of a block of -4 × 4 × 4 × 4 or 256 tesseracts. They are given in -four consecutive sections, each supposed to be taken one -inch apart in the fourth dimension, and so giving four<span class="pagenum" id="Page_167">[Pg 167]</span> -blocks of cubes, 64 in each block. Here we see, comparing -it with the figure of 81 tesseracts, that the number -of the different regions show a different tendency of -increase. By taking five blocks of five divisions each way -this would become even more clear.</p> - -<p>We see, <a href="#fig_102">fig. 102</a>, that starting from the point at any -corner, the white coloured regions only extend out in -a line. The same is true for the yellow, red, and blue. -With regard to the latter it should be noticed that the -line of blues does not consist in regions next to each -other in the drawing, but in portions which come in in -different cubes. The portions which lie next to one -another in the fourth dimension must always be represented -so, when we have a three-dimensional representation. -Again, those regions such as the pink one, go on increasing -in two dimensions. About the pink region this is seen -without going out of the cube itself, the pink regions -increase in length and height, but in no other dimension. -In examining these regions it is sufficient to take one as -a sample.</p> - -<p>The purple increases in the same manner, for it comes -in in a succession from below to above in block 2, and in -a succession from block to block in 2 and 3. Now, a -succession from below to above represents a continuous -extension upwards, and a succession from block to block -represents a continuous extension in the fourth dimension. -Thus the purple regions increase in two dimensions, the -upward and the fourth, so when we take a very great -many divisions, and let each become very small, the -purple region forms a two-dimensional extension.</p> - -<p>In the same way, looking at the regions marked l. b. or -light blue, which starts nearest a corner, we see that the -tesseracts occupying it increase in length from left to -right, forming a line, and that there are as many lines of -light blue tesseracts as there are sections between the<span class="pagenum" id="Page_168">[Pg 168]</span> -first and last section. Hence the light blue tesseracts -increase in number in two ways—in the right and left, -and in the fourth dimension. They ultimately form -what we may call a plane surface.</p> - -<p>Now all those regions which contain a mixture of two -simple colours, white, yellow, red, blue, increase in two -ways. On the other hand, those which contain a mixture -of three colours increase in three ways. Take, for instance, -the ochre region; this has three colours, white, yellow, -red; and in the cube itself it increases in three ways.</p> - -<p>Now regard the orange region; if we add blue to this -we get a brown. The region of the brown tesseracts -extends in two ways on the left of the second block, -No. 2 in the figure. It extends also from left to right in -succession from one section to another, from section 2 -to section 3 in our figure.</p> - -<p>Hence the brown tesseracts increase in number in three -dimensions upwards, to and fro, fourth dimension. Hence -they form a cubic, a three-dimensional region; this region -extends up and down, near and far, and in the fourth -direction, but is thin in the direction from left to right. -It is a cube which, when the complete tesseract is represented -in our space, appears as a series of faces on the -successive cubic sections of the tesseract. Compare fig. -103 in which the middle block, 2, stands as representing a -great number of sections intermediate between 1 and 3.</p> - -<p>In a similar way from the pink region by addition of -blue we have the light purple region, which can be seen -to increase in three ways as the number of divisions -becomes greater. The three ways in which this region of -tesseracts extends is up and down, right and left, fourth -dimension. Finally, therefore, it forms a cubic mass of -very small tesseracts, and when the tesseract is given in -space sections it appears on the faces containing the -upward and the right and left dimensions.</p> - -<p><span class="pagenum" id="Page_169">[Pg 169]</span></p> - -<p>We get then altogether, as three-dimensional regions, -ochre, brown, light purple, light green.</p> - -<p>Finally, there is the region which corresponds to a -mixture of all the colours; there is only one region such -as this. It is the one that springs from ochre by the -addition of blue—this colour we call light brown.</p> - -<p>Looking at the light brown region we see that it -increases in four ways. Hence, the tesseracts of which it -is composed increase in number in each of four dimensions, -and the shape they form does not remain thin in -any of the four dimensions. Consequently this region -becomes the solid content of the block of tesseracts, itself; -it is the real four-dimensional solid. All the other regions -are then boundaries of this light brown region. If we -suppose the process of increasing the number of tesseracts -and diminishing their size carried on indefinitely, then -the light brown coloured tesseracts become the whole -interior mass, the three-coloured tesseracts become three-dimensional -boundaries, thin in one dimension, and form -the ochre, the brown, the light purple, the light green. -The two-coloured tesseracts become two-dimensional -boundaries, thin in two dimensions, <i>e.g.</i>, the pink, the -green, the purple, the orange, the light blue, the light -yellow. The one-coloured tesseracts become bounding -lines, thin in three dimensions, and the null points become -bounding corners, thin in four dimensions. From these -thin real boundaries we can pass in thought to the -abstractions—points, lines, faces, solids—bounding the -four-dimensional solid, which in this case is light brown -coloured, and under this supposition the light brown -coloured region is the only real one, is the only one which -is not an abstraction.</p> - -<p>It should be observed that, in taking a square as the -representation of a cube on a plane, we only represent -one face, or the section between two faces. The squares,<span class="pagenum" id="Page_170">[Pg 170]</span> -as drawn by a plane being, are not the cubes themselves, -but represent the faces or the sections of a cube. Thus -in the plane being’s diagram a cube of twenty-seven cubes -“null” represents a cube, but is really, in the normal -position, the orange square of a null cube, and may be -called null, orange square.</p> - -<p>A plane being would save himself confusion if he named -his representative squares, not by using the names of the -cubes simply, but by adding to the names of the cubes a -word to show what part of a cube his representative square -was.</p> - -<p>Thus a cube null standing against his plane touches it -by null orange face, passing through his plane it has in -the plane a square as trace, which is null white section, if -we use the phrase white section to mean a section drawn -perpendicular to the white line. In the same way the -cubes which we take as representative of the tesseract are -not the tesseract itself, but definite faces or sections of it. -In the preceding figures we should say then, not null, but -“null tesseract ochre cube,” because the cube we actually -have is the one determined by the three axes, white, red, -yellow.</p> - -<p>There is another way in which we can regard the colour -nomenclature of the boundaries of a tesseract.</p> - -<p>Consider a null point to move tracing out a white line -one inch in length, and terminating in a null point, -see <a href="#fig_103">fig. 103</a> or in the coloured plate.</p> - -<p>Then consider this white line with its terminal points -itself to move in a second dimension, each of the points -traces out a line, the line itself traces out an area, and -gives two lines as well, its initial and its final position.</p> - -<p>Thus, if we call “a region” any element of the figure, -such as a point, or a line, etc., every “region” in moving -traces out a new kind of region, “a higher region,” and -gives two regions of its own kind, an initial and a final<span class="pagenum" id="Page_171">[Pg 171]</span> -position. The “higher region” means a region with -another dimension in it.</p> - -<p>Now the square can move and generate a cube. The -square light yellow moves and traces out the mass of the -cube. Letting the addition of red denote the region -made by the motion in the upward direction we get an -ochre solid. The light yellow face in its initial and -terminal positions give the two square boundaries of the -cube above and below. Then each of the four lines of the -light yellow square—white, yellow, and the white, yellow -opposite them—trace out a bounding square. So there -are in all six bounding squares, four of these squares being -designated in colour by adding red to the colour of the -generating lines. Finally, each point moving in the up -direction gives rise to a line coloured null + red, or red, -and then there are the initial and terminal positions of the -points giving eight points. The number of the lines is -evidently twelve, for the four lines of this light yellow -square give four lines in their initial, four lines in their -final position, while the four points trace out four lines, -that is altogether twelve lines.</p> - -<p>Now the squares are each of them separate boundaries -of the cube, while the lines belong, each of them, to two -squares, thus the red line is that which is common to the -orange and pink squares.</p> - -<p>Now suppose that there is a direction, the fourth -dimension, which is perpendicular alike to every one -of the space dimensions already used—a dimension -perpendicular, for instance, to up and to right hand, -so that the pink square moving in this direction traces -out a cube.</p> - -<p>A dimension, moreover, perpendicular to the up and -away directions, so that the orange square moving in this -direction also traces out a cube, and the light yellow -square, too, moving in this direction traces out a cube.<span class="pagenum" id="Page_172">[Pg 172]</span> -Under this supposition, the whole cube moving in the -unknown dimension, traces out something new—a new -kind of volume, a higher volume. This higher volume -is a four-dimensional volume, and we designate it in colour -by adding blue to the colour of that which by moving -generates it.</p> - -<p>It is generated by the motion of the ochre solid, and -hence it is of the colour we call light brown (white, yellow, -red, blue, mixed together). It is represented by a number -of sections like 2 in <a href="#fig_103">fig. 103</a>.</p> - -<p>Now this light brown higher solid has for boundaries: -first, the ochre cube in its initial position, second, the -same cube in its final position, 1 and 3, <a href="#fig_103">fig. 103</a>. Each -of the squares which bound the cube, moreover, by movement -in this new direction traces out a cube, so we have -from the front pink faces of the cube, third, a pink blue or -light purple cube, shown as a light purple face on cube 2 -in <a href="#fig_103">fig. 103</a>, this cube standing for any number of intermediate -sections; fourth, a similar cube from the opposite -pink face; fifth, a cube traced out by the orange face—this -is coloured brown and is represented by the brown -face of the section cube in <a href="#fig_103">fig. 103</a>; sixth, a corresponding -brown cube on the right hand; seventh, a cube -starting from the light yellow square below; the unknown -dimension is at right angles to this also. This cube is -coloured light yellow and blue or light green; and, -finally, eighth, a corresponding cube from the upper -light yellow face, shown as the light green square at the -top of the section cube.</p> - -<p>The tesseract has thus eight cubic boundaries. These -completely enclose it, so that it would be invisible to a -four-dimensional being. Now, as to the other boundaries, -just as the cube has squares, lines, points, as boundaries, -so the tesseract has cubes, squares, lines, points, as -boundaries.</p> - -<p><span class="pagenum" id="Page_173">[Pg 173]</span></p> - -<p>The number of squares is found thus—round the cube -are six squares, these will give six squares in their initial -and six in their final positions. Then each of the twelve -lines of the cube trace out a square in the motion in -the fourth dimension. Hence there will be altogether -12 + 12 = 24 squares.</p> - -<p>If we look at any one of these squares we see that it -is the meeting surface of two of the cubic sides. Thus, -the red line by its movement in the fourth dimension, -traces out a purple square—this is common to two -cubes, one of which is traced out by the pink square -moving in the fourth dimension, and the other is -traced out by the orange square moving in the same -way. To take another square, the light yellow one, this -is common to the ochre cube and the light green cube. -The ochre cube comes from the light yellow square -by moving it in the up direction, the light green cube -is made from the light yellow square by moving it in -the fourth dimension. The number of lines is thirty-two, -for the twelve lines of the cube give twelve lines -of the tesseract in their initial position, and twelve in -their final position, making twenty-four, while each of -the eight points traces out a line, thus forming thirty-two -lines altogether.</p> - -<p>The lines are each of them common to three cubes, or -to three square faces; take, for instance, the red line. -This is common to the orange face, the pink face, and -that face which is formed by moving the red line in the -sixth dimension, namely, the purple face. It is also -common to the ochre cube, the pale purple cube, and the -brown cube.</p> - -<p>The points are common to six square faces and to four -cubes; thus, the null point from which we start is common -to the three square faces—pink, light yellow, orange, and -to the three square faces made by moving the three lines<span class="pagenum" id="Page_174">[Pg 174]</span> -white, yellow, red, in the fourth dimension, namely, the -light blue, the light green, the purple faces—that is, to -six faces in all. The four cubes which meet in it are the -ochre cube, the light purple cube, the brown cube, and -the light green cube.</p> - -<div class="figcenter illowp100" id="fig_103" style="max-width: 62.5em;"> - <img src="images/fig_103.png" alt="" /> - <div class="caption">Fig. 103.</div> -</div> - - -<p>The tesseract, red, white, yellow axes in space. In the lower line the three rear faces -are shown, the interior being removed.]</p> - -<p><span class="pagenum" id="Page_175">[Pg 175]</span></p> - -<div class="figcenter illowp100" id="fig_104" style="max-width: 62.5em;"> - <img src="images/fig_104.png" alt="" /> - <div class="caption">Fig. 104.<br /> -The tesseract, red, yellow, blue axes in space, -the blue axis running to the left, -opposite faces are coloured identically.</div> -</div> - -<p>A complete view of the tesseract in its various space -presentations is given in the following figures or catalogue -cubes, figs. 103-106. The first cube in each figure<span class="pagenum" id="Page_176">[Pg 176]</span> -represents the view of a tesseract coloured as described as -it begins to pass transverse to our space. The intermediate -figure represents a sectional view when it is partly through, -and the final figure represents the far end as it is just -passing out. These figures will be explained in detail in -the next chapter.</p> - -<div class="figcenter illowp100" id="fig_105" style="max-width: 62.5em;"> - <img src="images/fig_105.png" alt="" /> - <div class="caption">Fig. 105.<br /> -The tesseract, with red, white, blue axes in space. Opposite faces are coloured identically.</div> -</div> - -<p><span class="pagenum" id="Page_177">[Pg 177]</span></p> - -<div class="figcenter illowp100" id="fig_106" style="max-width: 62.5em;"> - <img src="images/fig_106.png" alt="" /> - <div class="caption">Fig. 106.<br /> -The tesseract, with blue, white, yellow axes in space. The blue axis runs downward -from the base of the ochre cube as it stands originally. Opposite faces are coloured -identically.</div> -</div> - -<p>We have thus obtained a nomenclature for each of the -regions of a tesseract; we can speak of any one of the -eight bounding cubes, the twenty square faces, the thirty-two -lines, the sixteen points.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_178">[Pg 178]</span></p> - -<h2 class="nobreak" id="CHAPTER_XIII">CHAPTER XIII<br /> - -<small><i>REMARKS ON THE FIGURES</i></small></h2></div> - - -<p>An inspection of above figures will give an answer to -many questions about the tesseract. If we have a -tesseract one inch each way, then it can be represented -as a cube—a cube having white, yellow, red axes, and -from this cube as a beginning, a volume extending into -the fourth dimension. Now suppose the tesseract to pass -transverse to our space, the cube of the red, yellow, white -axes disappears at once, it is indefinitely thin in the -fourth dimension. Its place is occupied by those parts -of the tesseract which lie further away from our space -in the fourth dimension. Each one of these sections -will last only for one moment, but the whole of them -will take up some appreciable time in passing. If we -take the rate of one inch a minute the sections will take -the whole of the minute in their passage across our -space, they will take the whole of the minute except the -moment which the beginning cube and the end cube -occupy in their crossing our space. In each one of the -cubes, the section cubes, we can draw lines in all directions -except in the direction occupied by the blue line, the -fourth dimension; lines in that direction are represented -by the transition from one section cube to another. Thus -to give ourselves an adequate representation of the -tesseract we ought to have a limitless number of section -cubes intermediate between the first bounding cube, the<span class="pagenum" id="Page_179">[Pg 179]</span> -ochre cube, and the last bounding cube, the other ochre -cube. Practically three intermediate sectional cubes will -be found sufficient for most purposes. We will take then -a series of five figures—two terminal cubes, and three -intermediate sections—and show how the different regions -appear in our space when we take each set of three out -of the four axes of the tesseract as lying in our space.</p> - -<p>In <a href="#fig_107">fig. 107</a> initial letters are used for the colours. -A reference to <a href="#fig_103">fig. 103</a> will show the complete nomenclature, -which is merely indicated here.</p> - -<div class="figcenter illowp100" id="fig_107" style="max-width: 62.5em;"> - <img src="images/fig_107.png" alt="" /> - <div class="caption">Fig. 107.</div> -</div> - -<p>In this figure the tesseract is shown in five stages -distant from our space: first, zero; second, 1/4 in.; third, -2/4 in.; fourth, 3/4 in.; fifth, 1 in.; which are called <i>b</i>0, <i>b</i>1, -<i>b</i>2, <i>b</i>3, <i>b</i>4, because they are sections taken at distances -0, 1, 2, 3, 4 quarter inches along the blue line. All the -regions can be named from the first cube, the <i>b</i>0 cube, -as before, simply by remembering that transference along -the b axis gives the addition of blue to the colour of -the region in the ochre, the <i>b</i>0 cube. In the final cube -<i>b</i>4, the colouring of the original <i>b</i>0 cube is repeated. -Thus the red line moved along the blue axis gives a red -and blue or purple square. This purple square appears -as the three purple lines in the sections <i>b</i>1, <i>b</i>2, <i>b</i>3, taken -at 1/4, 2/4, 3/4 of an inch in the fourth dimension. If the -tesseract moves transverse to our space we have then in -this particular region, first of all a red line which lasts -for a moment, secondly a purple line which takes its<span class="pagenum" id="Page_180">[Pg 180]</span> -place. This purple line lasts for a minute—that is, all -of a minute, except the moment taken by the crossing -our space of the initial and final red line. The purple -line having lasted for this period is succeeded by a red -line, which lasts for a moment; then this goes and the -tesseract has passed across our space. The final red line -we call red bl., because it is separated from the initial -red line by a distance along the axis for which we use -the colour blue. Thus a line that lasts represents an -area duration; is in this mode of presentation equivalent -to a dimension of space. In the same way the white -line, during the crossing our space by the tesseract, is -succeeded by a light blue line which lasts for the inside -of a minute, and as the tesseract leaves our space, having -crossed it, the white bl. line appears as the final -termination.</p> - -<p>Take now the pink face. Moved in the blue direction -it traces out a light purple cube. This light purple -cube is shown in sections in <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, and the farther -face of this cube in the blue direction is shown in <i>b</i><sub>4</sub>—a -pink face, called pink <i>b</i> because it is distant from the -pink face we began with in the blue direction. Thus -the cube which we colour light purple appears as a lasting -square. The square face itself, the pink face, vanishes -instantly the tesseract begins to move, but the light -purple cube appears as a lasting square. Here also -duration is the equivalent of a dimension of space—a -lasting square is a cube. It is useful to connect these -diagrams with the views given in the coloured plate.</p> - -<p>Take again the orange face, that determined by the -red and yellow axes; from it goes a brown cube in the -blue direction, for red and yellow and blue are supposed -to make brown. This brown cube is shown in three -sections in the faces <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>. In <i>b</i><sub>4</sub> is the opposite -orange face of the brown cube, the face called orange <i>b</i>,<span class="pagenum" id="Page_181">[Pg 181]</span> -for it is distant in the blue direction from the orange -face. As the tesseract passes transverse to our space, -we have then in this region an instantly vanishing orange -square, followed by a lasting brown square, and finally -an orange face which vanishes instantly.</p> - -<p>Now, as any three axes will be in our space, let us send -the white axis out into the unknown, the fourth dimension, -and take the blue axis into our known space -dimension. Since the white and blue axes are perpendicular -to each other, if the white axis goes out into -the fourth dimension in the positive sense, the blue axis -will come into the direction the white axis occupied, -in the negative sense.</p> - -<div class="figcenter illowp100" id="fig_108" style="max-width: 62.5em;"> - <img src="images/fig_108.png" alt="" /> - <div class="caption">Fig. 108.</div> -</div> - -<p>Hence, not to complicate matters by having to think -of two senses in the unknown direction, let us send the -white line into the positive sense of the fourth dimension, -and take the blue one as running in the negative -sense of that direction which the white line has left; -let the blue line, that is, run to the left. We have -now the row of figures in <a href="#fig_108">fig. 108</a>. The dotted cube -shows where we had a cube when the white line ran -in our space—now it has turned out of our space, and -another solid boundary, another cubic face of the tesseract -comes into our space. This cube has red and yellow -axes as before; but now, instead of a white axis running -to the right, there is a blue axis running to the left. -Here we can distinguish the regions by colours in a perfectly -systematic way. The red line traces out a purple<span class="pagenum" id="Page_182">[Pg 182]</span> -square in the transference along the blue axis by which -this cube is generated from the orange face. This -purple square made by the motion of the red line is -the same purple face that we saw before as a series of -lines in the sections <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>. Here, since both red and -blue axes are in our space, we have no need of duration -to represent the area they determine. In the motion -of the tesseract across space this purple face would -instantly disappear.</p> - -<p>From the orange face, which is common to the initial -cubes in <a href="#fig_107">fig. 107</a> and <a href="#fig_108">fig. 108</a>, there goes in the blue -direction a cube coloured brown. This brown cube is -now all in our space, because each of its three axes run -in space directions, up, away, to the left. It is the same -brown cube which appeared as the successive faces on the -sections <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>. Having all its three axes in our -space, it is given in extension; no part of it needs to -be represented as a succession. The tesseract is now -in a new position with regard to our space, and when -it moves across our space the brown cube instantly -disappears.</p> - -<p>In order to exhibit the other regions of the tesseract -we must remember that now the white line runs in the -unknown dimension. Where shall we put the sections -at distances along the line? Any arbitrary position in -our space will do: there is no way by which we can -represent their real position.</p> - -<p>However, as the brown cube comes off from the orange -face to the left, let us put these successive sections to -the left. We can call them <i>wh</i><sub>0</sub>, <i>wh</i><sub>1</sub>, <i>wh</i><sub>2</sub>, <i>wh</i><sub>3</sub>, <i>wh</i><sub>4</sub>, -because they are sections along the white axis, which -now runs in the unknown dimension.</p> - -<p>Running from the purple square in the white direction -we find the light purple cube. This is represented in the -<span class="pagenum" id="Page_183">[Pg 183]</span>sections <i>wh</i><sub>1</sub>, <i>wh</i><sub>2</sub>, <i>wh</i><sub>3</sub>, <i>wh</i><sub>4</sub>, <a href="#fig_108">fig. 108</a>. It is the same cube -that is represented in the sections <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>: in <a href="#fig_107">fig. 107</a> -the red and white axes are in our space, the blue out of -it; in the other case, the red and blue are in our space, -the white out of it. It is evident that the face pink <i>y</i>, -opposite the pink face in <a href="#fig_107">fig. 107</a>, makes a cube shown -in squares in <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, <i>b</i><sub>4</sub>, on the opposite side to the <i>l</i> -purple squares. Also the light yellow face at the base -of the cube <i>b</i><sub>0</sub>, makes a light green cube, shown as a series -of base squares.</p> - -<p>The same light green cube can be found in <a href="#fig_107">fig. 107</a>. -The base square in <i>wh</i><sub>0</sub> is a green square, for it is enclosed -by blue and yellow axes. From it goes a cube in the -white direction, this is then a light green cube and the -same as the one just mentioned as existing in the sections -<i>b</i><sub>0</sub>, <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, <i>b</i><sub>4</sub>.</p> - -<p>The case is, however, a little different with the brown -cube. This cube we have altogether in space in the -section <i>wh</i><sub>0</sub>, <a href="#fig_108">fig. 108</a>, while it exists as a series of squares, -the left-hand ones, in the sections <i>b</i><sub>0</sub>, <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, <i>b</i><sub>4</sub>. The -brown cube exists as a solid in our space, as shown in -<a href="#fig_108">fig. 108</a>. In the mode of representation of the tesseract -exhibited in <a href="#fig_107">fig. 107</a>, the same brown cube appears as a -succession of squares. That is, as the tesseract moves -across space, the brown cube would actually be to us a -square—it would be merely the lasting boundary of another -solid. It would have no thickness at all, only extension -in two dimensions, and its duration would show its solidity -in three dimensions.</p> - -<p>It is obvious that, if there is a four-dimensional space, -matter in three dimensions only is a mere abstraction; all -material objects must then have a slight four-dimensional -thickness. In this case the above statement will undergo -modification. The material cube which is used as the -model of the boundary of a tesseract will have a slight -thickness in the fourth dimension, and when the cube is<span class="pagenum" id="Page_184">[Pg 184]</span> -presented to us in another aspect, it would not be a mere -surface. But it is most convenient to regard the cubes -we use as having no extension at all in the fourth -dimension. This consideration serves to bring out a point -alluded to before, that, if there is a fourth dimension, our -conception of a solid is the conception of a mere abstraction, -and our talking about real three-dimensional objects would -seem to a four-dimensional being as incorrect as a two-dimensional -being’s telling about real squares, real -triangles, etc., would seem to us.</p> - -<p>The consideration of the two views of the brown cube -shows that any section of a cube can be looked at by a -presentation of the cube in a different position in four-dimensional -space. The brown faces in <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, are the -very same brown sections that would be obtained by -cutting the brown cube, <i>wh</i><sub>0</sub>, across at the right distances -along the blue line, as shown in <a href="#fig_108">fig. 108</a>. But as these -sections are placed in the brown cube, <i>wh</i><sub>0</sub>, they come -behind one another in the blue direction. Now, in the -sections <i>wh</i><sub>1</sub>, <i>wh</i><sub>2</sub>, <i>wh</i><sub>3</sub>, we are looking at these sections -from the white direction—the blue direction does not -exist in these figures. So we see them in a direction at -right angles to that in which they occur behind one -another in <i>wh</i><sub>0</sub>. There are intermediate views, which -would come in the rotation of a tesseract. These brown -squares can be looked at from directions intermediate -between the white and blue axes. It must be remembered -that the fourth dimension is perpendicular equally to all -three space axes. Hence we must take the combinations -of the blue axis, with each two of our three axes, white, -red, yellow, in turn.</p> - -<p>In <a href="#fig_109">fig. 109</a> we take red, white, and blue axes in space, -sending yellow into the fourth dimension. If it goes into -the positive sense of the fourth dimension the blue line -will come in the opposite direction to that in which the<span class="pagenum" id="Page_185">[Pg 185]</span> -yellow line ran before. Hence, the cube determined by -the white, red, blue axes, will start from the pink plane -and run towards us. The dotted cube shows where the -ochre cube was. When it is turned out of space, the cube -coming towards from its front face is the one which comes -into our space in this turning. Since the yellow line now -runs in the unknown dimension we call the sections -<i>y</i><sub>0</sub>, <i>y</i><sub>1</sub>, <i>y</i><sub>2</sub>, <i>y</i><sub>3</sub>, <i>y</i><sub>4</sub>, as they are made at distances 0, 1, 2, 3, 4, -quarter inches along the yellow line. We suppose these -cubes arranged in a line coming towards us—not that -that is any more natural than any other arbitrary series -of positions, but it agrees with the plan previously adopted.</p> - -<div class="figcenter illowp100" id="fig_109" style="max-width: 62.5em;"> - <img src="images/fig_109.png" alt="" /> - <div class="caption">Fig. 109.</div> -</div> - -<p>The interior of the first cube, <i>y</i><sub>0</sub>, is that derived from -pink by adding blue, or, as we call it, light purple. The -faces of the cube are light blue, purple, pink. As drawn, -we can only see the face nearest to us, which is not the -one from which the cube starts—but the face on the -opposite side has the same colour name as the face -towards us.</p> - -<p>The successive sections of the series, <i>y</i><sub>0</sub>, <i>y</i><sub>1</sub>, <i>y</i><sub>2</sub>, etc., can -be considered as derived from sections of the <i>b</i><sub>0</sub> cube -made at distances along the yellow axis. What is distant -a quarter inch from the pink face in the yellow direction? -This question is answered by taking a section from a point -a quarter inch along the yellow axis in the cube <i>b</i><sub>0</sub>, <a href="#fig_107">fig. 107</a>. -It is an ochre section with lines orange and light yellow. -This section will therefore take the place of the pink face<span class="pagenum" id="Page_186">[Pg 186]</span> -in <i>y</i><sub>1</sub> when we go on in the yellow direction. Thus, the -first section, <i>y</i><sub>1</sub>, will begin from an ochre face with light -yellow and orange lines. The colour of the axis which -lies in space towards us is blue, hence the regions of this -section-cube are determined in nomenclature, they will be -found in full in <a href="#fig_105">fig. 105</a>.</p> - -<p>There remains only one figure to be drawn, and that is -the one in which the red axis is replaced by the blue. -Here, as before, if the red axis goes out into the positive -sense of the fourth dimension, the blue line must come -into our space in the negative sense of the direction which -the red line has left. Accordingly, the first cube will -come in beneath the position of our ochre cube, the one -we have been in the habit of starting with.</p> - -<div class="figcenter illowp100" id="fig_110" style="max-width: 62.5em;"> - <img src="images/fig_110.png" alt="" /> - <div class="caption">Fig. 110.</div> -</div> - -<p>To show these figures we must suppose the ochre cube -to be on a movable stand. When the red line swings out -into the unknown dimension, and the blue line comes in -downwards, a cube appears below the place occupied by -the ochre cube. The dotted cube shows where the ochre -cube was. That cube has gone and a different cube runs -downwards from its base. This cube has white, yellow, -and blue axes. Its top is a light yellow square, and hence -its interior is light yellow + blue or light green. Its front -face is formed by the white line moving along the blue -axis, and is therefore light blue, the left-hand side is -formed by the yellow line moving along the blue axis, and -therefore green.</p> - -<p><span class="pagenum" id="Page_187">[Pg 187]</span></p> - -<p>As the red line now runs in the fourth dimension, the -successive sections can he called <i>r</i><sub>0</sub>, <i>r</i><sub>1</sub>, <i>r</i><sub>2</sub>, <i>r</i><sub>3</sub>, <i>r</i><sub>4</sub>, these -letters indicating that at distances 0, 1/4, 2/4, 3/4, 1 inch along -the red axis we take all of the tesseract that can be found -in a three-dimensional space, this three-dimensional space -extending not at all in the fourth dimension, but up and -down, right and left, far and near.</p> - -<p>We can see what should replace the light yellow face of -<i>r</i><sub>0</sub>, when the section <i>r</i><sub>1</sub> comes in, by looking at the cube -<i>b</i><sub>0</sub>, <a href="#fig_107">fig. 107</a>. What is distant in it one-quarter of an inch -from the light yellow face in the red direction? It is an -ochre section with orange and pink lines and red points; -see also <a href="#fig_103">fig. 103</a>.</p> - -<p>This square then forms the top square of <i>r</i><sub>1</sub>. Now we -can determine the nomenclature of all the regions of <i>r</i><sub>1</sub> by -considering what would be formed by the motion of this -square along a blue axis.</p> - -<p>But we can adopt another plan. Let us take a horizontal -section of <i>r</i><sub>0</sub>, and finding that section in the figures, -of <a href="#fig_107">fig. 107</a> or <a href="#fig_103">fig. 103</a>, from them determine what will -replace it, going on in the red direction.</p> - -<p>A section of the <i>r</i><sub>0</sub> cube has green, light blue, green, -light blue sides and blue points.</p> - -<p>Now this square occurs on the base of each of the -section figures, <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, etc. In them we see that 1/4 inch in -the red direction from it lies a section with brown and -light purple lines and purple corners, the interior being -of light brown. Hence this is the nomenclature of the -section which in <i>r</i><sub>1</sub> replaces the section of <i>r</i><sub>0</sub> made from a -point along the blue axis.</p> - -<p>Hence the colouring as given can be derived.</p> - -<p>We have thus obtained a perfectly named group of -tesseracts. We can take a group of eighty-one of them -3 × 3 × 3 × 3, in four dimensions, and each tesseract will -have its name null, red, white, yellow, blue, etc., and<span class="pagenum" id="Page_188">[Pg 188]</span> -whatever cubic view we take of them we can say exactly -what sides of the tesseracts we are handling, and how -they touch each other.<a id="FNanchor_5" href="#Footnote_5" class="fnanchor">[5]</a></p> - -<div class="footnotes"> -<div class="footnote"> - -<p><a id="Footnote_5" href="#FNanchor_5" class="label">[5]</a> At this point the reader will find it advantageous, if he has the -models, to go through the manipulations described in the appendix.</p> - -</div></div> - -<p>Thus, for instance, if we have the sixteen tesseracts -shown below, we can ask how does null touch blue.</p> - -<div class="figcenter illowp100" id="fig_111" style="max-width: 62.5em;"> - <img src="images/fig_111.png" alt="" /> - <div class="caption">Fig. 111.</div> -</div> - -<p>In the arrangement given in <a href="#fig_111">fig. 111</a> we have the axes -white, red, yellow, in space, blue running in the fourth -dimension. Hence we have the ochre cubes as bases. -Imagine now the tesseractic group to pass transverse to -our space—we have first of all null ochre cube, white -ochre cube, etc.; these instantly vanish, and we get the -section shown in the middle cube in <a href="#fig_103">fig. 103</a>, and finally, -just when the tesseract block has moved one inch transverse -to our space, we have null ochre cube, and then -immediately afterwards the ochre cube of blue comes in. -Hence the tesseract null touches the tesseract blue by its -ochre cube, which is in contact, each and every point -of it, with the ochre cube of blue.</p> - -<p>How does null touch white, we may ask? Looking at -the beginning A, <a href="#fig_111">fig. 111</a>, where we have the ochre<span class="pagenum" id="Page_189">[Pg 189]</span> -cubes, we see that null ochre touches white ochre by an -orange face. Now let us generate the null and white -tesseracts by a motion in the blue direction of each of -these cubes. Each of them generates the corresponding -tesseract, and the plane of contact of the cubes generates -the cube by which the tesseracts are in contact. Now an -orange plane carried along a blue axis generates a brown -cube. Hence null touches white by a brown cube.</p> - -<div class="figcenter illowp100" id="fig_112" style="max-width: 62.5em;"> - <img src="images/fig_112.png" alt="" /> - <div class="caption">Fig. 112.</div> -</div> - -<p>If we ask again how red touches light blue tesseract, -let us rearrange our group, <a href="#fig_112">fig. 112</a>, or rather turn it -about so that we have a different space view of it; let -the red axis and the white axis run up and right, and let -the blue axis come in space towards us, then the yellow -axis runs in the fourth dimension. We have then two -blocks in which the bounding cubes of the tesseracts are -given, differently arranged with regard to us—the arrangement -is really the same, but it appears different to us. -Starting from the plane of the red and white axes we -have the four squares of the null, white, red, pink tesseracts -as shown in A, on the red, white plane, unaltered, only -from them now comes out towards us the blue axis.<span class="pagenum" id="Page_190">[Pg 190]</span> -Hence we have null, white, red, pink tesseracts in contact -with our space by their cubes which have the red, white, -blue axis in them, that is by the light purple cubes. -Following on these four tesseracts we have that which -comes next to them in the blue direction, that is the -four blue, light blue, purple, light purple. These are -likewise in contact with our space by their light purple -cubes, so we see a block as named in the figure, of which -each cube is the one determined by the red, white, blue, -axes.</p> - -<p>The yellow line now runs out of space; accordingly one -inch on in the fourth dimension we come to the tesseracts -which follow on the eight named in C, <a href="#fig_112">fig. 112</a>, in the -yellow direction.</p> - -<p>These are shown in C.y<sub>1</sub>, <a href="#fig_112">fig. 112</a>. Between figure C -and C.y<sub>1</sub> is that four-dimensional mass which is formed -by moving each of the cubes in C one inch in the fourth -dimension—that is, along a yellow axis; for the yellow -axis now runs in the fourth dimension.</p> - -<p>In the block C we observe that red (light purple -cube) touches light blue (light purple cube) by a point. -Now these two cubes moving together remain in contact -during the period in which they trace out the tesseracts -red and light blue. This motion is along the yellow -axis, consequently red and light blue touch by a yellow -line.</p> - -<p>We have seen that the pink face moved in a yellow -direction traces out a cube; moved in the blue direction it -also traces out a cube. Let us ask what the pink face -will trace out if it is moved in a direction within the -tesseract lying equally between the yellow and blue -directions. What section of the tesseract will it make?</p> - -<p>We will first consider the red line alone. Let us take -a cube with the red line in it and the yellow and blue -axes.</p> - -<p><span class="pagenum" id="Page_191">[Pg 191]</span></p> - -<div class="figleft illowp35" id="fig_113" style="max-width: 15.625em;"> - <img src="images/fig_113.png" alt="" /> - <div class="caption">Fig. 113.</div> -</div> - -<p>The cube with the yellow, red, blue axes is shown in -<a href="#fig_113">fig. 113</a>. If the red line is -moved equally in the yellow and -in the blue direction by four -equal motions of ¼ inch each, it -takes the positions 11, 22, 33, -and ends as a red line.</p> - -<p>Now, the whole of this red, -yellow, blue, or brown cube appears -as a series of faces on the -successive sections of the tesseract -starting from the ochre cube and letting the blue -axis run in the fourth dimension. Hence the plane -traced out by the red line appears as a series of lines in -the successive sections, in our ordinary way of representing -the tesseract; these lines are in different places in each -successive section.</p> - -<div class="figcenter illowp100" id="fig_114" style="max-width: 62.5em;"> - <img src="images/fig_114.png" alt="" /> - <div class="caption">Fig. 114.</div> -</div> - -<p>Thus drawing our initial cube and the successive -sections, calling them <i>b</i><sub>0</sub>, <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, <i>b</i><sub>4</sub>, <a href="#fig_115">fig. 115</a>, we have -the red line subject to this movement appearing in the -positions indicated.</p> - -<p>We will now investigate what positions in the tesseract -another line in the pink face assumes when it is moved in -a similar manner.</p> - -<p>Take a section of the original cube containing a vertical -line, 4, in the pink plane, <a href="#fig_115">fig. 115</a>. We have, in the -section, the yellow direction, but not the blue.</p> - -<p><span class="pagenum" id="Page_192">[Pg 192]</span></p> - -<p>From this section a cube goes off in the fourth dimension, -which is formed by moving each point of the section -in the blue direction.</p> - -<div class="figleft illowp40" id="fig_115" style="max-width: 15.625em;"> - <img src="images/fig_115.png" alt="" /> - <div class="caption">Fig. 115.</div> -</div> - -<div class="figright illowp40" id="fig_116" style="max-width: 18.75em;"> - <img src="images/fig_116.png" alt="" /> - <div class="caption">Fig. 116.</div> -</div> - -<p>Drawing this cube we have <a href="#fig_116">fig. 116</a>.</p> - -<p>Now this cube occurs as a series of sections in our -original representation of the tesseract. Taking four steps -as before this cube appears as the sections drawn in <i>b</i><sub>0</sub>, <i>b</i><sub>1</sub>, -<i>b</i><sub>2</sub>, <i>b</i><sub>3</sub>, <i>b</i><sub>4</sub>, <a href="#fig_117">fig. 117</a>, and if the line 4 is subjected to a -movement equal in the blue and yellow directions, it will -occupy the positions designated by 4, 4<sub>1</sub>, 4<sub>2</sub>, 4<sub>3</sub>, 4<sub>4</sub>.</p> - -<div class="figcenter illowp100" id="fig_117" style="max-width: 62.5em;"> - <img src="images/fig_117.png" alt="" /> - <div class="caption">Fig. 117.</div> -</div> - -<p>Hence, reasoning in a similar manner about every line, -it is evident that, moved equally in the blue and yellow -directions, the pink plane will trace out a space which is -shown by the series of section planes represented in the -diagram.</p> - -<p>Thus the space traced out by the pink face, if it is -moved equally in the yellow and blue directions, is represented -by the set of planes delineated in Fig. 118, pink<span class="pagenum" id="Page_193">[Pg 193]</span> -face or 0, then 1, 2, 3, and finally pink face or 4. This -solid is a diagonal solid of the tesseract, running from a -pink face to a pink face. Its length is the length of the -diagonal of a square, its side is a square.</p> - -<p>Let us now consider the unlimited space which springs -from the pink face extended.</p> - -<p>This space, if it goes off in the yellow direction, gives -us in it the ochre cube of the tesseract. Thus, if we have -the pink face given and a point in the ochre cube, we -have determined this particular space.</p> - -<p>Similarly going off from the pink face in the blue -direction is another space, which gives us the light purple -cube of the tesseract in it. And any point being taken in -the light purple cube, this space going off from the pink -face is fixed.</p> - -<div class="figcenter illowp100" id="fig_118" style="max-width: 62.5em;"> - <img src="images/fig_118.png" alt="" /> - <div class="caption">Fig. 118.</div> -</div> - -<p>The space we are speaking of can be conceived as -swinging round the pink face, and in each of its positions -it cuts out a solid figure from the tesseract, one of which -we have seen represented in <a href="#fig_118">fig. 118</a>.</p> - -<p>Each of these solid figures is given by one position of -the swinging space, and by one only. Hence in each of -them, if one point is taken, the particular one of the -slanting spaces is fixed. Thus we see that given a plane -and a point out of it a space is determined.</p> - -<p>Now, two points determine a line.</p> - -<p>Again, think of a line and a point outside it. Imagine -a plane rotating round the line. At some time in its -rotation it passes through the point. Thus a line and a<span class="pagenum" id="Page_194">[Pg 194]</span> -point, or three points, determine a plane. And finally -four points determine a space. We have seen that a -plane and a point determine a space, and that three -points determine a plane; so four points will determine -a space.</p> - -<p>These four points may be any points, and we can take, -for instance, the four points at the extremities of the red, -white, yellow, blue axes, in the tesseract. These will -determine a space slanting with regard to the section -spaces we have been previously considering. This space -will cut the tesseract in a certain figure.</p> - -<p>One of the simplest sections of a cube by a plane is -that in which the plane passes through the extremities -of the three edges which meet in a point. We see at -once that this plane would cut the cube in a triangle, but -we will go through the process by which a plane being -would most conveniently treat the problem of the determination -of this shape, in order that we may apply the -method to the determination of the figure in which a -space cuts a tesseract when it passes through the 4 -points at unit distance from a corner.</p> - -<p>We know that two points determine a line, three points -determine a plane, and given any two points in a plane -the line between them lies wholly in the plane.</p> -<div class="figleft illowp40" id="fig_119" style="max-width: 18.75em;"> - <img src="images/fig_119.png" alt="" /> - <div class="caption">Fig. 119.</div> -</div> - -<p>Let now the plane being study the section made by -a plane passing through the -null <i>r</i>, null <i>wh</i>, and null <i>y</i> -points, <a href="#fig_119">fig. 119</a>. Looking at -the orange square, which, as -usual, we suppose to be initially -in his plane, he sees -that the line from null <i>r</i> to -null <i>y</i>, which is a line in the -section plane, the plane, namely, through the three -extremities of the edges meeting in null, cuts the orange<span class="pagenum" id="Page_195">[Pg 195]</span> -face in an orange line with null points. This then is one -of the boundaries of the section figure.</p> - -<p>Let now the cube be so turned that the pink face -comes in his plane. The points null <i>r</i> and null <i>wh</i> -are now visible. The line between them is pink -with null points, and since this line is common to -the surface of the cube and the cutting plane, it is -a boundary of the figure in which the plane cuts the -cube.</p> - -<p>Again, suppose the cube turned so that the light -yellow face is in contact with the plane being’s plane. -He sees two points, the null <i>wh</i> and the null <i>y</i>. The -line between these lies in the cutting plane. Hence, -since the three cutting lines meet and enclose a portion -of the cube between them, he has determined the -figure he sought. It is a triangle with orange, pink, -and light yellow sides, all equal, and enclosing an -ochre area.</p> - -<p>Let us now determine in what figure the space, -determined by the four points, null <i>r</i>, null <i>y</i>, null -<i>wh</i>, null <i>b</i>, cuts the tesseract. We can see three -of these points in the primary position of the tesseract -resting against our solid sheet by the ochre cube. -These three points determine a plane which lies in -the space we are considering, and this plane cuts -the ochre cube in a triangle, the interior of which -is ochre (<a href="#fig_119">fig. 119</a> will serve for this view), with pink, -light yellow and orange sides, and null points. Going -in the fourth direction, in one sense, from this plane -we pass into the tesseract, in the other sense we pass -away from it. The whole area inside the triangle is -common to the cutting plane we see, and a boundary -of the tesseract. Hence we conclude that the triangle -drawn is common to the tesseract and the cutting -space.</p> - -<p><span class="pagenum" id="Page_196">[Pg 196]</span></p> - -<div class="figleft illowp50" id="fig_120" style="max-width: 21.875em;"> - <img src="images/fig_120.png" alt="" /> - <div class="caption">Fig. 120.</div> -</div> - -<p>Now let the ochre cube turn out and the brown cube -come in. The dotted lines -show the position the ochre -cube has left (<a href="#fig_120">fig. 120</a>).</p> - -<p>Here we see three out -of the four points through -which the cutting plane -passes, null <i>r</i>, null <i>y</i>, and -null <i>b</i>. The plane they -determine lies in the cutting space, and this plane -cuts out of the brown cube a triangle with orange, -purple and green sides, and null points. The orange -line of this figure is the same as the orange line in -the last figure.</p> - -<p>Now let the light purple cube swing into our space, -towards us, <a href="#fig_121">fig. 121</a>.</p> - -<div class="figleft illowp40" id="fig_121" style="max-width: 21.875em;"> - <img src="images/fig_121.png" alt="" /> - <div class="caption">Fig. 121.</div> -</div> - -<p>The cutting space which passes through the four points, -null <i>r</i>, <i>y</i>, <i>wh</i>, <i>b</i>, passes through -the null <i>r</i>, <i>wh</i>, <i>b</i>, and therefore -the plane these determine -lies in the cutting space.</p> - -<p>This triangle lies before us. -It has a light purple interior -and pink, light blue, and -purple edges with null points.</p> - -<p>This, since it is all of the -plane that is common to it, and this bounding of the -tesseract, gives us one of the bounding faces of our sectional -figure. The pink line in it is the same as the -pink line we found in the first figure—that of the ochre -cube.</p> - -<p>Finally, let the tesseract swing about the light yellow -plane, so that the light green cube comes into our space. -It will point downwards.</p> - -<div class="figleft illowp40" id="fig_122" style="max-width: 21.875em;"> - <img src="images/fig_122.png" alt="" /> - <div class="caption">Fig. 122.</div> -</div> - -<p>The three points, <i>n.y</i>, <i>n.wh</i>, <i>n.b</i>, are in the cutting<span class="pagenum" id="Page_197">[Pg 197]</span> -space, and the triangle they determine is common to -the tesseract and the cutting -space. Hence this -boundary is a triangle having -a light yellow line, -which is the same as the -light yellow line of the first -figure, a light blue line and -a green line.</p> - -<p>We have now traced the -cutting space between every -set of three that can be -made out of the four points -in which it cuts the tesseract, and have got four faces -which all join on to each other by lines.</p> - -<div class="figleft illowp35" id="fig_123" style="max-width: 18.75em;"> - <img src="images/fig_123.png" alt="" /> - <div class="caption">Fig. 123.</div> -</div> - -<p>The triangles are shown in <a href="#fig_123">fig. 123</a> as they join on to -the triangle in the ochre cube. But -they join on each to the other in an -exactly similar manner; their edges -are all identical two and two. They -form a closed figure, a tetrahedron, -enclosing a light brown portion which -is the portion of the cutting space -which lies inside the tesseract.</p> - -<p>We cannot expect to see this light brown portion, any -more than a plane being could expect to see the inside -of a cube if an angle of it were pushed through his -plane. All he can do is to come upon the boundaries -of it in a different way to that in which he would if it -passed straight through his plane.</p> - -<p>Thus in this solid section; the whole interior lies perfectly -open in the fourth dimension. Go round it as -we may we are simply looking at the boundaries of the -tesseract which penetrates through our solid sheet. If -the tesseract were not to pass across so far, the triangle<span class="pagenum" id="Page_198">[Pg 198]</span> -would be smaller; if it were to pass farther, we should -have a different figure, the outlines of which can be -determined in a similar manner.</p> - -<p>The preceding method is open to the objection that -it depends rather on our inferring what must be, than -our seeing what is. Let us therefore consider our -sectional space as consisting of a number of planes, each -very close to the last, and observe what is to be found -in each plane.</p> - -<div class="figleft illowp40" id="fig_124" style="max-width: 21.875em;"> - <img src="images/fig_124.png" alt="" /> - <div class="caption">Fig. 124.</div> -</div> - -<p>The corresponding method in the case of two dimensions -is as follows:—The plane -being can see that line of the -sectional plane through null <i>y</i>, -null <i>wh</i>, null <i>r</i>, which lies in -the orange plane. Let him -now suppose the cube and the -section plane to pass half way -through his plane. Replacing -the red and yellow axes are lines parallel to them, sections -of the pink and light yellow faces.</p> - -<p>Where will the section plane cut these parallels to -the red and yellow axes?</p> - -<p>Let him suppose the cube, in the position of the -drawing, <a href="#fig_124">fig. 124</a>, turned so that the pink face lies -against his plane. He can see the line from the null <i>r</i> -point to the null <i>wh</i> point, and can see (compare <a href="#fig_119">fig. 119</a>) -that it cuts <span class="allsmcap">AB</span> a parallel to his red axis, drawn at a point -half way along the white line, in a point <span class="allsmcap">B</span>, half way up. -I shall speak of the axis as having the length of an edge -of the cube. Similarly, by letting the cube turn so that -the light yellow square swings against his plane, he can -see (compare <a href="#fig_119">fig. 119</a>) that a parallel to his yellow axis -drawn from a point half-way along the white axis, is cut -at half its length by the trace of the section plane in the -light yellow face.</p> - -<p><span class="pagenum" id="Page_199">[Pg 199]</span></p> - -<p>Hence when the cube had passed half-way through he -would have—instead of the orange line with null points, -which he had at first—an ochre line of half its length, -with pink and light yellow points. Thus, as the cube -passed slowly through his plane, he would have a succession -of lines gradually diminishing in length and -forming an equilateral triangle. The whole interior would -be ochre, the line from which it started would be orange. -The succession of points at the ends of the succeeding -lines would form pink and light yellow lines and the -final point would be null. Thus looking at the successive -lines in the section plane as it and the cube passed across -his plane he would determine the figure cut out bit -by bit.</p> - -<p>Coming now to the section of the tesseract, let us -imagine that the tesseract and its cutting <i>space</i> pass -slowly across our space; we can examine portions of it, -and their relation to portions of the cutting space. Take -the section space which passes through the four points, -null <i>r</i>, <i>wh</i>, <i>y</i>, <i>b</i>; we can see in the ochre cube (<a href="#fig_119">fig. 119</a>) -the plane belonging to this section space, which passes -through the three extremities of the red, white, yellow -axes.</p> - -<p>Now let the tesseract pass half way through our space. -Instead of our original axes we have parallels to them, -purple, light blue, and green, each of the same length as -the first axes, for the section of the tesseract is of exactly -the same shape as its ochre cube.</p> - -<p>But the sectional space seen at this stage of the transference -would not cut the section of the tesseract in a -plane disposed as at first.</p> - -<p>To see where the sectional space would cut these -parallels to the original axes let the tesseract swing so -that, the orange face remaining stationary, the blue line -comes in to the left.</p> - -<p><span class="pagenum" id="Page_200">[Pg 200]</span></p> - -<div class="figleft illowp45" id="fig_125" style="max-width: 25em;"> - <img src="images/fig_125.png" alt="" /> - <div class="caption">Fig. 125.</div> -</div> - -<p>Here (<a href="#fig_125">fig. 125</a>) we have the null <i>r</i>, <i>y</i>, <i>b</i> points, and of -the sectional space all we -see is the plane through these -three points in it.</p> - -<p>In this figure we can draw -the parallels to the red and -yellow axes and see that, if -they started at a point half -way along the blue axis, they -would each be cut at a point so as to be half of their -previous length.</p> - -<p>Swinging the tesseract into our space about the pink -face of the ochre cube we likewise find that the parallel -to the white axis is cut at half its length by the sectional -space.</p> - -<div class="figleft illowp40" id="fig_126" style="max-width: 25em;"> - <img src="images/fig_126.png" alt="" /> - <div class="caption">Fig. 126.</div> -</div> - -<p>Hence in a section made when the tesseract had passed -half across our space the parallels to the red, white, yellow -axes, which are now in our -space, are cut by the section -space, each of them half way -along, and for this stage of -the traversing motion we -should have <a href="#fig_126">fig. 126</a>. The -section made of this cube by -the plane in which the sectional -space cuts it, is an -equilateral triangle with purple, l. blue, green points, and -l. purple, brown, l. green lines.</p> - -<p>Thus the original ochre triangle, with null points and -pink, orange, light yellow lines, would be succeeded by a -triangle coloured in manner just described.</p> - -<p>This triangle would initially be only a very little smaller -than the original triangle, it would gradually diminish, -until it ended in a point, a null point. Each of its -edges would be of the same length. Thus the successive<span class="pagenum" id="Page_201">[Pg 201]</span> -sections of the successive planes into which we analyse the -cutting space would be a tetrahedron of the description -shown (<a href="#fig_123">fig. 123</a>), and the whole interior of the tetrahedron -would be light brown.</p> - -<div class="figcenter illowp100" id="fig_127" style="max-width: 50em;"> - <img src="images/fig_127.png" alt="" /> - <div class="caption">Front view. <span class="gap8l"> The rear faces.</span><br /> -Fig. 127.</div> -</div> - - -<p>In <a href="#fig_127">fig. 127</a> the tetrahedron is represented by means of -its faces as two triangles which meet in the p. line, and -two rear triangles which join on to them, the diagonal -of the pink face being supposed to run vertically -upward.</p> - -<p>We have now reached a natural termination. The -reader may pursue the subject in further detail, but will -find no essential novelty. I conclude with an indication -as to the manner in which figures previously given may -be used in determining sections by the method developed -above.</p> - -<p>Applying this method to the tesseract, as represented -in Chapter IX., sections made by a space cutting the axes -equidistantly at any distance can be drawn, and also the -sections of tesseracts arranged in a block.</p> - -<p>If we draw a plane, cutting all four axes at a point -six units distance from null, we have a slanting space. -This space cuts the red, white, yellow axes in the<span class="pagenum" id="Page_202">[Pg 202]</span> -points <span class="allsmcap">LMN</span> (<a href="#fig_128">fig. 128</a>), and so in the region of our space -before we go off into -the fourth dimension, -we have the plane -represented by <span class="allsmcap">LMN</span> -extended. This is what -is common to the -slanting space and our -space.</p> - -<div class="figleft illowp50" id="fig_128" style="max-width: 31.25em;"> - <img src="images/fig_128.png" alt="" /> - <div class="caption">Fig. 128.</div> -</div> - -<p>This plane cuts the -ochre cube in the triangle <span class="allsmcap">EFG</span>.</p> - -<p>Comparing this with (<a href="#fig_72">fig. 72</a>) <i>oh</i>, we see that the -hexagon there drawn is part of the triangle <span class="allsmcap">EFG</span>.</p> - -<p>Let us now imagine the tesseract and the slanting -space both together to pass transverse to our space, a -distance of one unit, we have in 1<i>h</i> a section of the -tesseract, whose axes are parallels to the previous axes. -The slanting space cuts them at a distance of five units -along each. Drawing the plane through these points in -1<i>h</i> it will be found to cut the cubical section of the -tesseract in the hexagonal figure drawn. In 2<i>h</i> (<a href="#fig_72">fig. 72</a>) the -slanting space cuts the parallels to the axes at a distance -of four along each, and the hexagonal figure is the section -of this section of the tesseract by it. Finally when 3<i>h</i> -comes in the slanting space cuts the axes at a distance -of three along each, and the section is a triangle, of which -the hexagon drawn is a truncated portion. After this -the tesseract, which extends only three units in each of -the four dimensions, has completely passed transverse -of our space, and there is no more of it to be cut. Hence, -putting the plane sections together in the right relations, -we have the section determined by the particular slanting -space: namely an octahedron.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_203">[Pg 203]</span></p> - -<h2 class="nobreak" id="CHAPTER_XIV">CHAPTER XIV.<a id="FNanchor_6" href="#Footnote_6" class="fnanchor">[6]</a><br /> - -<small><i>A RECAPITULATION AND EXTENSION OF -THE PHYSICAL ARGUMENT</i></small></h2></div> - -<div class="footnotes"> -<div class="footnote"> - -<p><a id="Footnote_6" href="#FNanchor_6" class="label">[6]</a> The contents of this chapter are taken from a paper read before -the Philosophical Society of Washington. The mathematical portion -of the paper has appeared in part in the Proceedings of the Royal -Irish Academy under the title, “Cayley’s formulæ of orthogonal -transformation,” Nov. 29th, 1903.</p> - -</div></div> - -<p>There are two directions of inquiry in which the -research for the physical reality of a fourth dimension -can be prosecuted. One is the investigation of the -infinitely great, the other is the investigation of the -infinitely small.</p> - -<p>By the measurement of the angles of vast triangles, -whose sides are the distances between the stars, astronomers -have sought to determine if there is any deviation from -the values given by geometrical deduction. If the angles -of a celestial triangle do not together equal two right -angles, there would be an evidence for the physical reality -of a fourth dimension.</p> - -<p>This conclusion deserves a word of explanation. If -space is really four-dimensional, certain conclusions follow -which must be brought clearly into evidence if we are to -frame the questions definitely which we put to Nature. -To account for our limitation let us assume a solid material -sheet against which we move. This sheet must stretch -alongside every object in every direction in which it -visibly moves. Every material body must slip or slide -along this sheet, not deviating from contact with it in -any motion which we can observe.</p> - -<p><span class="pagenum" id="Page_204">[Pg 204]</span></p> - -<p>The necessity for this assumption is clearly apparent, if -we consider the analogous case of a suppositionary plane -world. If there were any creatures whose experiences -were confined to a plane, we must account for their -limitation. If they were free to move in every space -direction, they would have a three-dimensional motion; -hence they must be physically limited, and the only way -in which we can conceive such a limitation to exist is by -means of a material surface against which they slide. -The existence of this surface could only be known to -them indirectly. It does not lie in any direction from -them in which the kinds of motion they know of leads -them. If it were perfectly smooth and always in contact -with every material object, there would be no difference in -their relations to it which would direct their attention to it.</p> - -<p>But if this surface were curved—if it were, say, in the -form of a vast sphere—the triangles they drew would -really be triangles of a sphere, and when these triangles -are large enough the angles diverge from the magnitudes -they would have for the same lengths of sides if the -surface were plane. Hence by the measurement of -triangles of very great magnitude a plane being might -detect a difference from the laws of a plane world in his -physical world, and so be led to the conclusion that there -was in reality another dimension to space—a third -dimension—as well as the two which his ordinary experience -made him familiar with.</p> - -<p>Now, astronomers have thought it worth while to -examine the measurements of vast triangles drawn from -one celestial body to another with a view to determine if -there is anything like a curvature in our space—that is to -say, they have tried astronomical measurements to find<span class="pagenum" id="Page_205">[Pg 205]</span> -out if the vast solid sheet against which, on the supposition -of a fourth dimension, everything slides is -curved or not. These results have been negative. The -solid sheet, if it exists, is not curved or, being curved, has -not a sufficient curvature to cause any observable deviation -from the theoretical value of the angles calculated.</p> - -<p>Hence the examination of the infinitely great leads to -no decisive criterion. If it did we should have to decide -between the present theory and that of metageometry.</p> - -<p>Coming now to the prosecution of the inquiry in the -direction of the infinitely small, we have to state the -question thus: Our laws of movement are derived from -the examination of bodies which move in three-dimensional -space. All our conceptions are founded on the supposition -of a space which is represented analytically by -three independent axes and variations along them—that -is, it is a space in which there are three independent -movements. Any motion possible in it can be compounded -out of these three movements, which we may call: up, -right, away.</p> - -<p>To examine the actions of the very small portions of -matter with the view of ascertaining if there is any -evidence in the phenomena for the supposition of a fourth -dimension of space, we must commence by clearly defining -what the laws of mechanics would be on the supposition -of a fourth dimension. It is of no use asking if the -phenomena of the smallest particles of matter are like—we -do not know what. We must have a definite conception -of what the laws of motion would be on the -supposition of the fourth dimension, and then inquire if -the phenomena of the activity of the smaller particles of -matter resemble the conceptions which we have elaborated.</p> - -<p>Now, the task of forming these conceptions is by no -means one to be lightly dismissed. Movement in space -has many features which differ entirely from movement<span class="pagenum" id="Page_206">[Pg 206]</span> -on a plane; and when we set about to form the conception -of motion in four dimensions, we find that there -is at least as great a step as from the plane to three-dimensional -space.</p> - -<p>I do not say that the step is difficult, but I want to -point out that it must be taken. When we have formed -the conception of four-dimensional motion, we can ask a -rational question of Nature. Before we have elaborated -our conceptions we are asking if an unknown is like an -unknown—a futile inquiry.</p> - -<p>As a matter of fact, four-dimensional movements are in -every way simple and more easy to calculate than three-dimensional -movements, for four-dimensional movements -are simply two sets of plane movements put together.</p> - -<p>Without the formation of an experience of four-dimensional -bodies, their shapes and motions, the subject -can be but formal—logically conclusive, not intuitively -evident. It is to this logical apprehension that I must -appeal.</p> - -<p>It is perfectly simple to form an experiential familiarity -with the facts of four-dimensional movement. The -method is analogous to that which a plane being would -have to adopt to form an experiential familiarity with -three-dimensional movements, and may be briefly -summed up as the formation of a compound sense by -means of which duration is regarded as equivalent to -extension.</p> - -<p>Consider a being confined to a plane. A square enclosed -by four lines will be to him a solid, the interior of which -can only be examined by breaking through the lines. -If such a square were to pass transverse to his plane, it -would immediately disappear. It would vanish, going in -no direction to which he could point.</p> - -<p>If, now, a cube be placed in contact with his plane, its -surface of contact would appear like the square which we<span class="pagenum" id="Page_207">[Pg 207]</span> -have just mentioned. But if it were to pass transverse to -his plane, breaking through it, it would appear as a lasting -square. The three-dimensional matter will give a lasting -appearance in circumstances under which two-dimensional -matter will at once disappear.</p> - -<p>Similarly, a four-dimensional cube, or, as we may call -it, a tesseract, which is generated from a cube by a -movement of every part of the cube in a fourth direction -at right angles to each of the three visible directions in -the cube, if it moved transverse to our space, would -appear as a lasting cube.</p> - -<p>A cube of three-dimensional matter, since it extends to -no distance at all in the fourth dimension, would instantly -disappear, if subjected to a motion transverse to our space. -It would disappear and be gone, without it being possible -to point to any direction in which it had moved.</p> - -<p>All attempts to visualise a fourth dimension are futile. It -must be connected with a time experience in three space.</p> - -<p>The most difficult notion for a plane being to acquire -would be that of rotation about a line. Consider a plane -being facing a square. If he were told that rotation -about a line were possible, he would move his square this -way and that. A square in a plane can rotate about a -point, but to rotate about a line would seem to the plane -being perfectly impossible. How could those parts of his -square which were on one side of an edge come to the -other side without the edge moving? He could understand -their reflection in the edge. He could form an -idea of the looking-glass image of his square lying on the -opposite side of the line of an edge, but by no motion -that he knows of can he make the actual square assume -that position. The result of the rotation would be like -reflection in the edge, but it would be a physical impossibility -to produce it in the plane.</p> - -<p>The demonstration of rotation about a line must be to<span class="pagenum" id="Page_208">[Pg 208]</span> -him purely formal. If he conceived the notion of a cube -stretching out in an unknown direction away from his -plane, then he can see the base of it, his square in the -plane, rotating round a point. He can likewise apprehend -that every parallel section taken at successive intervals in -the unknown direction rotates in like manner round a -point. Thus he would come to conclude that the whole -body rotates round a line—the line consisting of the -succession of points round which the plane sections rotate. -Thus, given three axes, <i>x</i>, <i>y</i>, <i>z</i>, if <i>x</i> rotates to take -the place of <i>y</i>, and <i>y</i> turns so as to point to negative <i>x</i>, -then the third axis remaining unaffected by this turning -is the axis about which the rotation takes place. This, -then, would have to be his criterion of the axis of a -rotation—that which remains unchanged when a rotation -of every plane section of a body takes place.</p> - -<p>There is another way in which a plane being can think -about three-dimensional movements; and, as it affords -the type by which we can most conveniently think about -four-dimensional movements, it will be no loss of time to -consider it in detail.</p> -<div class="figleft illowp30" id="fig_129" style="max-width: 18.75em;"> - <img src="images/fig_129.png" alt="" /> - <div class="caption">Fig. 1 (129).</div> -</div> - -<p>We can represent the plane being and his object by -figures cut out of paper, which slip on a smooth surface. -The thickness of these bodies must be taken as so minute -that their extension in the third dimension -escapes the observation of the -plane being, and he thinks about them -as if they were mathematical plane -figures in a plane instead of being -material bodies capable of moving on -a plane surface. Let <span class="allsmcap">A</span><i>x</i>, <span class="allsmcap">A</span><i>y</i> be two -axes and <span class="allsmcap">ABCD</span> a square. As far as -movements in the plane are concerned, the square can -rotate about a point <span class="allsmcap">A</span>, for example. It cannot rotate -about a side, such as <span class="allsmcap">AC</span>.</p> - -<p><span class="pagenum" id="Page_209">[Pg 209]</span></p> - -<p>But if the plane being is aware of the existence of a -third dimension he can study the movements possible in -the ample space, taking his figure portion by portion.</p> - -<p>His plane can only hold two axes. But, since it can -hold two, he is able to represent a turning into the third -dimension if he neglects one of his axes and represents the -third axis as lying in his plane. He can make a drawing -in his plane of what stands up perpendicularly from his -plane. Let <span class="allsmcap">A</span><i>z</i> be the axis, which -stands perpendicular to his plane at -<span class="allsmcap">A</span>. He can draw in his plane two -lines to represent the two axes, <span class="allsmcap">A</span><i>x</i> -and <span class="allsmcap">A</span><i>z</i>. Let Fig. 2 be this drawing. -Here the <i>z</i> axis has taken -the place of the <i>y</i> axis, and the -plane of <span class="allsmcap">A</span><i>x</i> <span class="allsmcap">A</span><i>z</i> is represented in his -plane. In this figure all that exists of the square <span class="allsmcap">ABCD</span> -will be the line <span class="allsmcap">AB</span>.</p> - -<div class="figleft illowp30" id="fig_130" style="max-width: 18.75em;"> - <img src="images/fig_130.png" alt="" /> - <div class="caption">Fig. 2 (130).</div> -</div> - -<p>The square extends from this line in the <i>y</i> direction, -but more of that direction is represented in Fig. 2. The -plane being can study the turning of the line <span class="allsmcap">AB</span> in this -diagram. It is simply a case of plane turning around the -point <span class="allsmcap">A</span>. The line <span class="allsmcap">AB</span> occupies intermediate portions like <span class="allsmcap">AB</span><sub>1</sub> -and after half a revolution will lie on <span class="allsmcap">A</span><i>x</i> produced through <span class="allsmcap">A</span>.</p> - -<p>Now, in the same way, the plane being can take -another point, <span class="allsmcap">A´</span>, and another line, <span class="allsmcap">A´B´</span>, in his square. -He can make the drawing of the two directions at <span class="allsmcap">A´</span>, one -along <span class="allsmcap">A´B´</span>, the other perpendicular to his plane. He -will obtain a figure precisely similar to Fig. 2, and will -see that, as <span class="allsmcap">AB</span> can turn around <span class="allsmcap">A</span>, so <span class="allsmcap">A´C´</span> around <span class="allsmcap">A</span>.</p> - -<p>In this turning <span class="allsmcap">AB</span> and <span class="allsmcap">A´B´</span> would not interfere with -each other, as they would if they moved in the plane -around the separate points <span class="allsmcap">A</span> and <span class="allsmcap">A´</span>.</p> - -<p>Hence the plane being would conclude that a rotation -round a line was possible. He could see his square as it<span class="pagenum" id="Page_210">[Pg 210]</span> -began to make this turning. He could see it half way -round when it came to lie on the opposite side of the line -<span class="allsmcap">AC</span>. But in intermediate portions he could not see it, -for it runs out of the plane.</p> - -<p>Coming now to the question of a four-dimensional body, -let us conceive of it as a series of cubic sections, the first -in our space, the rest at intervals, stretching away from -our space in the unknown direction.</p> - -<p>We must not think of a four-dimensional body as -formed by moving a three-dimensional body in any -direction which we can see.</p> - -<p>Refer for a moment to Fig. 3. The point <span class="allsmcap">A</span>, moving to -the right, traces out the line <span class="allsmcap">AC</span>. The line <span class="allsmcap">AC</span>, moving -away in a new direction, traces out the square <span class="allsmcap">ACEG</span> at -the base of the cube. The square <span class="allsmcap">AEGC</span>, moving in a -new direction, will trace out the cube <span class="allsmcap">ACEGBDHF</span>. The -vertical direction of this last motion is not identical with -any motion possible in the plane of the base of the cube. -It is an entirely new direction, at right angles to every -line that can be drawn in the base. To trace out a -tesseract the cube must move in a new direction—a -direction at right angles to any and every line that can -be drawn in the space of the cube.</p> - -<p>The cubic sections of the tesseract are related to the -cube we see, as the square sections of the cube are related -to the square of its base which a plane being sees.</p> - -<p>Let us imagine the cube in our space, which is the base -of a tesseract, to turn about one of its edges. The rotation -will carry the whole body with it, and each of the cubic -sections will rotate. The axis we see in our space will -remain unchanged, and likewise the series of axes parallel -to it about which each of the parallel cubic sections -rotates. The assemblage of all of these is a plane.</p> - -<p>Hence in four dimensions a body rotates about a plane. -There is no such thing as rotation round an axis.</p> - -<p><span class="pagenum" id="Page_211">[Pg 211]</span></p> - -<p>We may regard the rotation from a different point of -view. Consider four independent axes each at right -angles to all the others, drawn in a four-dimensional body. -Of these four axes we can see any three. The fourth -extends normal to our space.</p> - -<p>Rotation is the turning of one axis into a second, and -the second turning to take the place of the negative of -the first. It involves two axes. Thus, in this rotation of -a four-dimensional body, two axes change and two remain -at rest. Four-dimensional rotation is therefore a turning -about a plane.</p> - -<p>As in the case of a plane being, the result of rotation -about a line would appear as the production of a looking-glass -image of the original object on the other side of the -line, so to us the result of a four-dimensional rotation -would appear like the production of a looking-glass image -of a body on the other side of a plane. The plane would -be the axis of the rotation, and the path of the body -between its two appearances would be unimaginable in -three-dimensional space.</p> - -<div class="figleft illowp30" id="fig_131" style="max-width: 18.75em;"> - <img src="images/fig_131.png" alt="" /> - <div class="caption">Fig. 3 (131).</div> -</div> - -<p>Let us now apply the method by which a plane being -could examine the nature of rotation -about a line in our examination -of rotation about a plane. Fig. 3 -represents a cube in our space, the -three axes <i>x</i>, <i>y</i>, <i>z</i> denoting its -three dimensions. Let <i>w</i> represent -the fourth dimension. Now, since -in our space we can represent any -three dimensions, we can, if we -choose, make a representation of what is in the space -determined by the three axes <i>x</i>, <i>z</i>, <i>w</i>. This is a three-dimensional -space determined by two of the axes we have -drawn, <i>x</i> and <i>z</i>, and in place of <i>y</i> the fourth axis, <i>w</i>. We -cannot, keeping <i>x</i> and <i>z</i>, have both <i>y</i> and <i>w</i> in our space;<span class="pagenum" id="Page_212">[Pg 212]</span> -so we will let <i>y</i> go and draw <i>w</i> in its place. What will be -our view of the cube?</p> - -<div class="figleft illowp30" id="fig_132" style="max-width: 18.75em;"> - <img src="images/fig_132.png" alt="" /> - <div class="caption">Fig. 4 (132).</div> -</div> - -<p>Evidently we shall have simply the square that is in -the plane of <i>xz</i>, the square <span class="allsmcap">ACDB</span>. -The rest of the cube stretches in -the <i>y</i> direction, and, as we have -none of the space so determined, -we have only the face of the cube. -This is represented in <a href="#fig_132">fig. 4</a>.</p> - -<p>Now, suppose the whole cube to -be turned from the <i>x</i> to the <i>w</i> -direction. Conformably with our method, we will not -take the whole of the cube into consideration at once, but -will begin with the face <span class="allsmcap">ABCD</span>.</p> - -<div class="figleft illowp30" id="fig_133" style="max-width: 18.75em;"> - <img src="images/fig_133.png" alt="" /> - <div class="caption">Fig. 5 (133).</div> -</div> - -<p>Let this face begin to turn. Fig. 5 -represents one of the positions it will -occupy; the line <span class="allsmcap">AB</span> remains on the -<i>z</i> axis. The rest of the face extends -between the <i>x</i> and the <i>w</i> direction.</p> - -<p>Now, since we can take any three -axes, let us look at what lies in -the space of <i>zyw</i>, and examine the -turning there. We must now let the <i>z</i> axis disappear -and let the <i>w</i> axis run in the direction in which the <i>z</i> ran.</p> - -<div class="figleft illowp30" id="fig_134" style="max-width: 18.75em;"> - <img src="images/fig_134.png" alt="" /> - <div class="caption">Fig. 6 (134).</div> -</div> - -<p>Making this representation, what -do we see of the cube? Obviously -we see only the lower face. The rest -of the cube lies in the space of <i>xyz</i>. -In the space of <i>xyz</i> we have merely -the base of the cube lying in the -plane of <i>xy</i>, as shown in <a href="#fig_134">fig. 6</a>.</p> - -<p>Now let the <i>x</i> to <i>w</i> turning take place. The square -<span class="allsmcap">ACEG</span> will turn about the line <span class="allsmcap">AE</span>. This edge will -remain along the <i>y</i> axis and will be stationary, however -far the square turns.</p> - -<p><span class="pagenum" id="Page_213">[Pg 213]</span></p> - -<div class="figleft illowp30" id="fig_135" style="max-width: 18.75em;"> - <img src="images/fig_135.png" alt="" /> - <div class="caption">Fig. 7 (135).</div> -</div> - -<p>Thus, if the cube be turned by an <i>x</i> to <i>w</i> turning, both -the edge <span class="allsmcap">AB</span> and the edge <span class="allsmcap">AC</span> remain -stationary; hence the whole face -<span class="allsmcap">ABEF</span> in the <i>yz</i> plane remains fixed. -The turning has taken place about -the face <span class="allsmcap">ABEF</span>.</p> - -<p>Suppose this turning to continue -till <span class="allsmcap">AC</span> runs to the left from <span class="allsmcap">A</span>. -The cube will occupy the position -shown in <a href="#fig_136">fig. 8</a>. This is the looking-glass image of the -cube in <a href="#fig_131">fig. 3</a>. By no rotation in three-dimensional space -can the cube be brought from -the position in <a href="#fig_131">fig. 3</a> to that -shown in <a href="#fig_136">fig. 8</a>.</p> - -<div class="figleft illowp40" id="fig_136" style="max-width: 21.875em;"> - <img src="images/fig_136.png" alt="" /> - <div class="caption">Fig. 8 (136).</div> -</div> - -<p>We can think of this turning -as a turning of the face <span class="allsmcap">ABCD</span> -about <span class="allsmcap">AB</span>, and a turning of each -section parallel to <span class="allsmcap">ABCD</span> round -the vertical line in which it -intersects the face <span class="allsmcap">ABEF</span>, the -space in which the turning takes place being a different -one from that in which the cube lies.</p> - -<p>One of the conditions, then, of our inquiry in the -direction of the infinitely small is that we form the conception -of a rotation about a plane. The production of a -body in a state in which it presents the appearance of a -looking-glass image of its former state is the criterion -for a four-dimensional rotation.</p> - -<p>There is some evidence for the occurrence of such transformations -of bodies in the change of bodies from those -which produce a right-handed polarisation of light to -those which produce a left-handed polarisation; but this -is not a point to which any very great importance can -be attached.</p> - -<p>Still, in this connection, let me quote a remark from<span class="pagenum" id="Page_214">[Pg 214]</span> -Prof. John G. McKendrick’s address on Physiology before -the British Association at Glasgow. Discussing the -possibility of the hereditary production of characteristics -through the material structure of the ovum, he estimates -that in it there exist 12,000,000,000 biophors, or ultimate -particles of living matter, a sufficient number to account -for hereditary transmission, and observes: “Thus it is -conceivable that vital activities may also be determined -by the kind of motion that takes place in the molecules -of that which we speak of as living matter. It may be -different in kind from some of the motions known to -physicists, and it is conceivable that life may be the -transmission to dead matter, the molecules of which have -already a special kind of motion, of a form of motion -<i>sui generis</i>.”</p> - -<p>Now, in the realm of organic beings symmetrical structures—those -with a right and left symmetry—are everywhere -in evidence. Granted that four dimensions exist, -the simplest turning produces the image form, and by a -folding-over structures could be produced, duplicated -right and left, just as is the case of symmetry in a -plane.</p> - -<p>Thus one very general characteristic of the forms of -organisms could be accounted for by the supposition that -a four-dimensional motion was involved in the process of -life.</p> - -<p>But whether four-dimensional motions correspond in -other respects to the physiologist’s demand for a special -kind of motion, or not, I do not know. Our business is -with the evidence for their existence in physics. For -this purpose it is necessary to examine into the significance -of rotation round a plane in the case of extensible -and of fluid matter.</p> - -<p>Let us dwell a moment longer on the rotation of a rigid -body. Looking at the cube in <a href="#fig_131">fig. 3</a>, which turns about<span class="pagenum" id="Page_215">[Pg 215]</span> -the face of <span class="allsmcap">ABFE</span>, we see that any line in the face can -take the place of the vertical and horizontal lines we have -examined. Take the diagonal line <span class="allsmcap">AF</span> and the section -through it to <span class="allsmcap">GH</span>. The portions of matter which were on -one side of <span class="allsmcap">AF</span> in this section in <a href="#fig_131">fig. 3</a> are on the -opposite side of it in <a href="#fig_136">fig. 8</a>. They have gone round the -line <span class="allsmcap">AF</span>. Thus the rotation round a face can be considered -as a number of rotations of sections round parallel lines -in it.</p> - -<p>The turning about two different lines is impossible in -three-dimensional space. To take another illustration, -suppose <span class="allsmcap">A</span> and <span class="allsmcap">B</span> are two parallel lines in the <i>xy</i> plane, -and let <span class="allsmcap">CD</span> and <span class="allsmcap">EF</span> be two rods crossing them. Now, in -the space of <i>xyz</i> if the rods turn round the lines <span class="allsmcap">A</span> and <span class="allsmcap">B</span> -in the same direction they -will make two independent -circles.</p> - -<div class="figleft illowp40" id="fig_137" style="max-width: 21.875em;"> - <img src="images/fig_137.png" alt="" /> - <div class="caption">Fig. 9 (137).</div> -</div> - -<p>When the end <span class="allsmcap">F</span> is going -down the end <span class="allsmcap">C</span> will be coming -up. They will meet and conflict.</p> - -<p>But if we rotate the rods -about the plane of <span class="allsmcap">AB</span> by the -<i>z</i> to <i>w</i> rotation these movements -will not conflict. Suppose -all the figure removed -with the exception of the plane <i>xz</i>, and from this plane -draw the axis of <i>w</i>, so that we are looking at the space -of <i>xzw</i>.</p> - -<p>Here, <a href="#fig_138">fig. 10</a>, we cannot see the lines <span class="allsmcap">A</span> and <span class="allsmcap">B</span>. We -see the points <span class="allsmcap">G</span> and <span class="allsmcap">H</span>, in which <span class="allsmcap">A</span> and <span class="allsmcap">B</span> intercept -the <i>x</i> axis, but we cannot see the lines themselves, for -they run in the <i>y</i> direction, and that is not in our -drawing.</p> - -<p>Now, if the rods move with the <i>z</i> to <i>w</i> rotation they will<span class="pagenum" id="Page_216">[Pg 216]</span> -turn in parallel planes, keeping their relative positions. -The point <span class="allsmcap">D</span>, for instance, will -describe a circle. At one time -it will be above the line <span class="allsmcap">A</span>, at -another time below it. Hence -it rotates round <span class="allsmcap">A</span>.</p> - -<div class="figleft illowp40" id="fig_138" style="max-width: 21.875em;"> - <img src="images/fig_138.png" alt="" /> - <div class="caption">Fig. 10 (138).</div> -</div> - -<p>Not only two rods but any -number of rods crossing the -plane will move round it harmoniously. -We can think of -this rotation by supposing the -rods standing up from one line -to move round that line and remembering that it is -not inconsistent with this rotation for the rods standing -up along another line also to move round it, the relative -positions of all the rods being preserved. Now, if the -rods are thick together, they may represent a disk of -matter, and we see that a disk of matter can rotate -round a central plane.</p> - -<p>Rotation round a plane is exactly analogous to rotation -round an axis in three dimensions. If we want a rod to -turn round, the ends must be free; so if we want a disk -of matter to turn round its central plane by a four-dimensional -turning, all the contour must be free. The whole -contour corresponds to the ends of the rod. Each point -of the contour can be looked on as the extremity of an -axis in the body, round each point of which there is a -rotation of the matter in the disk.</p> - -<p>If the one end of a rod be clamped, we can twist the -rod, but not turn it round; so if any part of the contour -of a disk is clamped we can impart a twist to the disk, -but not turn it round its central plane. In the case of -extensible materials a long, thin rod will twist round its -axis, even when the axis is curved, as, for instance, in the -case of a ring of India rubber.</p> - -<p><span class="pagenum" id="Page_217">[Pg 217]</span></p> - -<p>In an analogous manner, in four dimensions we can have -rotation round a curved plane, if I may use the expression. -A sphere can be turned inside out in four dimensions.</p> - -<div class="figleft illowp45" id="fig_139" style="max-width: 25em;"> - <img src="images/fig_139.png" alt="" /> - <div class="caption">Fig. 11 (139).</div> -</div> - -<p>Let <a href="#fig_139">fig. 11</a> represent a -spherical surface, on each -side of which a layer of -matter exists. The thickness -of the matter is represented -by the rods <span class="allsmcap">CD</span> and -<span class="allsmcap">EF</span>, extending equally without -and within.</p> - -<p>Now, take the section of -the sphere by the <i>yz</i> plane -we have a circle—<a href="#fig_140">fig. 12</a>. -Now, let the <i>w</i> axis be drawn -in place of the <i>x</i> axis so that -we have the space of <i>yzw</i> -represented. In this space all that there will be seen of -the sphere is the circle drawn.</p> - -<div class="figleft illowp45" id="fig_140" style="max-width: 25em;"> - <img src="images/fig_140.png" alt="" /> - <div class="caption">Fig. 12 (140).</div> -</div> - -<p>Here we see that there is no obstacle to prevent the -rods turning round. If -the matter is so elastic -that it will give enough -for the particles at <span class="allsmcap">E</span> and -<span class="allsmcap">C</span> to be separated as they -are at <span class="allsmcap">F</span> and <span class="allsmcap">D</span>, they -can rotate round to the -position <span class="allsmcap">D</span> and <span class="allsmcap">F</span>, and a -similar motion is possible -for all other particles. -There is no matter or -obstacle to prevent them -from moving out in the -<i>w</i> direction, and then on round the circumference as an -axis. Now, what will hold for one section will hold for<span class="pagenum" id="Page_218">[Pg 218]</span> -all, as the fourth dimension is at right angles to all the -sections which can be made of the sphere.</p> - -<p>We have supposed the matter of which the sphere is -composed to be three-dimensional. If the matter had a -small thickness in the fourth dimension, there would be -a slight thickness in <a href="#fig_140">fig. 12</a> above the plane of the paper—a -thickness equal to the thickness of the matter in the -fourth dimension. The rods would have to be replaced -by thin slabs. But this would make no difference as to -the possibility of the rotation. This motion is discussed -by Newcomb in the first volume of the <i>American Journal -of Mathematics</i>.</p> - -<p>Let us now consider, not a merely extensible body, but -a liquid one. A mass of rotating liquid, a whirl, eddy, -or vortex, has many remarkable properties. On first -consideration we should expect the rotating mass of -liquid immediately to spread off and lose itself in the -surrounding liquid. The water flies off a wheel whirled -round, and we should expect the rotating liquid to be -dispersed. But see the eddies in a river strangely persistent. -The rings that occur in puffs of smoke and last -so long are whirls or vortices curved round so that their -opposite ends join together. A cyclone will travel over -great distances.</p> - -<p>Helmholtz was the first to investigate the properties of -vortices. He studied them as they would occur in a perfect -fluid—that is, one without friction of one moving portion -or another. In such a medium vortices would be indestructible. -They would go on for ever, altering their -shape, but consisting always of the same portion of the -fluid. But a straight vortex could not exist surrounded -entirely by the fluid. The ends of a vortex must reach to -some boundary inside or outside the fluid.</p> - -<p>A vortex which is bent round so that its opposite ends -join is capable of existing, but no vortex has a free end in<span class="pagenum" id="Page_219">[Pg 219]</span> -the fluid. The fluid round the vortex is always in motion, -and one produces a definite movement in another.</p> - -<p>Lord Kelvin has proposed the hypothesis that portions -of a fluid segregated in vortices account for the origin of -matter. The properties of the ether in respect of its -capacity of propagating disturbances can be explained -by the assumption of vortices in it instead of by a property -of rigidity. It is difficult to conceive, however, -of any arrangement of the vortex rings and endless vortex -filaments in the ether.</p> - -<p>Now, the further consideration of four-dimensional -rotations shows the existence of a kind of vortex which -would make an ether filled with a homogeneous vortex -motion easily thinkable.</p> - -<p>To understand the nature of this vortex, we must go -on and take a step by which we accept the full significance -of the four-dimensional hypothesis. Granted four-dimensional -axes, we have seen that a rotation of one into -another leaves two unaltered, and these two form the -axial plane about which the rotation takes place. But -what about these two? Do they necessarily remain -motionless? There is nothing to prevent a rotation of -these two, one into the other, taking place concurrently -with the first rotation. This possibility of a double -rotation deserves the most careful attention, for it is the -kind of movement which is distinctly typical of four -dimensions.</p> - -<p>Rotation round a plane is analogous to rotation round -an axis. But in three-dimensional space there is no -motion analogous to the double rotation, in which, while -axis 1 changes into axis 2, axis 3 changes into axis 4.</p> - -<p>Consider a four-dimensional body, with four independent -axes, <i>x</i>, <i>y</i>, <i>z</i>, <i>w</i>. A point in it can move in only one -direction at a given moment. If the body has a velocity -of rotation by which the <i>x</i> axis changes into the <i>y</i> axis<span class="pagenum" id="Page_220">[Pg 220]</span> -and all parallel sections move in a similar manner, then -the point will describe a circle. If, now, in addition to -the rotation by which the <i>x</i> axis changes into the <i>y</i> axis the -body has a rotation by which the <i>z</i> axis turns into the -<i>w</i> axis, the point in question will have a double motion -in consequence of the two turnings. The motions will -compound, and the point will describe a circle, but not -the same circle which it would describe in virtue of either -rotation separately.</p> - -<p>We know that if a body in three-dimensional space is -given two movements of rotation they will combine into a -single movement of rotation round a definite axis. It is -in no different condition from that in which it is subjected -to one movement of rotation. The direction of -the axis changes; that is all. The same is not true about -a four-dimensional body. The two rotations, <i>x</i> to <i>y</i> and -<i>z</i> to <i>w</i>, are independent. A body subject to the two is in -a totally different condition to that which it is in when -subject to one only. When subject to a rotation such as -that of <i>x</i> to <i>y</i>, a whole plane in the body, as we have -seen, is stationary. When subject to the double rotation -no part of the body is stationary except the point common -to the two planes of rotation.</p> - -<p>If the two rotations are equal in velocity, every point -in the body describes a circle. All points equally distant -from the stationary point describe circles of equal size.</p> - -<p>We can represent a four-dimensional sphere by means -of two diagrams, in one of which we take the three axes, -<i>x</i>, <i>y</i>, <i>z</i>; in the other the axes <i>x</i>, <i>w</i>, and <i>z</i>. In <a href="#fig_141">fig. 13</a> we -have the view of a four-dimensional sphere in the space of -<i>xyz</i>. Fig. 13 shows all that we can see of the four -sphere in the space of <i>xyz</i>, for it represents all the -points in that space, which are at an equal distance from -the centre.</p> - -<p>Let us now take the <i>xz</i> section, and let the axis of <i>w</i><span class="pagenum" id="Page_221">[Pg 221]</span> -take the place of the <i>y</i> axis. Here, in <a href="#fig_142">fig. 14</a>, we have -the space of <i>xzw</i>. In this space we have to take all the -points which are at the same distance from the centre, -consequently we have another sphere. If we had a three-dimensional -sphere, as has been shown before, we should -have merely a circle in the <i>xzw</i> space, the <i>xz</i> circle seen -in the space of <i>xzw</i>. But now, taking the view in the -space of <i>xzw</i>, we have a sphere in that space also. In a -similar manner, whichever set of three axes we take, we -obtain a sphere.</p> - -<div class="figleft illowp40" id="fig_141" style="max-width: 28.125em;"> - <img src="images/fig_141.png" alt="" /> - <div class="caption"><i>Showing axes xyz</i><br /> -Fig. 13 (141).</div> -</div> - -<div class="figright illowp40" id="fig_142" style="max-width: 28.125em;"> - <img src="images/fig_142.png" alt="" /> - <div class="caption"><i>Showing axes xwz</i><br /> -Fig. 14 (142).</div> -</div> - -<p>In <a href="#fig_141">fig. 13</a>, let us imagine the rotation in the direction -<i>xy</i> to be taking place. The point <i>x</i> will turn to <i>y</i>, and <i>p</i> -to <i>p´</i>. The axis <i>zz´</i> remains stationary, and this axis is all -of the plane <i>zw</i> which we can see in the space section -exhibited in the figure.</p> - -<p>In <a href="#fig_142">fig. 14</a>, imagine the rotation from <i>z</i> to <i>w</i> to be taking -place. The <i>w</i> axis now occupies the position previously -occupied by the <i>y</i> axis. This does not mean that the -<i>w</i> axis can coincide with the <i>y</i> axis. It indicates that we -are looking at the four-dimensional sphere from a different -point of view. Any three-space view will show us three -axes, and in <a href="#fig_142">fig. 14</a> we are looking at <i>xzw</i>.</p> - -<p>The only part that is identical in the two diagrams is -the circle of the <i>x</i> and <i>z</i> axes, which axes are contained -in both diagrams. Thus the plane <i>zxz´</i> is the same in -both, and the point <i>p</i> represents the same point in both<span class="pagenum" id="Page_222">[Pg 222]</span> -diagrams. Now, in <a href="#fig_142">fig. 14</a> let the <i>zw</i> rotation take place, -the <i>z</i> axis will turn toward the point <i>w</i> of the <i>w</i> axis, and -the point <i>p</i> will move in a circle about the point <i>x</i>.</p> - -<p>Thus in <a href="#fig_141">fig. 13</a> the point <i>p</i> moves in a circle parallel to -the <i>xy</i> plane; in <a href="#fig_142">fig. 14</a> it moves in a circle parallel to the -<i>zw</i> plane, indicated by the arrow.</p> - -<p>Now, suppose both of these independent rotations compounded, -the point <i>p</i> will move in a circle, but this circle -will coincide with neither of the circles in which either -one of the rotations will take it. The circle the point <i>p</i> -will move in will depend on its position on the surface of -the four sphere.</p> - -<p>In this double rotation, possible in four-dimensional -space, there is a kind of movement totally unlike any -with which we are familiar in three-dimensional space. -It is a requisite preliminary to the discussion of the -behaviour of the small particles of matter, with a view to -determining whether they show the characteristics of four-dimensional -movements, to become familiar with the main -characteristics of this double rotation. And here I must -rely on a formal and logical assent rather than on the -intuitive apprehension, which can only be obtained by a -more detailed study.</p> - -<p>In the first place this double rotation consists in two -varieties or kinds, which we will call the A and B kinds. -Consider four axes, <i>x</i>, <i>y</i>, <i>z</i>, <i>w</i>. The rotation of <i>x</i> to <i>y</i> can -be accompanied with the rotation of <i>z</i> to <i>w</i>. Call this -the A kind.</p> - -<p>But also the rotation of <i>x</i> to <i>y</i> can be accompanied by -the rotation, of not <i>z</i> to <i>w</i>, but <i>w</i> to <i>z</i>. Call this the -B kind.</p> - -<p>They differ in only one of the component rotations. One -is not the negative of the other. It is the semi-negative. -The opposite of an <i>x</i> to <i>y</i>, <i>z</i> to <i>w</i> rotation would be <i>y</i> to <i>x</i>, -<i>w</i> to <i>z</i>. The semi-negative is <i>x</i> to <i>y</i> and <i>w</i> to <i>z</i>.</p> - -<p><span class="pagenum" id="Page_223">[Pg 223]</span></p> - -<p>If four dimensions exist and we cannot perceive them, -because the extension of matter is so small in the fourth -dimension that all movements are withheld from direct -observation except those which are three-dimensional, we -should not observe these double rotations, but only the -effects of them in three-dimensional movements of the -type with which we are familiar.</p> - -<p>If matter in its small particles is four-dimensional, -we should expect this double rotation to be a universal -characteristic of the atoms and molecules, for no portion -of matter is at rest. The consequences of this corpuscular -motion can be perceived, but only under the form -of ordinary rotation or displacement. Thus, if the theory -of four dimensions is true, we have in the corpuscles of -matter a whole world of movement, which we can never -study directly, but only by means of inference.</p> - -<p>The rotation A, as I have defined it, consists of two -equal rotations—one about the plane of <i>zw</i>, the other -about the plane of <i>xy</i>. It is evident that these rotations -are not necessarily equal. A body may be moving with a -double rotation, in which these two independent components -are not equal; but in such a case we can consider -the body to be moving with a composite rotation—a -rotation of the A or B kind and, in addition, a rotation -about a plane.</p> - -<p>If we combine an A and a B movement, we obtain a -rotation about a plane; for, the first being <i>x</i> to <i>y</i> and -<i>z</i> to <i>w</i>, and the second being <i>x</i> to <i>y</i> and <i>w</i> to <i>z</i>, when they -are put together the <i>z</i> to <i>w</i> and <i>w</i> to <i>z</i> rotations neutralise -each other, and we obtain an <i>x</i> to <i>y</i> rotation only, which -is a rotation about the plane of <i>zw</i>. Similarly, if we -take a B rotation, <i>y</i> to <i>x</i> and <i>z</i> to <i>w</i>, we get, on combining -this with the A rotation, a rotation of <i>z</i> to <i>w</i> about the -<i>xy</i> plane. In this case the plane of rotation is in the -three-dimensional space of <i>xyz</i>, and we have—what has<span class="pagenum" id="Page_224">[Pg 224]</span> -been described before—a twisting about a plane in our -space.</p> - -<p>Consider now a portion of a perfect liquid having an A -motion. It can be proved that it possesses the properties -of a vortex. It forms a permanent individuality—a -separated-out portion of the liquid—accompanied by a -motion of the surrounding liquid. It has properties -analogous to those of a vortex filament. But it is not -necessary for its existence that its ends should reach the -boundary of the liquid. It is self-contained and, unless -disturbed, is circular in every section.</p> - -<div class="figleft illowp50" id="fig_143" style="max-width: 28.125em;"> - <img src="images/fig_143.png" alt="" /> - <div class="caption">Fig. 15 (143).</div> -</div> - -<p>If we suppose the ether to have its properties of transmitting -vibration given it by such vortices, we must -inquire how they lie together in four-dimensional space. -Placing a circular disk on a plane and surrounding it by -six others, we find that if the central one is given a motion -of rotation, it imparts to the others a rotation which is -antagonistic in every two adjacent -ones. If <span class="allsmcap">A</span> goes round, -as shown by the arrow, <span class="allsmcap">B</span> and -<span class="allsmcap">C</span> will be moving in opposite -ways, and each tends to destroy -the motion of the other.</p> - -<p>Now, if we suppose spheres -to be arranged in a corresponding -manner in three-dimensional -space, they will -be grouped in figures which -are for three-dimensional space what hexagons are for -plane space. If a number of spheres of soft clay be -pressed together, so as to fill up the interstices, each will -assume the form of a fourteen-sided figure called a -tetrakaidecagon.</p> - -<p>Now, assuming space to be filled with such tetrakaidecagons, -and placing a sphere in each, it will be found<span class="pagenum" id="Page_225">[Pg 225]</span> -that one sphere is touched by eight others. The remaining -six spheres of the fourteen which surround the -central one will not touch it, but will touch three of -those in contact with it. Hence, if the central sphere -rotates, it will not necessarily drive those around it so -that their motions will be antagonistic to each other, -but the velocities will not arrange themselves in a -systematic manner.</p> - -<p>In four-dimensional space the figure which forms the -next term of the series hexagon, tetrakaidecagon, is a -thirty-sided figure. It has for its faces ten solid tetrakaidecagons -and twenty hexagonal prisms. Such figures -will exactly fill four-dimensional space, five of them meeting -at every point. If, now, in each of these figures we -suppose a solid four-dimensional sphere to be placed, any -one sphere is surrounded by thirty others. Of these it -touches ten, and, if it rotates, it drives the rest by means -of these. Now, if we imagine the central sphere to be -given an A or a B rotation, it will turn the whole mass of -sphere round in a systematic manner. Suppose four-dimensional -space to be filled with such spheres, each -rotating with a double rotation, the whole mass would -form one consistent system of motion, in which each one -drove every other one, with no friction or lagging behind.</p> - -<p>Every sphere would have the same kind of rotation. In -three-dimensional space, if one body drives another round -the second body rotates with the opposite kind of rotation; -but in four-dimensional space these four-dimensional -spheres would each have the double negative of the rotation -of the one next it, and we have seen that the double -negative of an A or B rotation is still an A or B rotation. -Thus four-dimensional space could be filled with a system -of self-preservative living energy. If we imagine the -four-dimensional spheres to be of liquid and not of solid -matter, then, even if the liquid were not quite perfect and<span class="pagenum" id="Page_226">[Pg 226]</span> -there were a slight retarding effect of one vortex on -another, the system would still maintain itself.</p> - -<p>In this hypothesis we must look on the ether as -possessing energy, and its transmission of vibrations, not -as the conveying of a motion imparted from without, but -as a modification of its own motion.</p> - -<p>We are now in possession of some of the conceptions of -four-dimensional mechanics, and will turn aside from the -line of their development to inquire if there is any -evidence of their applicability to the processes of nature.</p> - -<p>Is there any mode of motion in the region of the -minute which, giving three-dimensional movements for -its effect, still in itself escapes the grasp of our mechanical -theories? I would point to electricity. Through the -labours of Faraday and Maxwell we are convinced that the -phenomena of electricity are of the nature of the stress -and strain of a medium; but there is still a gap to be -bridged over in their explanation—the laws of elasticity, -which Maxwell assumes, are not those of ordinary matter. -And, to take another instance: a magnetic pole in the -neighbourhood of a current tends to move. Maxwell has -shown that the pressures on it are analogous to the -velocities in a liquid which would exist if a vortex took -the place of the electric current: but we cannot point out -the definite mechanical explanation of these pressures. -There must be some mode of motion of a body or of the -medium in virtue of which a body is said to be -electrified.</p> - -<p>Take the ions which convey charges of electricity 500 -times greater in proportion to their mass than are carried -by the molecules of hydrogen in electrolysis. In respect -of what motion can these ions be said to be electrified? -It can be shown that the energy they possess is not -energy of rotation. Think of a short rod rotating. If it -is turned over it is found to be rotating in the opposite<span class="pagenum" id="Page_227">[Pg 227]</span> -direction. Now, if rotation in one direction corresponds to -positive electricity, rotation in the opposite direction corresponds -to negative electricity, and the smallest electrified -particles would have their charges reversed by being -turned over—an absurd supposition.</p> - -<p>If we fix on a mode of motion as a definition of -electricity, we must have two varieties of it, one for -positive and one for negative; and a body possessing the -one kind must not become possessed of the other by any -change in its position.</p> - -<p>All three-dimensional motions are compounded of rotations -and translations, and none of them satisfy this first -condition for serving as a definition of electricity.</p> - -<p>But consider the double rotation of the A and B kinds. -A body rotating with the A motion cannot have its -motion transformed into the B kind by being turned over -in any way. Suppose a body has the rotation <i>x</i> to <i>y</i> and -<i>z</i> to <i>w</i>. Turning it about the <i>xy</i> plane, we reverse the -direction of the motion <i>x</i> to <i>y</i>. But we also reverse the -<i>z</i> to <i>w</i> motion, for the point at the extremity of the -positive <i>z</i> axis is now at the extremity of the negative <i>z</i> -axis, and since we have not interfered with its motion it -goes in the direction of position <i>w</i>. Hence we have <i>y</i> to -<i>x</i> and <i>w</i> to <i>z</i>, which is the same as <i>x</i> to <i>y</i> and <i>z</i> to <i>w</i>. -Thus both components are reversed, and there is the A -motion over again. The B kind is the semi-negative, -with only one component reversed.</p> - -<p>Hence a system of molecules with the A motion would -not destroy it in one another, and would impart it to a -body in contact with them. Thus A and B motions -possess the first requisite which must be demanded in -any mode of motion representative of electricity.</p> - -<p>Let us trace out the consequences of defining positive -electricity as an A motion and negative electricity as a B -motion. The combination of positive and negative<span class="pagenum" id="Page_228">[Pg 228]</span> -electricity produces a current. Imagine a vortex in the -ether of the A kind and unite with this one of the B kind. -An A motion and B motion produce rotation round a plane, -which is in the ether a vortex round an axial surface. -It is a vortex of the kind we represent as a part of a -sphere turning inside out. Now such a vortex must have -its rim on a boundary of the ether—on a body in the -ether.</p> - -<p>Let us suppose that a conductor is a body which has -the property of serving as the terminal abutment of such -a vortex. Then the conception we must form of a closed -current is of a vortex sheet having its edge along the -circuit of the conducting wire. The whole wire will then -be like the centres on which a spindle turns in three-dimensional -space, and any interruption of the continuity -of the wire will produce a tension in place of a continuous -revolution.</p> - -<p>As the direction of the rotation of the vortex is from a -three-space direction into the fourth dimension and back -again, there will be no direction of flow to the current; -but it will have two sides, according to whether <i>z</i> goes -to <i>w</i> or <i>z</i> goes to negative <i>w</i>.</p> - -<p>We can draw any line from one part of the circuit to -another; then the ether along that line is rotating round -its points.</p> - -<p>This geometric image corresponds to the definition of -an electric circuit. It is known that the action does not -lie in the wire, but in the medium, and it is known that -there is no direction of flow in the wire.</p> - -<p>No explanation has been offered in three-dimensional -mechanics of how an action can be impressed throughout -a region and yet necessarily run itself out along a closed -boundary, as is the case in an electric current. But this -phenomenon corresponds exactly to the definition of a -four-dimensional vortex.</p> - -<p><span class="pagenum" id="Page_229">[Pg 229]</span></p> - -<p>If we take a very long magnet, so long that one of its -poles is practically isolated, and put this pole in the -vicinity of an electric circuit, we find that it moves.</p> - -<p>Now, assuming for the sake of simplicity that the wire -which determines the current is in the form of a circle, -if we take a number of small magnets and place them all -pointing in the same direction normal to the plane of the -circle, so that they fill it and the wire binds them round, -we find that this sheet of magnets has the same effect on -the magnetic pole that the current has. The sheet of -magnets may be curved, but the edge of it must coincide -with the wire. The collection of magnets is then -equivalent to the vortex sheet, and an elementary magnet -to a part of it. Thus, we must think of a magnet as -conditioning a rotation in the ether round the plane -which bisects at right angles the line joining its poles.</p> - -<p>If a current is started in a circuit, we must imagine -vortices like bowls turning themselves inside out, starting -from the contour. In reaching a parallel circuit, if the -vortex sheet were interrupted and joined momentarily to -the second circuit by a free rim, the axis plane would lie -between the two circuits, and a point on the second circuit -opposite a point on the first would correspond to a point -opposite to it on the first; hence we should expect a -current in the opposite direction in the second circuit. -Thus the phenomena of induction are not inconsistent -with the hypothesis of a vortex about an axial plane.</p> - -<p>In four-dimensional space, in which all four dimensions -were commensurable, the intensity of the action transmitted -by the medium would vary inversely as the cube of the -distance. Now, the action of a current on a magnetic -pole varies inversely as the square of the distance; hence, -over measurable distances the extension of the ether in -the fourth dimension cannot be assumed as other than -small in comparison with those distances.</p> - -<p><span class="pagenum" id="Page_230">[Pg 230]</span></p> - -<p>If we suppose the ether to be filled with vortices in the -shape of four-dimensional spheres rotating with the A -motion, the B motion would correspond to electricity in -the one-fluid theory. There would thus be a possibility -of electricity existing in two forms, statically, by itself, -and, combined with the universal motion, in the form of -a current.</p> - -<p>To arrive at a definite conclusion it will be necessary to -investigate the resultant pressures which accompany the -collocation of solid vortices with surface ones.</p> - -<p>To recapitulate:</p> - -<p>The movements and mechanics of four-dimensional -space are definite and intelligible. A vortex with a -surface as its axis affords a geometric image of a closed -circuit, and there are rotations which by their polarity -afford a possible definition of statical electricity.<a id="FNanchor_7" href="#Footnote_7" class="fnanchor">[7]</a></p> - -<div class="footnotes"> -<div class="footnote"> - -<p><a id="Footnote_7" href="#FNanchor_7" class="label">[7]</a> These double rotations of the A and B kinds I should like to call -Hamiltons and co-Hamiltons, for it is a singular fact that in his -“Quaternions” Sir Wm. Rowan Hamilton has given the theory of -either the A or the B kind. They follow the laws of his symbols, -I, J, K.</p> - -<p>Hamiltons and co-Hamiltons seem to be natural units of geometrical -expression. In the paper in the “Proceedings of the Royal Irish -Academy,” Nov. 1903, already alluded to, I have shown something of -the remarkable facility which is gained in dealing with the composition -of three- and four-dimensional rotations by an alteration in Hamilton’s -notation, which enables his system to be applied to both the A and B -kinds of rotations.</p> - -<p>The objection which has been often made to Hamilton’s system, -namely, that it is only under special conditions of application that his -processes give geometrically interpretable results, can be removed, if -we assume that he was really dealing with a four-dimensional motion, -and alter his notation to bring this circumstance into explicit -recognition.</p> - -</div></div> - -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_231">[Pg 231]</span></p> - -<h2 class="nobreak" id="APPENDIX_I">APPENDIX I<br /> - -<small><i>THE MODELS</i></small></h2></div> - - -<p>In Chapter XI. a description has been given which will -enable any one to make a set of models illustrative of the -tesseract and its properties. The set here supposed to be -employed consists of:—</p> - -<div class="blockquote"> - -<p>1. Three sets of twenty-seven cubes each.</p> - -<p>2. Twenty-seven slabs.</p> - -<p>3. Twelve cubes with points, lines, faces, distinguished -by colours, which will be called the catalogue cubes.</p> -</div> - -<p>The preparation of the twelve catalogue cubes involves -the expenditure of a considerable amount of time. It is -advantageous to use them, but they can be replaced by -the drawing of the views of the tesseract or by a reference -to figs. <a href="#fig_103">103</a>, <a href="#fig_104">104</a>, <a href="#fig_105">105</a>, <a href="#fig_106">106</a> of the text.</p> - -<p>The slabs are coloured like the twenty-seven cubes of -the first cubic block in <a href="#fig_101">fig. 101</a>, the one with red, -white, yellow axes.</p> - -<p>The colours of the three sets of twenty-seven cubes are -those of the cubes shown in <a href="#fig_101">fig. 101</a>.</p> - -<p>The slabs are used to form the representation of a cube -in a plane, and can well be dispensed with by any one -who is accustomed to deal with solid figures. But the -whole theory depends on a careful observation of how the -cube would be represented by these slabs.</p> - -<p>In the first step, that of forming a clear idea how a<span class="pagenum" id="Page_232">[Pg 232]</span> -plane being would represent three-dimensional space, only -one of the catalogue cubes and one of the three blocks is -needed.</p> - - -<h3><span class="smcap">Application to the Step from Plane to Solid.</span></h3> - -<p>Look at <a href="#fig_1">fig. 1</a> of the views of the tesseract, or, what -comes to the same thing, take catalogue cube No. 1 and -place it before you with the red line running up, the -white line running to the right, the yellow line running -away. The three dimensions of space are then marked -out by these lines or axes. Now take a piece of cardboard, -or a book, and place it so that it forms a wall -extending up and down not opposite to you, but running -away parallel to the wall of the room on your -left hand.</p> - -<p>Placing the catalogue cube against this wall we see -that it comes into contact with it by the red and yellow -lines, and by the included orange face.</p> - -<p>In the plane being’s world the aspect he has of the -cube would be a square surrounded by red and yellow -lines with grey points.</p> - -<p>Now, keeping the red line fixed, turn the cube about it -so that the yellow line goes out to the right, and the -white line comes into contact with the plane.</p> - -<p>In this case a different aspect is presented to the plane -being, a square, namely, surrounded by red and white -lines and grey points. You should particularly notice -that when the yellow line goes out, at right angles to the -plane, and the white comes in, the latter does not run in -the same sense that the yellow did.</p> - -<p>From the fixed grey point at the base of the red line -the yellow line ran away from you. The white line now -runs towards you. This turning at right angles makes -the line which was out of the plane before, come into it<span class="pagenum" id="Page_233">[Pg 233]</span> -in an opposite sense to that in which the line ran which -has just left the plane. If the cube does not break -through the plane this is always the rule.</p> - -<p>Again turn the cube back to the normal position with -red running up, white to the right, and yellow away, and -try another turning.</p> - -<p>You can keep the yellow line fixed, and turn the cube -about it. In this case the red line going out to the -right the white line will come in pointing downwards.</p> - -<p>You will be obliged to elevate the cube from the table -in order to carry out this turning. It is always necessary -when a vertical axis goes out of a space to imagine a -movable support which will allow the line which ran out -before to come in below.</p> - -<p>Having looked at the three ways of turning the cube -so as to present different faces to the plane, examine what -would be the appearance if a square hole were cut in the -piece of cardboard, and the cube were to pass through it. -A hole can be actually cut, and it will be seen that in the -normal position, with red axis running up, yellow away, -and white to the right, the square first perceived by the -plane being—the one contained by red and yellow lines—would -be replaced by another square of which the line -towards you is pink—the section line of the pink face. -The line above is light yellow, below is light yellow and -on the opposite side away from you is pink.</p> - -<p>In the same way the cube can be pushed through a -square opening in the plane from any of the positions -which you have already turned it into. In each case -the plane being will perceive a different set of contour -lines.</p> - -<p>Having observed these facts about the catalogue cube, -turn now to the first block of twenty-seven cubes.</p> - -<p>You notice that the colour scheme on the catalogue cube -and that of this set of blocks is the same.</p> - -<p><span class="pagenum" id="Page_234">[Pg 234]</span></p> - -<p>Place them before you, a grey or null cube on the -table, above it a red cube, and on the top a null cube -again. Then away from you place a yellow cube, and -beyond it a null cube. Then to the right place a white -cube and beyond it another null. Then complete the -block, according to the scheme of the catalogue cube, -putting in the centre of all an ochre cube.</p> - -<p>You have now a cube like that which is described in -the text. For the sake of simplicity, in some cases, this -cubic block can be reduced to one of eight cubes, by -leaving out the terminations in each direction. Thus, -instead of null, red, null, three cubes, you can take null, -red, two cubes, and so on.</p> - -<p>It is useful, however, to practise the representation in -a plane of a block of twenty-seven cubes. For this -purpose take the slabs, and build them up against the -piece of cardboard, or the book in such a way as to -represent the different aspects of the cube.</p> - -<p>Proceed as follows:—</p> - -<p>First, cube in normal position.</p> - -<p>Place nine slabs against the cardboard to represent the -nine cubes in the wall of the red and yellow axes, facing -the cardboard; these represent the aspect of the cube as it -touches the plane.</p> - -<p>Now push these along the cardboard and make a -different set of nine slabs to represent the appearance -which the cube would present to a plane being, if it were -to pass half way through the plane.</p> - -<p>There would be a white slab, above it a pink one, above -that another white one, and six others, representing what -would be the nature of a section across the middle of the -block of cubes. The section can be thought of as a thin -slice cut out by two parallel cuts across the cube. -Having arranged these nine slabs, push them along the -plane, and make another set of nine to represent what<span class="pagenum" id="Page_235">[Pg 235]</span> -would be the appearance of the cube when it had almost -completely gone through. This set of nine will be the -same as the first set of nine.</p> - -<p>Now we have in the plane three sets of nine slabs -each, which represent three sections of the twenty-seven -block.</p> - -<p>They are put alongside one another. We see that it -does not matter in what order the sets of nine are put. -As the cube passes through the plane they represent appearances -which follow the one after the other. If they -were what they represented, they could not exist in the -same plane together.</p> - -<p>This is a rather important point, namely, to notice that -they should not co-exist on the plane, and that the order -in which they are placed is indifferent. When we -represent a four-dimensional body our solid cubes are to -us in the same position that the slabs are to the plane -being. You should also notice that each of these slabs -represents only the very thinnest slice of a cube. The -set of nine slabs first set up represents the side surface of -the block. It is, as it were, a kind of tray—a beginning -from which the solid cube goes off. The slabs as we use -them have thickness, but this thickness is a necessity of -construction. They are to be thought of as merely of the -thickness of a line.</p> - -<p>If now the block of cubes passed through the plane at -the rate of an inch a minute the appearance to a plane -being would be represented by:—</p> - -<p>1. The first set of nine slabs lasting for one minute.</p> - -<p>2. The second set of nine slabs lasting for one minute.</p> - -<p>3. The third set of nine slabs lasting for one minute.</p> - -<p>Now the appearances which the cube would present -to the plane being in other positions can be shown by -means of these slabs. The use of such slabs would be -the means by which a plane being could acquire a<span class="pagenum" id="Page_236">[Pg 236]</span> -familiarity with our cube. Turn the catalogue cube (or -imagine the coloured figure turned) so that the red line -runs up, the yellow line out to the right, and the white -line towards you. Then turn the block of cubes to -occupy a similar position.</p> - -<p>The block has now a different wall in contact with -the plane. Its appearance to a plane being will not be -the same as before. He has, however, enough slabs to -represent this new set of appearances. But he must -remodel his former arrangement of them.</p> - -<p>He must take a null, a red, and a null slab from the first -of his sets of slabs, then a white, a pink, and a white from -the second, and then a null, a red, and a null from the -third set of slabs.</p> - -<p>He takes the first column from the first set, the first -column from the second set, and the first column from -the third set.</p> - -<p>To represent the half-way-through appearance, which -is as if a very thin slice were cut out half way through the -block, he must take the second column of each of his -sets of slabs, and to represent the final appearance, the -third column of each set.</p> - -<p>Now turn the catalogue cube back to the normal -position, and also the block of cubes.</p> - -<p>There is another turning—a turning about the yellow -line, in which the white axis comes below the support.</p> - -<p>You cannot break through the surface of the table, so -you must imagine the old support to be raised. Then -the top of the block of cubes in its new position is at the -level at which the base of it was before.</p> - -<p>Now representing the appearance on the plane, we must -draw a horizontal line to represent the old base. The -line should be drawn three inches high on the cardboard.</p> - -<p>Below this the representative slabs can be arranged.</p> - -<p>It is easy to see what they are. The old arrangements<span class="pagenum" id="Page_237">[Pg 237]</span> -have to be broken up, and the layers taken in order, the -first layer of each for the representation of the aspect of -the block as it touches the plane.</p> - -<p>Then the second layers will represent the appearance -half way through, and the third layers will represent the -final appearance.</p> - -<p>It is evident that the slabs individually do not represent -the same portion of the cube in these different presentations.</p> - -<p>In the first case each slab represents a section or a face -perpendicular to the white axis, in the second case a -face or a section which runs perpendicularly to the yellow -axis, and in the third case a section or a face perpendicular -to the red axis.</p> - -<p>But by means of these nine slabs the plane being can -represent the whole of the cubic block. He can touch -and handle each portion of the cubic block, there is no -part of it which he cannot observe. Taking it bit by bit, -two axes at a time, he can examine the whole of it.</p> - - -<h3><span class="smcap">Our Representation of a Block of Tesseracts.</span></h3> - -<p>Look at the views of the tesseract 1, 2, 3, or take the -catalogue cubes 1, 2, 3, and place them in front of you, -in any order, say running from left to right, placing 1 in -the normal position, the red axis running up, the white -to the right, and yellow away.</p> - -<p>Now notice that in catalogue cube 2 the colours of each -region are derived from those of the corresponding region -of cube 1 by the addition of blue. Thus null + blue = -blue, and the corners of number 2 are blue. Again, -red + blue = purple, and the vertical lines of 2 are purple. -Blue + yellow = green, and the line which runs away is -coloured green.</p> - -<p>By means of these observations you may be sure that<span class="pagenum" id="Page_238">[Pg 238]</span> -catalogue cube 2 is rightly placed. Catalogue cube 3 is -just like number 1.</p> - -<p>Having these cubes in what we may call their normal -position, proceed to build up the three sets of blocks.</p> - -<p>This is easily done in accordance with the colour scheme -on the catalogue cubes.</p> - -<p>The first block we already know. Build up the second -block, beginning with a blue corner cube, placing a purple -on it, and so on.</p> - -<p>Having these three blocks we have the means of -representing the appearances of a group of eighty-one -tesseracts.</p> - -<p>Let us consider a moment what the analogy in the case -of the plane being is.</p> - -<p>He has his three sets of nine slabs each. We have our -three sets of twenty-seven cubes each.</p> - -<p>Our cubes are like his slabs. As his slabs are not the -things which they represent to him, so our cubes are not -the things they represent to us.</p> - -<p>The plane being’s slabs are to him the faces of cubes.</p> - -<p>Our cubes then are the faces of tesseracts, the cubes by -which they are in contact with our space.</p> - -<p>As each set of slabs in the case of the plane being -might be considered as a sort of tray from which the solid -contents of the cubes came out, so our three blocks of -cubes may be considered as three-space trays, each of -which is the beginning of an inch of the solid contents -of the four-dimensional solids starting from them.</p> - -<p>We want now to use the names null, red, white, etc., -for tesseracts. The cubes we use are only tesseract faces. -Let us denote that fact by calling the cube of null colour, -null face; or, shortly, null f., meaning that it is the face -of a tesseract.</p> - -<p>To determine which face it is let us look at the catalogue -cube 1 or the first of the views of the tesseract, which<span class="pagenum" id="Page_239">[Pg 239]</span> -can be used instead of the models. It has three axes, -red, white, yellow, in our space. Hence the cube determined -by these axes is the face of the tesseract which we -now have before us. It is the ochre face. It is enough, -however, simply to say null f., red f. for the cubes which -we use.</p> - -<p>To impress this in your mind, imagine that tesseracts -do actually run from each cube. Then, when you move the -cubes about, you move the tesseracts about with them. -You move the face but the tesseract follows with it, as the -cube follows when its face is shifted in a plane.</p> - -<p>The cube null in the normal position is the cube which -has in it the red, yellow, white axes. It is the face -having these, but wanting the blue. In this way you can -define which face it is you are handling. I will write an -“f.” after the name of each tesseract just as the plane -being might call each of his slabs null slab, yellow slab, -etc., to denote that they were representations.</p> - -<p>We have then in the first block of twenty-seven cubes, -the following—null f., red f., null f., going up; white f., null -f., lying to the right, and so on. Starting from the null -point and travelling up one inch we are in the null region, -the same for the away and the right-hand directions. -And if we were to travel in the fourth dimension for an -inch we should still be in a null region. The tesseract -stretches equally all four ways. Hence the appearance we -have in this first block would do equally well if the -tesseract block were to move across our space for a certain -distance. For anything less than an inch of their transverse -motion we should still have the same appearance. -You must notice, however, that we should not have null -face after the motion had begun.</p> - -<p>When the tesseract, null for instance, had moved ever -so little we should not have a face of null but a section of -null in our space. Hence, when we think of the motion<span class="pagenum" id="Page_240">[Pg 240]</span> -across our space we must call our cubes tesseract sections. -Thus on null passing across we should see first null f., then -null s., and then, finally, null f. again.</p> - -<p>Imagine now the whole first block of twenty-seven -tesseracts to have moved tranverse to our space a distance -of one inch. Then the second set of tesseracts, which -originally were an inch distant from our space, would be -ready to come in.</p> - -<p>Their colours are shown in the second block of twenty-seven -cubes which you have before you. These represent -the tesseract faces of the set of tesseracts that lay before -an inch away from our space. They are ready now to -come in, and we can observe their colours. In the place -which null f. occupied before we have blue f., in place of -red f. we have purple f., and so on. Each tesseract is -coloured like the one whose place it takes in this motion -with the addition of blue.</p> - -<p>Now if the tesseract block goes on moving at the rate -of an inch a minute, this next set of tesseracts will occupy -a minute in passing across. We shall see, to take the null -one for instance, first of all null face, then null section, -then null face again.</p> - -<p>At the end of the second minute the second set of -tesseracts has gone through, and the third set comes in. -This, as you see, is coloured just like the first. Altogether, -these three sets extend three inches in the fourth dimension, -making the tesseract block of equal magnitude in all -dimensions.</p> - -<p>We have now before us a complete catalogue of all the -tesseracts in our group. We have seen them all, and we -shall refer to this arrangement of the blocks as the -“normal position.” We have seen as much of each -tesseract at a time as could be done in a three-dimensional -space. Each part of each tesseract has been in -our space, and we could have touched it.</p> - -<p><span class="pagenum" id="Page_241">[Pg 241]</span></p> - -<p>The fourth dimension appeared to us as the duration -of the block.</p> - -<p>If a bit of our matter were to be subjected to the same -motion it would be instantly removed out of our space. -Being thin in the fourth dimension it is at once taken -out of our space by a motion in the fourth dimension.</p> - -<p>But the tesseract block we represent having length in -the fourth dimension remains steadily before our eyes for -three minutes, when it is subjected to this transverse -motion.</p> - -<p>We have now to form representations of the other -views of the same tesseract group which are possible in -our space.</p> - -<p>Let us then turn the block of tesseracts so that another -face of it comes into contact with our space, and then -by observing what we have, and what changes come when -the block traverses our space, we shall have another view -of it. The dimension which appeared as duration before -will become extension in one of our known dimensions, -and a dimension which coincided with one of our space -dimensions will appear as duration.</p> - -<p>Leaving catalogue cube 1 in the normal position, -remove the other two, or suppose them removed. We -have in space the red, the yellow, and the white axes. -Let the white axis go out into the unknown, and occupy -the position the blue axis holds. Then the blue axis, -which runs in that direction now will come into space. -But it will not come in pointing in the same way that -the white axis does now. It will point in the opposite -sense. It will come in running to the left instead of -running to the right as the white axis does now.</p> - -<p>When this turning takes place every part of the cube 1 -will disappear except the left-hand face—the orange face.</p> - -<p>And the new cube that appears in our space will run to -the left from this orange face, having axes, red, yellow, blue.</p> - -<p><span class="pagenum" id="Page_242">[Pg 242]</span></p> - -<p>Take models 4, 5, 6. Place 4, or suppose No. 4 of the -tesseract views placed, with its orange face coincident with -the orange face of 1, red line to red line, and yellow line -to yellow line, with the blue line pointing to the left. -Then remove cube 1 and we have the tesseract face -which comes in when the white axis runs in the positive -unknown, and the blue axis comes into our space.</p> - -<p>Now place catalogue cube 5 in some position, it does -not matter which, say to the left; and place it so that -there is a correspondence of colour corresponding to the -colour of the line that runs out of space. The line that -runs out of space is white, hence, every part of this -cube 5 should differ from the corresponding part of 4 by -an alteration in the direction of white.</p> - -<p>Thus we have white points in 5 corresponding to the -null points in 4. We have a pink line corresponding to -a red line, a light yellow line corresponding to a yellow -line, an ochre face corresponding to an orange face. This -cube section is completely named in Chapter XI. Finally -cube 6 is a replica of 1.</p> - -<p>These catalogue cubes will enable us to set up our -models of the block of tesseracts.</p> - -<p>First of all for the set of tesseracts, which beginning -in our space reach out one inch in the unknown, we have -the pattern of catalogue cube 4.</p> - -<p>We see that we can build up a block of twenty-seven -tesseract faces after the colour scheme of cube 4, by -taking the left-hand wall of block 1, then the left-hand -wall of block 2, and finally that of block 3. We take, -that is, the three first walls of our previous arrangement -to form the first cubic block of this new one.</p> - -<p>This will represent the cubic faces by which the group -of tesseracts in its new position touches our space. -We have running up, null f., red f., null f. In the next -vertical line, on the side remote from us, we have yellow f.,<span class="pagenum" id="Page_243">[Pg 243]</span> -orange f., yellow f., and then the first colours over again. -Then the three following columns are, blue f., purple f., -blue f.; green f., brown f., green f.; blue f., purple f., blue f. -The last three columns are like the first.</p> - -<p>These tesseracts touch our space, and none of them are -by any part of them distant more than an inch from it. -What lies beyond them in the unknown?</p> - -<p>This can be told by looking at catalogue cube 5. -According to its scheme of colour we see that the second -wall of each of our old arrangements must be taken. -Putting them together we have, as the corner, white f. -above it, pink f. above it, white f. The column next to -this remote from us is as follows:—light yellow f., ochre f., -light yellow f., and beyond this a column like the first. -Then for the middle of the block, light blue f., above -it light purple, then light blue. The centre column has, -at the bottom, light green f., light brown f. in the centre -and at the top light green f. The last wall is like the -first.</p> - -<p>The third block is made by taking the third walls of -our previous arrangement, which we called the normal -one.</p> - -<p>You may ask what faces and what sections our cubes -represent. To answer this question look at what axes -you have in our space. You have red, yellow, blue. -Now these determine brown. The colours red, -yellow, blue are supposed by us when mixed to produce -a brown colour. And that cube which is determined -by the red, yellow, blue axes we call the brown cube.</p> - -<p>When the tesseract block in its new position begins to -move across our space each tesseract in it gives a section -in our space. This section is transverse to the white -axis, which now runs in the unknown.</p> - -<p>As the tesseract in its present position passes across -our space, we should see first of all the first of the blocks<span class="pagenum" id="Page_244">[Pg 244]</span> -of cubic faces we have put up—these would last for a -minute, then would come the second block and then the -third. At first we should have a cube of tesseract faces, -each of which would be brown. Directly the movement -began, we should have tesseract sections transverse to the -white line.</p> - -<p>There are two more analogous positions in which the -block of tesseracts can be placed. To find the third -position, restore the blocks to the normal arrangement.</p> - -<p>Let us make the yellow axis go out into the positive -unknown, and let the blue axis, consequently, come in -running towards us. The yellow ran away, so the blue -will come in running towards us.</p> - -<p>Put catalogue cube 1 in its normal position. Take -catalogue cube 7 and place it so that its pink face -coincides with the pink face of cube 1, making also its -red axis coincide with the red axis of 1 and its white -with the white. Moreover, make cube 7 come -towards us from cube 1. Looking at it we see in our -space, red, white, and blue axes. The yellow runs out. -Place catalogue cube 8 in the neighbourhood of -7—observe that every region in 8 has a change in -the direction of yellow from the corresponding region -in 7. This is because it represents what you come -to now in going in the unknown, when the yellow axis -runs out of our space. Finally catalogue cube 9, -which is like number 7, shows the colours of the third -set of tesseracts. Now evidently, starting from the -normal position, to make up our three blocks of tesseract -faces we have to take the near wall from the first block, -the near wall from the second, and then the near wall -from the third block. This gives us the cubic block -formed by the faces of the twenty-seven tesseracts which -are now immediately touching our space.</p> - -<p>Following the colour scheme of catalogue cube 8,<span class="pagenum" id="Page_245">[Pg 245]</span> -we make the next set of twenty-seven tesseract faces, -representing the tesseracts, each of which begins one inch -off from our space, by putting the second walls of our -previous arrangement together, and the representation -of the third set of tesseracts is the cubic block formed of -the remaining three walls.</p> - -<p>Since we have red, white, blue axes in our space to -begin with, the cubes we see at first are light purple -tesseract faces, and after the transverse motion begins -we have cubic sections transverse to the yellow line.</p> - -<p>Restore the blocks to the normal position, there -remains the case in which the red axis turns out of -space. In this case the blue axis will come in downwards, -opposite to the sense in which the red axis ran.</p> - -<p>In this case take catalogue cubes 10, 11, 12. Lift up -catalogue cube 1 and put 10 underneath it, imagining -that it goes down from the previous position of 1.</p> - -<p>We have to keep in space the white and the yellow -axes, and let the red go out, the blue come in.</p> - -<p>Now, you will find on cube 10 a light yellow face; this -should coincide with the base of 1, and the white and -yellow lines on the two cubes should coincide. Then the -blue axis running down you have the catalogue cube -correctly placed, and it forms a guide for putting up the -first representative block.</p> - -<p>Catalogue cube 11 will represent what lies in the fourth -dimension—now the red line runs in the fourth dimension. -Thus the change from 10 to 11 should be towards -red, corresponding to a null point is a red point, to a -white line is a pink line, to a yellow line an orange -line, and so on.</p> - -<p>Catalogue cube 12 is like 10. Hence we see that to -build up our blocks of tesseract faces we must take the -bottom layer of the first block, hold that up in the air, -underneath it place the bottom layer of the second block,<span class="pagenum" id="Page_246">[Pg 246]</span> -and finally underneath this last the bottom layer of the -last of our normal blocks.</p> - -<p>Similarly we make the second representative group by -taking the middle courses of our three blocks. The last -is made by taking the three topmost layers. The three -axes in our space before the transverse motion begins are -blue, white, yellow, so we have light green tesseract -faces, and after the motion begins sections transverse to -the red light.</p> - -<p>These three blocks represent the appearances as the -tesseract group in its new position passes across our space. -The cubes of contact in this case are those determinal by -the three axes in our space, namely, the white, the -yellow, the blue. Hence they are light green.</p> - -<p>It follows from this that light green is the interior -cube of the first block of representative cubic faces.</p> - -<p>Practice in the manipulations described, with a -realization in each case of the face or section which -is in our space, is one of the best means of a thorough -comprehension of the subject.</p> - -<p>We have to learn how to get any part of these four-dimensional -figures into space, so that we can look at -them. We must first learn to swing a tesseract, and a -group of tesseracts about in any way.</p> - -<p>When these operations have been repeated and the -method of arrangement of the set of blocks has become -familiar, it is a good plan to rotate the axes of the normal -cube 1 about a diagonal, and then repeat the whole series -of turnings.</p> - -<p>Thus, in the normal position, red goes up, white to the -right, yellow away. Make white go up, yellow to the right, -and red away. Learn the cube in this position by putting -up the set of blocks of the normal cube, over and over -again till it becomes as familiar to you as in the normal -position. Then when this is learned, and the corre<span class="pagenum" id="Page_247">[Pg 247]</span>sponding -changes in the arrangements of the tesseract -groups are made, another change should be made: let, -in the normal cube, yellow go up, red to the right, and -white away.</p> - -<p>Learn the normal block of cubes in this new position -by arranging them and re-arranging them till you know -without thought where each one goes. Then carry out -all the tesseract arrangements and turnings.</p> - -<p>If you want to understand the subject, but do not see -your way clearly, if it does not seem natural and easy to -you, practise these turnings. Practise, first of all, the -turning of a block of cubes round, so that you know it -in every position as well as in the normal one. Practise -by gradually putting up the set of cubes in their new -arrangements. Then put up the tesseract blocks in their -arrangements. This will give you a working conception -of higher space, you will gain the feeling of it, whether -you take up the mathematical treatment of it or not.</p> -<hr class="chap x-ebookmaker-drop" /> - -<div class="chapter"> -<p><span class="pagenum" id="Page_248">[Pg 248]</span></p> - -<h2 class="nobreak" id="APPENDIX_II">APPENDIX II<br /> - -<small><i>A LANGUAGE OF SPACE</i></small></h2></div> - - -<p>The mere naming the parts of the figures we consider -involves a certain amount of time and attention. This -time and attention leads to no result, for with each -new figure the nomenclature applied is completely -changed, every letter or symbol is used in a different -significance.</p> - -<p>Surely it must be possible in some way to utilise the -labour thus at present wasted!</p> - -<p>Why should we not make a language for space itself, so -that every position we want to refer to would have its own -name? Then every time we named a figure in order to -demonstrate its properties we should be exercising -ourselves in the vocabulary of place.</p> - -<p>If we use a definite system of names, and always refer -to the same space position by the same name, we create -as it were a multitude of little hands, each prepared to -grasp a special point, position, or element, and hold it -for us in its proper relations.</p> - -<p>We make, to use another analogy, a kind of mental -paper, which has somewhat of the properties of a sensitive -plate, in that it will register, without effort, complex, -visual, or tactual impressions.</p> - -<p>But of far more importance than the applications of a -space language to the plane and to solid space is the<span class="pagenum" id="Page_249">[Pg 249]</span> -facilitation it brings with it to the study of four-dimensional -shapes.</p> - -<p>I have delayed introducing a space language because -all the systems I made turned out, after giving them a -fair trial, to be intolerable. I have now come upon one -which seems to present features of permanence, and I will -here give an outline of it, so that it can be applied to -the subject of the text, and in order that it may be -subjected to criticism.</p> - -<p>The principle on which the language is constructed is -to sacrifice every other consideration for brevity.</p> - -<p>It is indeed curious that we are able to talk and -converse on every subject of thought except the fundamental -one of space. The only way of speaking about -the spatial configurations that underlie every subject -of discursive thought is a co-ordinate system of numbers. -This is so awkward and incommodious that it is never -used. In thinking also, in realising shapes, we do not -use it; we confine ourselves to a direct visualisation.</p> - -<p>Now, the use of words corresponds to the storing up -of our experience in a definite brain structure. A child, -in the endless tactual, visual, mental manipulations it -makes for itself, is best left to itself, but in the course -of instruction the introduction of space names would -make the teachers work more cumulative, and the child’s -knowledge more social.</p> - -<p>Their full use can only be appreciated, if they are -introduced early in the course of education; but in a -minor degree any one can convince himself of their -utility, especially in our immediate subject of handling -four-dimensional shapes. The sum total of the results -obtained in the preceding pages can be compendiously and -accurately expressed in nine words of the Space Language.</p> - -<p>In one of Plato’s dialogues Socrates makes an experiment -on a slave boy standing by. He makes certain<span class="pagenum" id="Page_250">[Pg 250]</span> -perceptions of space awake in the mind of Meno’s slave -by directing his close attention on some simple facts of -geometry.</p> - -<p>By means of a few words and some simple forms we can -repeat Plato’s experiment on new ground.</p> - -<p>Do we by directing our close attention on the facts of -four dimensions awaken a latent faculty in ourselves? -The old experiment of Plato’s, it seems to me, has come -down to us as novel as on the day he incepted it, and its -significance not better understood through all the discussion -of which it has been the subject.</p> - -<p>Imagine a voiceless people living in a region where -everything had a velvety surface, and who were thus -deprived of all opportunity of experiencing what sound is. -They could observe the slow pulsations of the air caused -by their movements, and arguing from analogy, they -would no doubt infer that more rapid vibrations were -possible. From the theoretical side they could determine -all about these more rapid vibrations. They merely differ, -they would say, from slower ones, by the number that -occur in a given time; there is a merely formal difference.</p> - -<p>But suppose they were to take the trouble, go to the -pains of producing these more rapid vibrations, then a -totally new sensation would fall on their rudimentary ears. -Probably at first they would only be dimly conscious of -Sound, but even from the first they would become aware -that a merely formal difference, a mere difference in point -of number in this particular respect, made a great difference -practically, as related to them. And to us the difference -between three and four dimensions is merely formal, -numerical. We can tell formally all about four dimensions, -calculate the relations that would exist. But that the -difference is merely formal does not prove that it is a -futile and empty task, to present to ourselves as closely as -we can the phenomena of four dimensions. In our formal<span class="pagenum" id="Page_251">[Pg 251]</span> -knowledge of it, the whole question of its actual relation -to us, as we are, is left in abeyance.</p> - -<p>Possibly a new apprehension of nature may come to us -through the practical, as distinguished from the mathematical -and formal, study of four dimensions. As a child -handles and examines the objects with which he comes in -contact, so we can mentally handle and examine four-dimensional -objects. The point to be determined is this. -Do we find something cognate and natural to our faculties, -or are we merely building up an artificial presentation of -a scheme only formally possible, conceivable, but which -has no real connection with any existing or possible -experience?</p> - -<p>This, it seems to me, is a question which can only be -settled by actually trying. This practical attempt is the -logical and direct continuation of the experiment Plato -devised in the “Meno.”</p> - -<p>Why do we think true? Why, by our processes of -thought, can we predict what will happen, and correctly -conjecture the constitution of the things around us? -This is a problem which every modern philosopher has -considered, and of which Descartes, Leibnitz, Kant, to -name a few, have given memorable solutions. Plato was -the first to suggest it. And as he had the unique position -of being the first devisor of the problem, so his solution -is the most unique. Later philosophers have talked about -consciousness and its laws, sensations, categories. But -Plato never used such words. Consciousness apart from a -conscious being meant nothing to him. His was always -an objective search. He made man’s intuitions the basis -of a new kind of natural history.</p> - -<p>In a few simple words Plato puts us in an attitude -with regard to psychic phenomena—the mind—the ego—“what -we are,” which is analogous to the attitude scientific -men of the present day have with regard to the phenomena<span class="pagenum" id="Page_252">[Pg 252]</span> -of outward nature. Behind this first apprehension of ours -of nature, there is an infinite depth to be learned and -known. Plato said that behind the phenomena of mind -that Meno’s slave boy exhibited, there was a vast, an -infinite perspective. And his singularity, his originality, -comes out most strongly marked in this, that the perspective, -the complex phenomena beyond were, according -to him, phenomena of personal experience. A footprint -in the sand means a man to a being that has the conception -of a man. But to a creature that has no such -conception, it means a curious mark, somehow resulting -from the concatenation of ordinary occurrences. Such a -being would attempt merely to explain how causes known -to him could so coincide as to produce such a result; -he would not recognise its significance.</p> - -<p>Plato introduced the conception which made a new -kind of natural history possible. He said that Meno’s -slave boy thought true about things he had never -learned, because his “soul” had experience. I know this -will sound absurd to some people, and it flies straight -in the face of the maxim, that explanation consists in -showing how an effect depends on simple causes. But -what a mistaken maxim that is! Can any single instance -be shown of a simple cause? Take the behaviour of -spheres for instance; say those ivory spheres, billiard balls, -for example. We can explain their behaviour by supposing -they are homogeneous elastic solids. We can give formulæ -which will account for their movements in every variety. -But are they homogeneous elastic solids? No, certainly -not. They are complex in physical and molecular structure, -and atoms and ions beyond open an endless vista. Our -simple explanation is false, false as it can be. The balls -act as if they were homogeneous elastic spheres. There is -a statistical simplicity in the resultant of very complex -conditions, which makes that artificial conception useful.<span class="pagenum" id="Page_253">[Pg 253]</span> -But its usefulness must not blind us to the fact that it is -artificial. If we really look deep into nature, we find a -much greater complexity than we at first suspect. And -so behind this simple “I,” this myself, is there not a -parallel complexity? Plato’s “soul” would be quite -acceptable to a large class of thinkers, if by “soul” and -the complexity he attributes to it, he meant the product -of a long course of evolutionary changes, whereby simple -forms of living matter endowed with rudimentary sensation -had gradually developed into fully conscious beings.</p> - -<p>But Plato does not mean by “soul” a being of such a -kind. His soul is a being whose faculties are clogged by -its bodily environment, or at least hampered by the -difficulty of directing its bodily frame—a being which -is essentially higher than the account it gives of itself -through its organs. At the same time Plato’s soul is -not incorporeal. It is a real being with a real experience. -The question of whether Plato had the conception of non-spatial -existence has been much discussed. The verdict -is, I believe, that even his “ideas” were conceived by him -as beings in space, or, as we should say, real. Plato’s -attitude is that of Science, inasmuch as he thinks of a -world in Space. But, granting this, it cannot be denied -that there is a fundamental divergence between Plato’s -conception and the evolutionary theory, and also an -absolute divergence between his conception and the -genetic account of the origin of the human faculties. -The functions and capacities of Plato’s “soul” are not -derived by the interaction of the body and its environment.</p> - -<p>Plato was engaged on a variety of problems, and his -religious and ethical thoughts were so keen and fertile -that the experimental investigation of his soul appears -involved with many other motives. In one passage Plato -will combine matter of thought of all kinds and from all -sources, overlapping, interrunning. And in no case is he<span class="pagenum" id="Page_254">[Pg 254]</span> -more involved and rich than in this question of the soul. -In fact, I wish there were two words, one denoting that -being, corporeal and real, but with higher faculties than -we manifest in our bodily actions, which is to be taken as -the subject of experimental investigation; and the other -word denoting “soul” in the sense in which it is made -the recipient and the promise of so much that men desire. -It is the soul in the former sense that I wish to investigate, -and in a limited sphere only. I wish to find out, in continuation -of the experiment in the Meno, what the “soul” -in us thinks about extension, experimenting on the -grounds laid down by Plato. He made, to state the -matter briefly, the hypothesis with regard to the thinking -power of a being in us, a “soul.” This soul is not accessible -to observation by sight or touch, but it can be -observed by its functions; it is the object of a new kind -of natural history, the materials for constructing which -lie in what it is natural to us to think. With Plato -“thought” was a very wide-reaching term, but still I -would claim in his general plan of procedure a place for -the particular question of extension.</p> - -<p>The problem comes to be, “What is it natural to us to -think about matter <i>qua</i> extended?”</p> - -<p>First of all, I find that the ordinary intuition of any -simple object is extremely imperfect. Take a block of -differently marked cubes, for instance, and become acquainted -with them in their positions. You may think -you know them quite well, but when you turn them round—rotate -the block round a diagonal, for instance—you -will find that you have lost track of the individuals in -their new positions. You can mentally construct the -block in its new position, by a rule, by taking the remembered -sequences, but you don’t know it intuitively. By -observation of a block of cubes in various positions, and -very expeditiously by a use of Space names applied to the<span class="pagenum" id="Page_255">[Pg 255]</span> -cubes in their different presentations, it is possible to get -an intuitive knowledge of the block of cubes, which is not -disturbed by any displacement. Now, with regard to this -intuition, we moderns would say that I had formed it by -my tactual visual experiences (aided by hereditary pre-disposition). -Plato would say that the soul had been -stimulated to recognise an instance of shape which it -knew. Plato would consider the operation of learning -merely as a stimulus; we as completely accounting for -the result. The latter is the more common-sense view. -But, on the other hand, it presupposes the generation of -experience from physical changes. The world of sentient -experience, according to the modern view, is closed and -limited; only the physical world is ample and large and -of ever-to-be-discovered complexity. Plato’s world of soul, -on the other hand, is at least as large and ample as the -world of things.</p> - -<p>Let us now try a crucial experiment. Can I form an -intuition of a four-dimensional object? Such an object -is not given in the physical range of my sense contacts. -All I can do is to present to myself the sequences of solids, -which would mean the presentation to me under my conditions -of a four-dimensional object. All I can do is to -visualise and tactualise different series of solids which are -alternative sets of sectional views of a four-dimensional -shape.</p> - -<p>If now, on presenting these sequences, I find a power -in me of intuitively passing from one of these sets of -sequences to another, of, being given one, intuitively -constructing another, not using a rule, but directly apprehending -it, then I have found a new fact about my soul, -that it has a four-dimensional experience; I have observed -it by a function it has.</p> - -<p>I do not like to speak positively, for I might occasion -a loss of time on the part of others, if, as may very well<span class="pagenum" id="Page_256">[Pg 256]</span> -be, I am mistaken. But for my own part, I think there -are indications of such an intuition; from the results of -my experiments, I adopt the hypothesis that that which -thinks in us has an ample experience, of which the intuitions -we use in dealing with the world of real objects -are a part; of which experience, the intuition of four-dimensional -forms and motions is also a part. The process -we are engaged in intellectually is the reading the obscure -signals of our nerves into a world of reality, by means of -intuitions derived from the inner experience.</p> - -<p>The image I form is as follows. Imagine the captain -of a modern battle-ship directing its course. He has -his charts before him; he is in communication with his -associates and subordinates; can convey his messages and -commands to every part of the ship, and receive information -from the conning-tower and the engine-room. Now -suppose the captain immersed in the problem of the -navigation of his ship over the ocean, to have so absorbed -himself in the problem of the direction of his craft over -the plane surface of the sea that he forgets himself. All -that occupies his attention is the kind of movement that -his ship makes. The operations by which that movement -is produced have sunk below the threshold of his consciousness, -his own actions, by which he pushes the buttons, -gives the orders, are so familiar as to be automatic, his -mind is on the motion of the ship as a whole. In such -a case we can imagine that he identifies himself with his -ship; all that enters his conscious thought is the direction -of its movement over the plane surface of the ocean.</p> - -<p>Such is the relation, as I imagine it, of the soul to the -body. A relation which we can imagine as existing -momentarily in the case of the captain is the normal -one in the case of the soul with its craft. As the captain -is capable of a kind of movement, an amplitude of motion, -which does not enter into his thoughts with regard to the<span class="pagenum" id="Page_257">[Pg 257]</span> -directing the ship over the plane surface of the ocean, so -the soul is capable of a kind of movement, has an amplitude -of motion, which is not used in its task of directing -the body in the three-dimensional region in which the -body’s activity lies. If for any reason it became necessary -for the captain to consider three-dimensional motions with -regard to his ship, it would not be difficult for him to -gain the materials for thinking about such motions; all -he has to do is to call his own intimate experience into -play. As far as the navigation of the ship, however, is -concerned, he is not obliged to call on such experience. -The ship as a whole simply moves on a surface. The -problem of three-dimensional movement does not ordinarily -concern its steering. And thus with regard to ourselves -all those movements and activities which characterise our -bodily organs are three-dimensional; we never need to -consider the ampler movements. But we do more than -use the movements of our body to effect our aims by -direct means; we have now come to the pass when we act -indirectly on nature, when we call processes into play -which lie beyond the reach of any explanation we can -give by the kind of thought which has been sufficient for -the steering of our craft as a whole. When we come to -the problem of what goes on in the minute, and apply -ourselves to the mechanism of the minute, we find our -habitual conceptions inadequate.</p> - -<p>The captain in us must wake up to his own intimate -nature, realise those functions of movement which are his -own, and in virtue of his knowledge of them apprehend -how to deal with the problems he has come to.</p> - -<p>Think of the history of man. When has there been a -time, in which his thoughts of form and movement were -not exclusively of such varieties as were adapted for his -bodily performance? We have never had a demand to -conceive what our own most intimate powers are. But,<span class="pagenum" id="Page_258">[Pg 258]</span> -just as little as by immersing himself in the steering of -his ship over the plane surface of the ocean, a captain -can lose the faculty of thinking about what he actually -does, so little can the soul lose its own nature. It -can be roused to an intuition that is not derived from -the experience which the senses give. All that is -necessary is to present some few of those appearances -which, while inconsistent with three-dimensional matter, -are yet consistent with our formal knowledge of four-dimensional -matter, in order for the soul to wake up and -not begin to learn, but of its own intimate feeling fill up -the gaps in the presentiment, grasp the full orb of possibilities -from the isolated points presented to it. In relation -to this question of our perceptions, let me suggest another -illustration, not taking it too seriously, only propounding -it to exhibit the possibilities in a broad and general way.</p> - -<p>In the heavens, amongst the multitude of stars, there -are some which, when the telescope is directed on them, -seem not to be single stars, but to be split up into two. -Regarding these twin stars through a spectroscope, an -astronomer sees in each a spectrum of bands of colour and -black lines. Comparing these spectrums with one another, -he finds that there is a slight relative shifting of the dark -lines, and from that shifting he knows that the stars are -rotating round one another, and can tell their relative -velocity with regard to the earth. By means of his -terrestrial physics he reads this signal of the skies. This -shifting of lines, the mere slight variation of a black line -in a spectrum, is very unlike that which the astronomer -knows it means. But it is probably much more like what -it means than the signals which the nerves deliver are -like the phenomena of the outer world.</p> - -<p>No picture of an object is conveyed through the nerves. -No picture of motion, in the sense in which we postulate -its existence, is conveyed through the nerves. The actual<span class="pagenum" id="Page_259">[Pg 259]</span> -deliverances of which our consciousness takes account are -probably identical for eye and ear, sight and touch.</p> - -<p>If for a moment I take the whole earth together and -regard it as a sentient being, I find that the problem of -its apprehension is a very complex one, and involves a -long series of personal and physical events. Similarly the -problem of our apprehension is a very complex one. I -only use this illustration to exhibit my meaning. It has -this especial merit, that, as the process of conscious -apprehension takes place in our case in the minute, so, -with regard to this earth being, the corresponding process -takes place in what is relatively to it very minute.</p> - -<p>Now, Plato’s view of a soul leads us to the hypothesis -that that which we designate as an act of apprehension -may be a very complex event, both physically and personally. -He does not seek to explain what an intuition -is; he makes it a basis from whence he sets out on a -voyage of discovery. Knowledge means knowledge; he -puts conscious being to account for conscious being. He -makes an hypothesis of the kind that is so fertile in -physical science—an hypothesis making no claim to -finality, which marks out a vista of possible determination -behind determination, like the hypothesis of space itself, -the type of serviceable hypotheses.</p> - -<p>And, above all, Plato’s hypothesis is conducive to experiment. -He gives the perspective in which real objects -can be determined; and, in our present enquiry, we are -making the simplest of all possible experiments—we are -enquiring what it is natural to the soul to think of matter -as extended.</p> - -<p>Aristotle says we always use a “phantasm” in thinking, -a phantasm of our corporeal senses a visualisation or a -tactualisation. But we can so modify that visualisation -or tactualisation that it represents something not known -by the senses. Do we by that representation wake up an<span class="pagenum" id="Page_260">[Pg 260]</span> -intuition of the soul? Can we by the presentation of -these hypothetical forms, that are the subject of our -present discussion, wake ourselves up to higher intuitions? -And can we explain the world around by a motion that we -only know by our souls?</p> - -<p>Apart from all speculation, however, it seems to me -that the interest of these four-dimensional shapes and -motions is sufficient reason for studying them, and that -they are the way by which we can grow into a fuller -apprehension of the world as a concrete whole.</p> - - -<h3><span class="smcap">Space Names.</span></h3> - -<p>If the words written in the squares drawn in <a href="#fig_144">fig. 1</a> are -used as the names of the squares in the positions in -which they are placed, it is evident that -a combination of these names will denote -a figure composed of the designated -squares. It is found to be most convenient -to take as the initial square that -marked with an asterisk, so that the -directions of progression are towards the -observer and to his right. The directions -of progression, however, are arbitrary, and can be chosen -at will.</p> - -<div class="figleft illowp25" id="fig_144" style="max-width: 12.5em;"> - <img src="images/fig_144.png" alt="" /> - <div class="caption">Fig. 1.</div> -</div> - -<p>Thus <i>et</i>, <i>at</i>, <i>it</i>, <i>an</i>, <i>al</i> will denote a figure in the form -of a cross composed of five squares.</p> - -<p>Here, by means of the double sequence, <i>e</i>, <i>a</i>, <i>i</i> and <i>n</i>, <i>t</i>, <i>l</i>, it -is possible to name a limited collection of space elements.</p> - -<p>The system can obviously be extended by using letter -sequences of more members.</p> - -<p>But, without introducing such a complexity, the -principles of a space language can be exhibited, and a -nomenclature obtained adequate to all the considerations -of the preceding pages.</p> - -<p><span class="pagenum" id="Page_261">[Pg 261]</span></p> - - -<p>1. <i>Extension.</i></p> - -<div class="figleft illowp35" id="fig_145" style="max-width: 15.625em;"> - <img src="images/fig_145.png" alt="" /> - <div class="caption">Fig. 2.</div> -</div> - -<p>Call the large squares in <a href="#fig_145">2</a> by the name written -in them. It is evident that each -can be divided as shown in <a href="#fig_144">fig. 1</a>. -Then the small square marked 1 -will be “en” in “En,” or “Enen.” -The square marked 2 will be “et” -in “En” or “Enet,” while the -square marked 4 will be “en” in -“Et” or “Eten.” Thus the square -5 will be called “Ilil.”</p> - -<p>This principle of extension can -be applied in any number of dimensions.</p> - - -<p>2. <i>Application to Three-Dimensional Space.</i></p> - -<div class="figleft illowp25" id="fig_146" style="max-width: 12.5em;"> - <img src="images/fig_146.png" alt="Three cube faces" /> -</div> - -<p>To name a three-dimensional collocation of cubes take -the upward direction first, secondly the -direction towards the observer, thirdly the -direction to his right hand.</p> - -<p>These form a word in which the first -letter gives the place of the cube upwards, -the second letter its place towards the -observer, the third letter its place to the -right.</p> - -<p>We have thus the following scheme, -which represents the set of cubes of -column 1, <a href="#fig_101">fig. 101</a>, page 165.</p> - -<p>We begin with the remote lowest cube -at the left hand, where the asterisk is -placed (this proves to be by far the most -convenient origin to take for the normal -system).</p> - -<p>Thus “nen” is a “null” cube, “ten” -a red cube on it, and “len” a “null” -cube above “ten.”</p> - -<p><span class="pagenum" id="Page_262">[Pg 262]</span></p> - -<p>By using a more extended sequence of consonants and -vowels a larger set of cubes can be named.</p> - -<p>To name a four-dimensional block of tesseracts it is -simply necessary to prefix an “e,” an “a,” or an “i” to -the cube names.</p> - -<p>Thus the tesseract blocks schematically represented on -page 165, <a href="#fig_101">fig. 101</a> are named as follows:—</p> - -<div class="figcenter illowp80" id="fig_147" style="max-width: 62.5em;"> - <img src="images/fig_147.png" alt="Nine cube faces" /> -</div> - -<p>2. <span class="smcap">Derivation of Point, Line, Face, etc., Names.</span></p> - -<p>The principle of derivation can be shown as follows: -Taking the square of squares<span class="pagenum" id="Page_263">[Pg 263]</span></p> - -<div class="figcenter illowp35" id="fig_148" style="max-width: 15.625em;"> - <img src="images/fig_148.png" alt="Cube face" /> -</div> -<p class="pnind">the number of squares in it can be enlarged and the -whole kept the same size.</p> - -<div class="figcenter illowp35" id="fig_149" style="max-width: 15.625em;"> - <img src="images/fig_149.png" alt="Cube face" /> -</div> - -<p>Compare <a href="#fig_79">fig. 79</a>, p. 138, for instance, or the bottom layer -of <a href="#fig_84">fig. 84</a>.</p> - -<p>Now use an initial “s” to denote the result of carrying -this process on to a great extent, and we obtain the limit -names, that is the point, line, area names for a square. -“Sat” is the whole interior. The corners are “sen,” -“sel,” “sin,” “sil,” while the lines -are “san,” “sal,” “set,” “sit.”</p> - -<div class="figleft illowp30" id="fig_150" style="max-width: 15.625em;"> - <img src="images/fig_150.png" alt="see para above" /> -</div> - -<p>I find that by the use of the -initial “s” these names come to be -practically entirely disconnected with -the systematic names for the square -from which they are derived. They -are easy to learn, and when learned -can be used readily with the axes running in any -direction.</p> - -<p>To derive the limit names for a four-dimensional rectangular -figure, like the tesseract, is a simple extension of -this process. These point, line, etc., names include those -which apply to a cube, as will be evident on inspection -of the first cube of the diagrams which follow.</p> - -<p>All that is necessary is to place an “s” before each of the -names given for a tesseract block. We then obtain -apellatives which, like the colour names on page 174, -<a href="#fig_103">fig. 103</a>, apply to all the points, lines, faces, solids, and to<span class="pagenum" id="Page_264">[Pg 264]</span> -the hyper-solid of the tesseract. These names have the -advantage over the colour marks that each point, line, etc., -has its own individual name.</p> - -<p>In the diagrams I give the names corresponding to -the positions shown in the coloured plate or described on -p. 174. By comparing cubes 1, 2, 3 with the first row of -cubes in the coloured plate, the systematic names of each -of the points, lines, faces, etc., can be determined. The -asterisk shows the origin from which the names run.</p> - -<p>These point, line, face, etc., names should be used in -connection with the corresponding colours. The names -should call up coloured images of the parts named in their -right connection.</p> - -<p>It is found that a certain abbreviation adds vividness of -distinction to these names. If the final “en” be dropped -wherever it occurs the system is improved. Thus instead -of “senen,” “seten,” “selen,” it is preferable to abbreviate -to “sen,” “set,” “sel,” and also use “san,” “sin” for -“sanen,” “sinen.”</p> -<div class="figcenter illowp100" id="fig_151" style="max-width: 62.5em;"> - <img src="images/fig_151.png" alt="See above" /> -</div> -<p><span class="pagenum" id="Page_265">[Pg 265]</span></p> - -<div class="figcenter illowp100" id="fig_152" style="max-width: 62.5em;"> - <img src="images/fig_152.png" alt="see above" /> -</div> - -<div class="figcenter illowp100" id="fig_153" style="max-width: 62.5em;"> - <img src="images/fig_153.png" alt="see above" /> -</div> - -<p><span class="pagenum" id="Page_266">[Pg 266]</span></p> - -<div class="figcenter illowp100" id="fig_154" style="max-width: 62.5em;"> - <img src="images/fig_154.png" alt="see above" /> -</div> - -<p>We can now name any section. Take <i>e.g.</i> the line in -the first cube from senin to senel, we should call the line -running from senin to senel, senin senat senel, a line -light yellow in colour with null points.</p> - -<p>Here senat is the name for all of the line except its ends. -Using “senat” in this way does not mean that the line is -the whole of senat, but what there is of it is senat. It is -a part of the senat region. Thus also the triangle, which -has its three vertices in senin, senel, selen, is named thus:</p> - - -<ul> -<li>Area: setat.</li> -<li>Sides: setan, senat, setet.</li> -<li>Vertices: senin, senel, sel.</li> -</ul> - -<p>The tetrahedron section of the tesseract can be thought -of as a series of plane sections in the successive sections of -the tesseract shown in <a href="#fig_114">fig. 114</a>, p. 191. In b<sub>0</sub> the section -<span class="pagenum" id="Page_267">[Pg 267]</span>is the one written above. In b<sub>1</sub> the section is made by a -plane which cuts the three edges from sanen intermediate -of their lengths and thus will be:</p> - - -<ul> -<li>Area: satat.</li> -<li>Sides: satan, sanat, satet.</li> -<li>Vertices: sanan, sanet, sat.</li> -</ul> - - -<p>The sections in b<sub>2</sub>, b<sub>3</sub> will be like the section in b<sub>1</sub> but -smaller.</p> - -<p>Finally in b<sub>4</sub> the section plane simply passes through the -corner named sin.</p> - -<p>Hence, putting these sections together in their right -relation, from the face setat, surrounded by the lines and -points mentioned above, there run:</p> - - -<ul> -<li>3 faces: satan, sanat, satet</li> -<li>3 lines: sanan, sanet, sat</li> -</ul> - - -<p>and these faces and lines run to the point sin. Thus -the tetrahedron is completely named.</p> - -<p>The octahedron section of the tesseract, which can be -traced from <a href="#fig_72">fig. 72</a>, p. 129 by extending the lines there -drawn, is named:</p> - -<p>Front triangle selin, selat, selel, setal, senil, setit, selin -with area setat.</p> - -<p>The sections between the front and rear triangle, of -which one is shown in 1b, another in 2b, are thus named, -points and lines, salan, salat, salet, satet, satel, satal, sanal, -sanat, sanit, satit, satin, satan, salan.</p> - -<p>The rear triangle found in 3b by producing lines is sil, -sitet, sinel, sinat, sinin, sitan, sil.</p> - -<p>The assemblage of sections constitute the solid body of -the octahedron satat with triangular faces. The one from -the line selat to the point sil, for instance, is named<span class="pagenum" id="Page_268">[Pg 268]</span> -selin, selat, selel, salet, salat, salan, sil. The whole -interior is salat.</p> - -<p>Shapes can easily be cut out of cardboard which, when -folded together, form not only the tetrahedron and the -octahedron, but also samples of all the sections of the -tesseract taken as it passes cornerwise through our space. -To name and visualise with appropriate colours a series of -these sections is an admirable exercise for obtaining -familiarity with the subject.</p> - - -<h3><span class="smcap">Extension and Connection with Numbers.</span></h3> - -<p>By extending the letter sequence it is of course possible -to name a larger field. By using the limit names the -corners of each square can be named.</p> - -<p>Thus “en sen,” “an sen,” etc., will be the names of the -points nearest the origin in “en” and in “an.”</p> - -<p>A field of points of which each one is indefinitely small -is given by the names written below.</p> - -<div class="figcenter illowp30" id="fig_155" style="max-width: 12.5em;"> - <img src="images/fig_155.png" alt="Field of points" /> -</div> - -<p>The squares are shown in dotted lines, the names -denote the points. These points are not mathematical -points, but really minute areas.</p> - -<p>Instead of starting with a set of squares and naming -them, we can start with a set of points.</p> - -<p>By an easily remembered convention we can give -names to such a region of points.</p> - -<p><span class="pagenum" id="Page_269">[Pg 269]</span></p> - -<p>Let the space names with a final “e” added denote the -mathematical points at the corner of each square nearest -the origin. We have then</p> - -<div class="figcenter illowp25" id="i_269" style="max-width: 15.625em;"> - <img src="images/i_269.png" alt="illustrating immediate text" /> -</div> -<p class="pnind">for the set of mathematical points indicated. This -system is really completely independent of the area -system and is connected with it merely for the purpose -of facilitating the memory processes. The word “ene” is -pronounced like “eny,” with just sufficient attention to -the final vowel to distinguish it from the word “en.”</p> - -<p>Now, connecting the numbers 0, 1, 2 with the sequence -e, a, i, and also with the sequence n, t, l, we have a set of -points named as with numbers in a co-ordinate system. -Thus “ene” is (0, 0) “ate” is (1, 1) “ite” is (2, 1). -To pass to the area system the rule is that the name of -the square is formed from the name of its point nearest -to the origin by dropping the final e.</p> - -<p>By using a notation analogous to the decimal system -a larger field of points can be named. It remains to -assign a letter sequence to the numbers from positive 0 -to positive 9, and from negative 0 to negative 9, to obtain -a system which can be used to denote both the usual -co-ordinate system of mapping and a system of named -squares. The names denoting the points all end with e. -Those that denote squares end with a consonant.</p> - -<p>There are many considerations which must be attended -to in extending the sequences to be used, such as -uniqueness in the meaning of the words formed, ease -of pronunciation, avoidance of awkward combinations.</p> - -<p><span class="pagenum" id="Page_270">[Pg 270]</span></p> - -<p>I drop “s” altogether from the consonant series and -short “u” from the vowel series. It is convenient to -have unsignificant letters at disposal. A double consonant -like “st” for instance can be referred to without giving it -a local significance by calling it “ust.” I increase the -number of vowels by considering a sound like “ra” to -be a vowel, using, that is, the letter “r” as forming a -compound vowel.</p> - -<p>The series is as follows:—</p> - -<table class="standard" summary=""> -<tr> -<td class="tdc" colspan="11"><span class="smcap">Consonants.</span></td> -</tr> -<tr> -<td class="tdc"></td> -<td class="tdc">0</td> -<td class="tdc">1</td> -<td class="tdc">2</td> -<td class="tdc">3</td> -<td class="tdc">4</td> -<td class="tdc">5</td> -<td class="tdc">6</td> -<td class="tdc">7</td> -<td class="tdc">8</td> -<td class="tdc">9</td> -</tr> -<tr> -<td class="tdl">positive</td> -<td class="tdc">n</td> -<td class="tdc">t</td> -<td class="tdc">l</td> -<td class="tdc">p</td> -<td class="tdc">f</td> -<td class="tdc">sh</td> -<td class="tdc">k</td> -<td class="tdc">ch</td> -<td class="tdc">nt</td> -<td class="tdc">st</td> -</tr> -<tr> -<td class="tdl">negative</td> -<td class="tdc">z</td> -<td class="tdc">d</td> -<td class="tdc">th</td> -<td class="tdc">b</td> -<td class="tdc">v</td> -<td class="tdc">m</td> -<td class="tdc">g</td> -<td class="tdc">j</td> -<td class="tdc">nd</td> -<td class="tdc">sp</td> -</tr> -<tr> -<td class="tdc" colspan="11"><span class="smcap">Vowels.</span></td> -</tr> -<tr> -<td class="tdc"></td> -<td class="tdc">0</td> -<td class="tdc">1</td> -<td class="tdc">2</td> -<td class="tdc">3</td> -<td class="tdc">4</td> -<td class="tdc">5</td> -<td class="tdc">6</td> -<td class="tdc">7</td> -<td class="tdc">8</td> -<td class="tdc">9</td> -</tr> -<tr> -<td class="tdc">positive</td> -<td class="tdc">e</td> -<td class="tdc">a</td> -<td class="tdc">i</td> -<td class="tdc">ee</td> -<td class="tdc">ae</td> -<td class="tdc">ai</td> -<td class="tdc">ar</td> -<td class="tdc">ra</td> -<td class="tdc">ri</td> -<td class="tdc">ree</td> -</tr> -<tr> -<td class="tdc">negative</td> -<td class="tdc">er</td> -<td class="tdc">o</td> -<td class="tdc">oo</td> -<td class="tdc">io</td> -<td class="tdc">oe</td> -<td class="tdc">iu</td> -<td class="tdc">or</td> -<td class="tdc">ro</td> -<td class="tdc">roo rio</td> -</tr> -</table> - - -<p><i>Pronunciation.</i>—e as in men; a as in man; i as in in; -ee as in between; ae as ay in may; ai as i in mine; ar as -in art; er as ear in earth; o as in on; oo as oo in soon; -io as in clarion; oe as oa in oat; iu pronounced like yew.</p> - -<p>To name a point such as (23, 41) it is considered as -(3, 1) on from (20, 40) and is called “ifeete.” It is the -initial point of the square ifeet of the area system.</p> - -<p>The preceding amplification of a space language has -been introduced merely for the sake of completeness. As -has already been said nine words and their combinations, -applied to a few simple models suffice for the purposes of -our present enquiry.</p> - - -<p class="center small"><i>Printed by Hazell, Watson & Viney, Ld., London and Aylesbury.</i></p> - -<div style='display:block; margin-top:4em'>*** END OF THE PROJECT GUTENBERG EBOOK THE FOURTH DIMENSION ***</div> -<div style='text-align:left'> - -<div style='display:block; margin:1em 0'> -Updated editions will replace the previous one—the old editions will -be renamed. -</div> - -<div style='display:block; margin:1em 0'> -Creating the works from print editions not protected by U.S. copyright -law means that no one owns a United States copyright in these works, -so the Foundation (and you!) can copy and distribute it in the United -States without permission and without paying copyright -royalties. 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