,transcriber's ,notes ,obvious typographical errors have been silently corrected4 ,all other spelling and punctuation remains unchanged4 ,italics are represented thus italic1 bold thus bold and superscripts thus yen4 ,it should be noted that much of the text is a discussion centred on the many illustrations which have not been included4 ,,the ,,fourth ,,dimension ,,some ,,opinions ,,of ,,the ,,press 8,mr4 ,c4 ,h4 ,hinton discusses the subject of the higher dimensionality of space1 his aim being to avoid mathematical subtleties and technicalities1 and thus enable his argument to be followed by readers who are not sufficiently conversant with mathematics to follow these processes of reasoning40--,,notts ,,guardian4 8,the fourth dimension is a subject which has had a great fascination for many teachers1 and though one cannot pretend to have quite grasped ,mr4 ,hinton's conceptions and arguments1 yet it must be admitted that he reveals the elusive idea in quite a fascinating light4 ,quite apart from the main thesis of the book many chapters are of great independent interest4 ,altogether an interesting1 clever and ingenious book40--,,dundee ,,courier4 8,the book will well repay the study of men who like to exercise their wits upon the problems of abstract thought40--,,scotsman4 8,professor ,hinton has done well to attempt a treatise of moderate size1 which shall at once be clear in method and free from technicalities of the schools40--,,pall ,,mall ,,gazette4 8,a very interesting book he has made of it40--,,publishers' ,,circular4 8,mr4 ,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 it4 ,if he is not altogether successful it is not from want of clearness on his part1 but because the whole theory comes as such an absolute shock to all one's preconceived ideas40--,,bristol ,,times4 8,mr4 ,hinton's enthusiasm is only the result of an exhaustive study1 which has enabled him to set his subject before the reader with far more than the amount of lucidity to which it is accustomed40--,,pall ,,mall ,,gazette4 8,the book throughout is a very solid piece of reasoning in the domain of higher mathematics40--,,glasgow ,,herald4 8,those who wish to grasp the meaning of this somewhat difficult subject would do well to read ,the ,fourth ,dimension4 ,no mathematical knowledge is demanded of the reader1 and any one1 who is not afraid of a little hard thinking1 should be able to follow the argument40--,,light4 8,a splendidly clear re-statement of the old problem of the fourth dimension4 ,all who are interested in this subject will find the work not only fascinating1 but lucid1 it being written in a style easily understandable4 ,the illustrations make still more clear the letterpress1 and the whole is most admirably adapted to the requirements of the novice or the student40--,,two ,,worlds4 8,those in search of mental gymnastics will find abundance of exercise in ,mr4 ,c4 ,h4 ,hinton's ,fourth ,dimension40--,,westminster ,,review4 ,,first ,,edition1 ,april #aijd2 ,,second ,,edition1 ,may #aijf4 ,views of the ,tessaract4 ,no4 #a4 ,no4 #b4 ,no4 #c4 ,no4 #d4 ,no4 #e4 ,no4 #f4 ,no4 #g4 ,no4 #h4 ,no4 #i4 ,no4 #aj4 ,no4 #aa4 ,no4 #ab4 ,,the ,,fourth ,,dimension ,,by ,c4 ,,howard ,,hinton1 ,m4,a4 ,,author ,,of 8,,scientific ,,romances0 8,a ,,new ,,era ,,of ,,thought10 ,,etc41 ,,etc4 ,illustration3 ,colophon ,,london ,,swan ,,sonnenschein ,,co41 ,,limited #be ,,high ,,street1 ,,bloomsbury #aijf ,,printed ,,by ,,hazell1 ,,watson ,,and ,,viney1 ,,ld41 ,,london ,,and ,,aylesbury4 ,,preface ,i have endeavoured to present the subject of the higher dimensionality of space in a clear manner1 devoid of mathematical subtleties and technicalities4 ,in order to engage the interest of the reader1 ,i have in the earlier chapters dwelt on the perspective the hypothesis of a fourth dimension opens1 and have treated of the many connections there are between this hypothesis and the ordinary topics of our thoughts4 ,a lack of mathematical knowledge will prove of no disadvantage to the reader1 for ,i have used no mathematical processes of reasoning4 ,i have taken the view that the space which we ordinarily think of1 the space of real things 7which ,i would call permeable matter71 is different from the space treated of by mathematics4 ,mathematics will tell us a great deal about space1 just as the atomic theory will tell us a great deal about the chemical combinations of bodies4 ,but after all1 a theory is not precisely equivalent to the subject with regard to which it is held4 ,there is an opening1 therefore1 from the side of our ordinary space perceptions for a simple1 altogether rational1 mechanical1 and observational way of treating this subject of higher space1 and of this opportunity ,i have availed myself4 ,the details introduced in the earlier chapters1 especially in ,chapters ,,viii41 ,,ix41 ,x41 may perhaps be found wearisome4 ,they are of no essential importance in the main line of argument1 and if left till ,chapters ,,xi4 and ,,xii4 have been read1 will be found to afford interesting and obvious illustrations of the properties discussed in the later chapters4 ,my thanks are due to the friends who have assisted me in designing and preparing the modifications of my previous models1 and in no small degree to the publisher of this volume1 ,mr4 ,sonnenschein1 to whose unique appreciation of the line of thought of this1 as of my former essays1 their publication is owing4 ,by the provision of a coloured plate1 in addition to the other illustrations1 he has added greatly to the convenience of the reader4 ,c4 ,,howard ,,hinton4 ,,contents ,,chap4 ,,page ,i4 ,,four-,,dimensional ,,space #a ,,ii4 ,,the ,,analogy ,,of ,a ,,plane ,,world #f ,,iii4 ,,the ,,significance ,,of ,a ,,four-,,dimensional ,,existence #ae ,,iv4 ,,the ,,first ,,chapter ,,in ,,the ,,history ,,of ,,four ,,space #bc ,v4 ,,the ,,second ,,chapter ,,in ,,the ,,history ,,of ,,four ,,space #da ,lobatchewsky1 ,bolyai1 and ,gauss ,metageometry ,,vi4 ,,the ,,higher ,,world #fa ,,vii4 ,,the ,,evidence ,,for ,a ,,fourth ,,dimension #gf ,,viii4 ,,the ,,use ,,of ,,four ,,dimensions ,,in ,,thought #he ,,ix4 ,,application ,,to ,,kant',s ,,theory ,,of ,,experience #ajg ,x4 ,a ,,four-,,dimensional ,,figure #abb ,,xi4 ,,nomenclature ,,and ,,analogies #acf ,,xii4 ,,the ,,simplest ,,four-,,dimensional ,,solid #aeg ,,xiii4 ,,remarks ,,on ,,the ,,figures #agh ,,xiv4 ,a ,,recapitulation ,,and ,,extension ,,of ,,the ,,physical ,,argument #bjc ,,appendix ,i4--,,the ,,models #bca ,,ii4--,a ,,language ,,of ,,space #bdh ,,the ,,fourth ,,dimension ,,chapter ,i ,,four-,,dimensional ,,space ,there is nothing more indefinite1 and at the same time more real1 than that which we indicate when we speak of the 8higher40 ,in our social life we see it evidenced in a greater complexity of relations4 ,but this complexity is not all4 ,there is1 at the same time1 a contact with1 an apprehension of1 something more fundamental1 more real4 ,with the greater development of man there comes a consciousness of something more than all the forms in which it shows itself4 ,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 representation4 ,the physical life of civilised man and of a mere savage are practically the same1 but the civilised man has discovered a depth in his existence1 which makes him feel that that which appears all to the savage is a mere externality and appurtenage to his true being4 ,now1 this higher--how shall we apprehend it8 ,it is generally embraced by our religious faculties1 by our idealising tendency4 ,but the higher existence has two sides4 ,it has a being as well as qualities4 ,and in trying to realise it through our emotions we are always taking the subjective view4 ,our attention is always fixed on what we feel1 what we think4 ,is there any way of apprehending the higher after the purely objective method of a natural science8 ,i think that there is4 ,plato1 in a wonderful allegory1 speaks of some men living in such a condition that they were practically reduced to be the denizens of a shadow world4 ,they were chained1 and perceived but the shadows of themselves and all real objects projected on a wall1 towards which their faces were turned4 ,all movements to them were but movements on the surface1 all shapes but the shapes of outlines with no substantiality4 ,plato uses this illustration to portray the relation between true being and the illusions of the sense world4 ,he says that just as a man liberated from his chains could learn and discover that the world was solid and real1 and could go back and tell his bound companions of this greater higher reality1 so the philosopher who has been liberated1 who has gone into the thought of the ideal world1 into the world of ideas greater and more real than the things of sense1 can come and tell his fellow men of that which is more true than the visible sun--more noble than ,athens1 the visible state4 ,now1 ,i take ,plato's suggestion2 but literally1 not metaphorically4 ,he imagines a world which is lower than this world1 in that shadow figures and shadow motions are its constituents2 and to it he contrasts the real world4 ,as the real world is to this shadow world1 so is the higher world to our world4 ,i accept his analogy4 ,as our world in three dimensions is to a shadow or plane world1 so is the higher world to our three-dimensional world4 ,that is1 the higher world is four-dimensional2 the higher being is1 so far as its existence is concerned apart from its qualities1 to be sought through the conception of an actual existence spatially higher than that which we realise with our senses4 ,here you will observe ,i necessarily leave out all that gives its charm and interest to ,plato's writings4 ,all those conceptions of the beautiful and good which live immortally in his pages4 ,all that ,i keep from his great storehouse of wealth is this one thing simply--a world spatially higher than this world1 a world which can only be approached through the stocks and stones of it1 a world which must be apprehended laboriously1 patiently1 through the material things of it1 the shapes1 the movements1 the figures of it4 ,we must learn to realise the shapes of objects in this world of the higher man2 we must become familiar with the movements that objects make in his world1 so that we can learn something about his daily experience1 his thoughts of material objects1 his machinery4 ,the means for the prosecution of this enquiry are given in the conception of space itself4 ,it often happens that that which we consider to be unique and unrelated gives us1 within itself1 those relations by means of which we are able to see it as related to others1 determining and determined by them4 ,thus1 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 planets4 ,our terrestrial globe was determined in regard to other bodies of the solar system by means of a relation which subsisted on the earth itself4 ,and so space itself bears within it relations of which we can determine it as related to other space4 ,for within space are given the conceptions of point and line1 line and plane1 which really involve the relation of space to a higher space4 ,where one segment of a straight line leaves off and another begins is a point1 and the straight line itself can be generated by the motion of the point4 ,one portion of a plane is bounded from another by a straight line1 and the plane itself can be generated by the straight line moving in a direction not contained in itself4 ,again1 two portions of solid space are limited with regard to each other by a plane2 and the plane1 moving in a direction not contained in itself1 can generate solid space4 ,thus1 going on1 we may say that space is that which limits two portions of higher space from each other1 and that our space will generate the higher space by moving in a direction not contained in itself4 ,another indication of the nature of four-dimensional space can be gained by considering the problem of the arrangement of objects4 ,if ,i have a number of swords of varying degrees of brightness1 ,i can represent them in respect of this quality by points arranged along a straight line4 ,if ,i place a sword at ,a1 fig4 #a1 and regard it as having a certain brightness1 then the other swords can be arranged in a series along the line1 as at ,a1 ,b1 ,c1 etc41 according to their degrees of brightness4 ,illustration3 ,fig4 #a4 ,if now ,i take account of another quality1 say length1 they can be arranged in a plane4 ,starting from ,a1 ,b1 ,c1 ,i can find points to represent different degrees of length along such lines as ,,af1 ,,bd1 ,,ce1 drawn from ,a and ,b and ,c4 ,points on these lines represent different degrees of length with the same degree of brightness4 ,thus the whole plane is occupied by points representing all conceivable varieties of brightness and length4 ,illustration3 ,fig4 #b4 ,bringing in a third quality1 say sharpness1 ,i can draw1 as in fig4 #c1 any number of upright lines4 ,let distances along these upright lines represent degrees of sharpness1 thus the points ,f and ,g will represent swords of certain definite degrees of the three qualities mentioned1 and the whole of space will serve to represent all conceivable degrees of these three qualities4 ,illustration3 ,fig4 #c4 ,if now ,i bring in a fourth quality1 such as weight1 and try to find a means of representing it as ,i did the other three qualities1 ,i find a difficulty4 ,every point in space is taken up by some conceivable combination of the three qualities already taken4 ,to represent four qualities in the same way as that in which ,i have represented three1 ,i should need another dimension of space4 ,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 qualities1 as three-dimensional space gives positions representative of three qualities4 ,,chapter ,,ii ,,the ,,analogy ,,of ,a ,,plane ,,world ,at the risk of some prolixity ,i will go fully into the experience of a hypothetical creature confined to motion on a plane surface4 ,by so doing ,i shall obtain an analogy which will serve in our subsequent enquiries1 because the change in our conception1 which we make in passing from the shapes and motions in two dimensions to those in three1 affords a pattern by which we can pass on still further to the conception of an existence in four-dimensional space4 ,a piece of paper on a smooth table affords a ready image of a two-dimensional existence4 ,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 table1 it is obvious that he can have no knowledge of objects of a similar description1 except by the contact with their edges4 ,his body and the objects in his world have a thickness of which however1 he has no consciousness4 ,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 only4 ,figures are to him completely bounded by their lines1 just as solid objects are to us by their surfaces4 ,he cannot conceive of approaching the centre of a circle1 except by breaking through the circumference1 for the circumference encloses the centre in the directions in which motion is possible to him4 ,the plane surface over which he slips and with which he is always in contact will be unknown to him2 there are no differences by which he can recognise its existence4 ,but for the purposes of our analogy this representation is deficient4 ,a being as thus described has nothing about him to push off from1 the surface over which he slips affords no means by which he can move in one direction rather than another4 ,placed on a surface over which he slips freely1 he is in a condition analogous to that in which we should be if we were suspended free in space4 ,there is nothing which he can push off from in any direction known to him4 ,let us therefore modify our representation4 ,let us suppose a vertical plane against which particles of thin matter slip1 never leaving the surface4 ,let these particles possess an attractive force and cohere together into a disk2 this disk will represent the globe of a plane being4 ,he must be conceived as existing on the rim4 ,illustration3 ,fig4 #d4 ,let #a represent this vertical disk of flat matter and #b the plane being on it1 standing upon its rim as we stand on the surface of our earth4 ,the direction of the attractive force of his matter will give the creature a knowledge of up and down1 determining for him one direction in his plane space4 ,also1 since he can move along the surface of his earth1 he will have the sense of a direction parallel to its surface1 which we may call forwards and backwards4 ,he will have no sense of right and left--that is1 of the direction which we recognise as extending out from the plane to our right and left4 ,the distinction of right and left is the one that we must suppose to be absent1 in order to project ourselves into the condition of a plane being4 ,let the reader imagine himself1 as he looks along the plane1 fig4 #d1 to become more and more identified with the thin body on it1 till he finally looks along parallel to the surface of the plane earth1 and up and down1 losing the sense of the direction which stretches right and left4 ,this direction will be an unknown dimension to him4 ,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 being1 without picturing him as in material surroundings with a definite direction of up and down4 ,hence the necessity of our somewhat elaborate scheme of representation1 which1 when its import has been grasped1 can be dispensed with for the simpler one of a thin object slipping over a smooth surface1 which lies in front of us4 ,it is obvious that we must suppose some means by which the plane being is kept in contact with the surface on which he slips4 ,the simplest supposition to make is that there is a transverse gravity1 which keeps him to the plane4 ,this gravity must be thought of as different to the attraction exercised by his matter1 and as unperceived by him4 ,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 dimension1 but simply to investigate his plane consciousness4 ,it is obvious that the existence of a plane being must be very limited4 ,a straight line standing up from the surface of his earth affords a bar to his progress4 ,an object like a wheel which rotates round an axis would be unknown to him1 for there is no conceivable way in which he can get to the centre without going through the circumference4 ,he would have spinning disks1 but could not get to the centre of them4 ,the plane being can represent the motion from any one point of his space to any other1 by means of two straight lines drawn at right angles to each other4 ,let ,,ax and ,,ay be two such axes4 ,he can accomplish the translation from ,a to ,b by going along ,,ax to ,c1 and then from ,c along ,,cb parallel to ,,ay4 ,the same result can of course be obtained by moving to ,d along ,,ay and then parallel to ,,ax from ,d to ,b1 or of course by any diagonal movement compounded by these axial movements4 ,illustration3 ,fig4 #e4 ,by means of movements parallel to these two axes he can proceed 7except for material obstacles7 from any one point of his space to any other4 ,if now we suppose a third line drawn out from ,a at right angles to the plane it is evident that no motion in either of the two dimensions he knows will carry him in the least degree in the direction represented by ,,az4 ,illustration3 ,fig4 #f4 ,the lines ,,az and ,,ax determine a plane4 ,if he could be taken off his plane1 and transferred to the plane ,,axz1 he would be in a world exactly like his own4 ,from every line in his world there goes off a space world exactly like his own4 ,illustration3 ,fig4 #g4 ,from every point in his world a line can be drawn parallel to ,,az in the direction unknown to him4 ,if we suppose the square in fig4 #g to be a geometrical square from every point of it1 inside as well as on the contour1 a straight line can be drawn parallel to ,,az4 ,the assemblage of these lines constitute a solid figure1 of which the square in the plane is the base4 ,if we consider the square to represent an object in the plane being's world then we must attribute to it a very small thickness1 for every real thing must possess all three dimensions4 ,this thickness he does not perceive1 but thinks of this real object as a geometrical square4 ,he thinks of it as possessing area only1 and no degree of solidity4 ,the edges which project from the plane to a very small extent he thinks of as having merely length and no breadth--as being1 in fact1 geometrical lines4 ,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 objects4 ,he had conceived them as geometrical figures of two dimensions only4 ,if a third dimension exists1 such figures are incapable of real existence4 ,thus he would admit that all his real objects had a certain1 though very small thickness in the unknown dimension1 and that the conditions of his existence demanded the supposition of an extended sheet of matter1 from contact with which in their motion his objects never diverge4 ,analogous conceptions must be formed by us on the supposition of a four-dimensional existence4 ,we must suppose a direction in which we can never point extending from every point of our space4 ,we must draw a distinction between a geometrical cube and a cube of real matter4 ,the cube of real matter we must suppose to have an extension in an unknown direction1 real1 but so small as to be imperceptible by us4 ,from every point of a cube1 interior as well as exterior1 we must imagine that it is possible to draw a line in the unknown direction4 ,the assemblage of these lines would constitute a higher solid4 ,the lines going off in the unknown direction from the face of a cube would constitute a cube starting from that face4 ,of this cube all that we should see in our space would be the face4 ,again1 just as the plane being can represent any motion in his space by two axes1 so we can represent any motion in our three-dimensional space by means of three axes4 ,there is no point in our space to which we cannot move by some combination of movements on the directions marked out by these axes4 ,on the assumption of a fourth dimension we have to suppose a fourth axis1 which we will call ,,aw4 ,it must be supposed to be at right angles to each and every one of the three axes ,,ax1 ,,ay1 ,,az4 ,just as the two axes1 ,,ax1 ,,az1 determine a plane which is similar to the original plane on which we supposed the plane being to exist1 but which runs off from it1 and only meets it in a line2 so in our space if we take any three axes such as ,,ax1 ,,ay1 and ,,aw1 they determine a space like our space world4 ,this space runs off from our space1 and if we were transferred to it we should find ourselves in a space exactly similar to our own4 ,we must give up any attempt to picture this space in its relation to ours1 just as a plane being would have to give up any attempt to picture a plane at right angles to his plane4 ,such a space and ours run in different directions from the plane of ,,ax and ,,ay4 ,they meet in this plane but have nothing else in common1 just as the plane space of ,,ax and ,,ay and that of ,,ax and ,,az run in different directions and have but the line ,,ax in common4 ,omitting all discussion of the manner on which a plane being might be conceived to form a theory of a three-dimensional existence1 let us examine how1 with the means at his disposal1 he could represent the properties of three-dimensional objects4 ,there are two ways in which the plane being can think of one of our solid bodies4 ,he can think of the cube1 fig4 #h1 as composed of a number of sections parallel to his plane1 each lying in the third dimension a little further off from his plane than the preceding one4 ,these sections he can represent as a series of plane figures lying in his plane1 but in so representing them he destroys the coherence of them in the higher figure4 ,the set of squares1 ,a1 ,b1 ,c1 ,d1 represents the section parallel to the plane of the cube shown in figure1 but they are not in their proper relative positions4 ,illustration3 ,fig4 #h4 ,the plane being can trace out a movement in the third dimension by assuming discontinuous leaps from one section to another4 ,thus1 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 ,a1 ,b1 ,c1 ,d4 ,a point moving from ,a through ,,bcd in our space must be represented in the plane as appearing in ,a1 then in ,b1 and so on1 without passing through the intervening plane space4 ,in these sections the plane being leaves out1 of course1 the extension in the third dimension2 the distance between any two sections is not represented4 ,in order to realise this distance the conception of motion can be employed4 ,illustration3 ,fig4 #i4 ,let fig4 #i represent a cube passing transverse to the plane4 ,it will appear to the plane being as a square object1 but the matter of which this object is composed will be continually altering4 ,one material particle takes the place of another1 but it does not come from anywhere or go anywhere in the space which the plane being knows4 ,the analogous manner of representing a higher solid in our case1 is to conceive it as composed of a number of sections1 each lying a little further off in the unknown direction than the preceding4 ,illustration3 ,fig4 #aj4 ,we can represent these sections as a number of solids4 ,thus the cubes ,a1 ,b1 ,c1 ,d1 may be considered as the sections at different intervals in the unknown dimension of a higher cube4 ,arranged thus their coherence in the higher figure is destroyed1 they are mere representations4 ,a motion in the fourth dimension from ,a through ,b1 ,c1 etc41 would be continuous1 but we can only represent it as the occupation of the positions ,a1 ,b1 ,c1 etc41 in succession4 ,we can exhibit the results of the motion at different stages1 but no more4 ,in this representation we have left out the distance between one section and another2 we have considered the higher body merely as a series of sections1 and so left out its contents4 ,the only way to exhibit its contents is to call in the aid of the conception of motion4 ,illustration3 ,fig4 #aa4 ,if a higher cube passes transverse to our space1 it will appear as a cube isolated in space1 the part that has not come into our space and the part that has passed through will not be visible4 ,the gradual passing through our space would appear as the change of the matter of the cube before us4 ,one material particle in it is succeeded by another1 neither coming nor going in any direction we can point to4 ,in this manner1 by the duration of the figure1 we can exhibit the higher dimensionality of it2 a cube of our matter1 under the circumstances supposed1 namely1 that it has a motion transverse to our space1 would instantly disappear4 ,a higher cube would last till it had passed transverse to our space by its whole distance of extension in the fourth dimension4 ,as the plane being can think of the cube as consisting of sections1 each like a figure he knows1 extending away from his plane1 so we can think of a higher solid as composed of sections1 each like a solid which we know1 but extending away from our space4 ,thus1 taking a higher cube1 we can look on it as starting from a cube in our space and extending in the unknown dimension4 ,illustration3 ,fig4 #ab4 ,take the face ,a and conceive it to exist as simply a face1 a square with no thickness4 ,from this face the cube in our space extends by the occupation of space which we can see4 ,but from this face there extends equally a cube in the unknown dimension4 ,we can think of the higher cube1 then1 by taking the set of sections ,a1 ,b1 ,c1 ,d1 etc41 and considering that from each of them there runs a cube4 ,these cubes have nothing in common with each other1 and of each of them in its actual position all that we can have in our space is an isolated square4 ,it is obvious that we can take our series of sections in any manner we please4 ,we can take them parallel1 for instance1 to any one of the three isolated faces shown in the figure4 ,corresponding to the three series of sections at right angles to each other1 which we can make of the cube in space1 we must conceive of the higher cube1 as composed of cubes starting from squares parallel to the faces of the cube1 and of these cubes all that exist in our space are the isolated squares from which they start4 ,,chapter ,,iii ,,the ,,significance ,,of ,a ,,four-,,dimensional ,,existence ,having now obtained the conception of a four-dimensional space1 and having formed the analogy which1 without any further geometrical difficulties1 enables us to enquire into its properties1 ,i will refer the reader1 whose interest is principally in the mechanical aspect1 to ,chapters ,,vi4 and ,,vii4 ,in the present chapter ,i will deal with the general significance of the enquiry1 and in the next with the historical origin of the idea4 ,first1 with regard to the question of whether there is any evidence that we are really in four-dimensional space1 ,i will go back to the analogy of the plane world4 ,a being in a plane world could not have any experience of three-dimensional shapes1 but he could have an experience of three-dimensional movements4 ,we have seen that his matter must be supposed to have an extension1 though a very small one1 in the third dimension4 ,and thus1 in the small particles of his matter1 three-dimensional movements may well be conceived to take place4 ,of these movements he would only perceive the resultants4 ,since all movements of an observable size in the plane world are two-dimensional1 he would only perceive the resultants in two dimensions of the small three-dimensional movements4 ,thus1 there would be phenomena which he could not explain by his theory of mechanics--motions would take place which he could not explain by his theory of motion4 ,hence1 to determine if we are in a four-dimensional world1 we must examine the phenomena of motion in our space4 ,if movements occur which are not explicable on the suppositions of our three-dimensional mechanics1 we should have an indication of a possible four-dimensional motion1 and if1 moreover1 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 ether1 we should have a strong presumption in favour of the reality of the fourth dimension4 ,by proceeding in the direction of finer and finer subdivision1 we come to forms of matter possessing properties different to those of the larger masses4 ,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 dimensions4 ,this form of matter ,i speak of as four-dimensional ether1 and attribute to it properties approximating to those of a perfect liquid4 ,deferring the detailed discussion of this form of matter to ,chapter ,,vi41 we will now examine the means by which a plane being would come to the conclusion that three-dimensional movements existed in his world1 and point out the analogy by which we can conclude the existence of four-dimensional movements in our world4 ,since the dimensions of the matter in his world are small in the third direction1 the phenomena in which he would detect the motion would be those of the small particles of matter4 ,suppose that there is a ring in his plane4 ,we can imagine currents flowing round the ring in either of two opposite directions4 ,these would produce unlike effects1 and give rise to two different fields of influence4 ,if the ring with a current in it in one direction be taken up and turned over1 and put down again on the plane1 it would be identical with the ring having a current in the opposite direction4 ,an operation of this kind would be impossible to the plane being4 ,hence he would have in his space two irreconcilable objects1 namely1 the two fields of influence due to the two rings with currents in them in opposite directions4 ,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 plane4 ,instead of currents flowing in the rings we can imagine a different kind of current4 ,imagine a number of small rings strung on the original ring4 ,a current round these secondary rings would give two varieties of effect1 or two different fields of influence1 according to its direction4 ,these two varieties of current could be turned one into the other by taking one of the rings up1 turning it over1 and putting it down again in the plane4 ,this operation is impossible to the plane being1 hence in this case also there would be two irreconcilable fields in the plane4 ,now1 if the plane being found two such irreconcilable fields and could prove that they could not be accounted for by currents in the rings1 he would have to admit the existence of currents round the rings--that is1 in rings strung on the primary ring4 ,thus he would come to admit the existence of a three-dimensional motion1 for such a disposition of currents is in three dimensions4 ,now in our space there are two fields of different properties1 which can be produced by an electric current flowing in a closed circuit or ring4 ,these two fields can be changed one into the other by reversing the currents1 but they cannot be changed one into the other by any turning about of the rings in our space2 for the disposition of the field with regard to the ring itself is different when we turn the ring1 over and when we reverse the direction of the current in the ring4 ,as hypotheses to explain the differences of these two fields and their effects we can suppose the following kinds of space motions3--,first1 a current along the conductor2 second1 a current round the conductor--that is1 of rings of currents strung on the conductor as an axis4 ,neither of these suppositions accounts for facts of observation4 ,hence we have to make the supposition of a four-dimensional motion4 ,we find that a four-dimensional rotation of the nature explained in a subsequent chapter1 has the following characteristics3--,first1 it would give us two fields of influence1 the one of which could be turned into the other by taking the circuit up into the fourth dimension1 turning it over1 and putting it down in our space again1 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 space4 ,second1 it involves a phenomenon precisely identical with that most remarkable and mysterious feature of an electric current1 namely that it is a field of action1 the rim of which necessarily abuts on a continuous boundary formed by a conductor4 ,hence1 on the assumption of a four-dimensional movement in the region of the minute particles of matter1 we should expect to find a motion analogous to electricity4 ,now1 a phenomenon of such universal occurrence as electricity cannot be due to matter and motion in any very complex relation1 but ought to be seen as a simple and natural consequence of their properties4 ,i infer that the difficulty in its theory is due to the attempt to explain a four-dimensional phenomenon by a three-dimensional geometry4 ,in view of this piece of evidence we cannot disregard that afforded by the existence of symmetry4 ,in this connection ,i will allude to the simple way of producing the images of insects1 sometimes practised by children4 ,they put a few blots of ink in a straight line on a piece of paper1 fold the paper along the blots1 and on opening it the lifelike presentment of an insect is obtained4 ,if we were to find a multitude of these figures1 we should conclude that they had originated from a process of folding over2 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 way4 ,the production of the symmetrical forms of organised beings1 though not of course due to a turning over of bodies of any appreciable size in four-dimensional space1 can well be imagined as due to a disposition in that manner of the smallest living particles from which they are built up4 ,thus1 not only electricity1 but life1 and the processes by which we think and feel1 must be attributed to that region of magnitude in which four-dimensional movements take place4 ,i do not mean1 however1 that life can be explained as a four-dimensional movement4 ,it seems to me that the whole bias of thought1 which tends to explain the phenomena of life and volition1 as due to matter and motion in some peculiar relation1 is adopted rather in the interests of the explicability of things than with any regard to probability4 ,of course1 if we could show that life were a phenomenon of motion1 we should be able to explain a great deal that is at present obscure4 ,but there are two great difficulties in the way4 ,it would be necessary to show that in a germ capable of developing into a living being1 there were modifications of structure capable of determining in the developed germ all the characteristics of its form1 and not only this1 but of determining those of all the descendants of such a form in an infinite series4 ,such a complexity of mechanical relations1 undeniable though it be1 cannot surely be the best way of grouping the phenomena and giving a practical account of them4 ,and another difficulty is this1 that no amount of mechanical adaptation would give that element of consciousness which we possess1 and which is shared in to a modified degree by the animal world4 ,in those complex structures which men build up and direct1 such as a ship or a railway train 7and which1 if seen by an observer of such a size that the men guiding them were invisible1 would seem to present some of the phenomena of life7 the appearance of animation is not due to any diffusion of life in the material parts of the structure1 but to the presence of a living being4 ,the old hypothesis of a soul1 a living organism within the visible one1 appears to me much more rational than the attempt to explain life as a form of motion4 ,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 disappears4 ,lord ,kelvin supposes that matter is formed from the ether4 ,we may very well suppose that the living organisms directing the material ones are co-ordinate with them1 not composed of matter1 but consisting of etherial bodies1 and as such capable of motion through the ether1 and able to originate material living bodies throughout the mineral4 ,hypotheses such as these find no immediate ground for proof or disproof in the physical world4 ,let us1 therefore1 turn to a different field1 and1 assuming that the human soul is a four-dimensional being1 capable in itself of four dimensional movements1 but in its experiences through the senses limited to three dimensions1 ask if the history of thought1 of these productivities which characterise man1 correspond to our assumption4 ,let us pass in review those steps by which man1 presumably a four-dimensional being1 despite his bodily environment1 has come to recognise the fact of four-dimensional existence4 ,deferring this enquiry to another chapter1 ,i will here recapitulate the argument in order to show that our purpose is entirely practical and independent of any philosophical or metaphysical considerations4 ,if two shots are fired at a target1 and the second bullet hits it at a different place to the first1 we suppose that there was some difference in the conditions under which the second shot was fired from those affecting the first shot4 ,the force of the powder1 the direction of aim1 the strength of the wind1 or some condition must have been different in the second case1 if the course of the bullet was not exactly the same as in the first case4 ,corresponding to every difference in a result there must be some difference in the antecedent material conditions4 ,by tracing out this chain of relations we explain nature4 ,but there is also another mode of explanation which we apply4 ,if we ask what was the cause that a certain ship was built1 or that a certain structure was erected1 we might proceed to investigate the changes in the brain cells of the men who designed the works4 ,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 designers4 ,but practically this would be a very long task4 ,a more effective mode of explaining the production of the ship or building would be to enquire into the motives1 plans1 and aims of the men who constructed them4 ,we obtain a cumulative and consistent body of knowledge much more easily and effectively in the latter way4 ,sometimes we apply the one1 sometimes the other mode of explanation4 ,but it must be observed that the method of explanation founded on aim1 purpose1 volition1 always presupposes a mechanical system on which the volition and aim works4 ,the conception of man as willing and acting from motives involves that of a number of uniform processes of nature which he can modify1 and of which he can make application4 ,in the mechanical conditions of the three-dimensional world1 the only volitional agency which we can demonstrate is the human agency4 ,but when we consider the four-dimensional world the conclusion remains perfectly open4 ,the method of explanation founded on purpose and aim does not1 surely1 suddenly begin with man and end with him4 ,there is as much behind the exhibition of will and motive which we see in man as there is behind the phenomena of movement2 they are co-ordinate1 neither to be resolved into the other4 ,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 organism1 which expresses its higher physical being in the symmetry of the body1 and gives the aims and motives of human existence4 ,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 universe4 ,but a subsidiary contribution towards the verification of the hypothesis may be made by passing in review the history of human thought1 and enquiring if it presents such features as would be naturally expected on this assumption4 ,,chapter ,,iv ,,the ,,first ,,chapter ,,in ,,the ,,history ,,of ,,four ,,space ,parmenides1 and the ,asiatic thinkers with whom he is in close affinity1 propound a theory of existence which is in close accord with a conception of a possible relation between a higher and a lower dimensional space4 ,this theory1 prior and in marked contrast to the main stream of thought1 which we shall afterwards describe1 forms a closed circle by itself4 ,it is one which in all ages has had a strong attraction for pure intellect1 and is the natural mode of thought for those who refrain from projecting their own volition into nature under the guise of causality4 ,according to ,parmenides of the school of ,elea the all is one1 unmoving and unchanging4 ,the permanent amid the transient--that foothold for thought1 that solid ground for feeling on the discovery of which depends all our life--is no phantom2 it is the image amidst deception of true being1 the eternal1 the unmoved1 the one4 ,thus says ,parmenides4 ,but how explain the shifting scene1 these mutations of things6 8,illusion10 answers ,parmenides4 ,distinguishing between truth and error1 he tells of the true doctrine of the one--the false opinion of a changing world4 ,he is no less memorable for the manner of his advocacy than for the cause he advocates4 ,it is as if from his firm foothold of being he could play with the thoughts under the burden of which others laboured1 for from him springs that fluency of supposition and hypothesis which forms the texture of ,plato's dialectic4 ,can the mind conceive a more delightful intellectual picture than that of ,parmenides1 pointing to the one1 the true1 the unchanging1 and yet on the other hand ready to discuss all manner of false opinion1 forming a cosmogony too1 false 8but mine own0 after the fashion of the time8 ,in support of the true opinion he proceeded by the negative way of showing the self-contradictions in the ideas of change and motion4 ,it is doubtful if his criticism1 save in minor points1 has ever been successfully refuted4 ,to express his doctrine in the ponderous modern way we must make the statement that motion is phenomenal1 not real4 ,let us represent his doctrine4 ,illustration3 ,fig4 #ac4 ,imagine a sheet of still water into which a slanting stick is being lowered with a motion vertically downwards4 ,let #a1 #b1 #c 7,fig4 #ac71 be three consecutive positions of the stick4 ,a1 ,b1 ,c1 will be three consecutive positions of the meeting of the stick1 with the surface of the water4 ,as the stick passes down1 the meeting will move from ,a on to ,b and ,c4 ,suppose now all the water to be removed except a film4 ,at the meeting of the film and the stick there will be an interruption of the film4 ,if we suppose the film to have a property1 like that of a soap bubble1 of closing up round any penetrating object1 then as the stick goes vertically downwards the interruption in the film will move on4 ,illustration3 ,fig4 #ad4 ,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 figure4 ,suppose now the spiral to be still and the film to move vertically upwards1 the whole spiral will be represented in the film of the consecutive positions of the point of intersection4 ,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 circle4 ,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 experience1 we see that we shall have in the film a point moving in a circle1 conscious of its motion1 knowing nothing of that real spiral the record of the successive intersections of which by the film is the motion of the point4 ,it is easy to imagine complicated structures of the nature of the spiral1 structures consisting of filaments1 and to suppose also that these structures are distinguishable from each other at every section4 ,if we consider the intersections of these filaments with the film as it passes to be the atoms constituting a filmar universe1 we shall have in the film a world of apparent motion2 we shall have bodies corresponding to the filamentary structure1 and the positions of these structures with regard to one another will give rise to bodies in the film moving amongst one another4 ,this mutual motion is apparent merely4 ,the reality is of permanent structures stationary1 and all the relative motions accounted for by one steady movement of the film as a whole4 ,thus we can imagine a plane world1 in which all the variety of motion is the phenomenon of structures consisting of filamentary atoms traversed by a plane of consciousness4 ,passing to four dimensions and our space1 we can conceive that all things and movements in our world are the reading off of a permanent reality by a space of consciousness4 ,each atom at every moment is not what it was1 but a new part of that endless line which is itself4 ,and all this system successively revealed in the time which is but the succession of consciousness1 separate as it is in parts1 in its entirety is one vast unity4 ,representing ,parmenides' doctrine thus1 we gain a firmer hold on it than if we merely let his words rest1 grand and massive1 in our minds4 ,and we have gained the means also of representing phases of that ,eastern thought to which ,parmenides was no stranger4 ,modifying his uncompromising doctrine1 let us suppose1 to go back to the plane of consciousness and the structure of filamentary atoms1 that these structures are themselves moving--are acting1 living4 ,then1 in the transverse motion of the film1 there would be two phenomena of motion1 one due to the reading off in the film of the permanent existences as they are in themselves1 and another phenomenon of motion due to the modification of the record of the things themselves1 by their proper motion during the process of traversing them4 ,thus a conscious being in the plane would have1 as it were1 a two-fold experience4 ,in the complete traversing of the structure1 the intersection of which with the film gives his conscious all1 the main and principal movements and actions which he went through would be the record of his higher self as it existed unmoved and unacting4 ,slight modifications and deviations from these movements and actions would represent the activity and self-determination of the complete being1 of his higher self4 ,it is admissible to suppose that the consciousness in the plane has a share in that volition by which the complete existence determines itself4 ,thus the motive and will1 the initiative and life1 of the higher being1 would be represented in the case of the being in the film by an initiative and a will capable1 not of determining any great things or important movements in his existence1 but only of small and relatively insignificant activities4 ,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 on4 ,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 lives4 ,an alteration of the higher being would correspond to a different life history for him4 ,let us now make the supposition that film after film traverses these higher structures1 that the life of the real being is read off again and again in successive waves of consciousness4 ,there would be a succession of lives in the different advancing planes of consciousness1 each differing from the preceding1 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 life1 but the minute and apparently unimportant4 ,in all great things the being of the film shares in the existence of his higher self as it is at any one time4 ,in the small things he shares in that volition by which the higher being alters and changes1 acts and lives4 ,thus we gain the conception of a life changing and developing as a whole1 a life in which our separation and cessation and fugitiveness are merely apparent1 but which in its events and course alters1 changes1 develops2 and the power of altering and changing this whole lies in the will and power the limited being has of directing1 guiding1 altering himself in the minute things of his existence4 ,transferring our conceptions to those of an existence in a higher dimensionality traversed by a space of consciousness1 we have an illustration of a thought which has found frequent and varied expression4 ,when1 however1 we ask ourselves what degree of truth there lies in it1 we must admit that1 as far as we can see1 it is merely symbolical4 ,the true path in the investigation of a higher dimensionality lies in another direction4 ,the significance of the ,parmenidean doctrine lies in this that here1 as again and again1 we find that those conceptions which man introduces of himself1 which he does not derive from the mere record of his outward experience1 have a striking and significant correspondence to the conception of a physical existence in a world of a higher space4 ,how close we come to ,parmenides' thought by this manner of representation it is impossible to say4 ,what ,i want to point out is the adequateness of the illustration1 not only to give a static model of his doctrine1 but one capable as it were1 of a plastic modification into a correspondence into kindred forms of thought4 ,either one of two things must be true--that four-dimensional conceptions give a wonderful power of representing the thought of the ,east1 or that the thinkers of the ,east must have been looking at and regarding four-dimensional existence4 ,coming now to the main stream of thought we must dwell in some detail on ,pythagoras1 not because of his direct relation to the subject1 but because of his relation to investigators who came later4 ,pythagoras invented the two-way counting4 ,let us represent the single-way counting by the posits aa1 ab1 ac1 ad1 using these pairs of letters instead of the numbers #a1 #b1 #c1 #d4 ,i put an a in each case first for a reason which will immediately appear4 ,we have a sequence and order4 ,there is no conception of distance necessarily involved4 ,the difference between the posits is one of order not of distance--only when identified with a number of equal material things in juxtaposition does the notion of distance arise4 ,now1 besides the simple series ,i can have1 starting from aa1 ba1 ca1 da1 from ab1 bb1 cb1 db1 and so on1 and forming a scheme3 da db dc dd ca cb cc cd ba bb bc bd aa ab ac ad ,this complex or manifold gives a two-way order4 ,i can represent it by a set of points1 if ,i am on my guard against assuming any relation of distance4 ,illustration3 ,fig4 #ae4 ,pythagoras studied this two-fold way of counting in reference to material bodies1 and discovered that most remarkable property of the combination of number and matter that bears his name4 ,the ,pythagorean property of an extended material system can be exhibited in a manner which will be of use to us afterwards1 and which therefore ,i will employ now instead of using the kind of figure which he himself employed4 ,consider a two-fold field of points arranged in regular rows4 ,such a field will be presupposed in the following argument4 ,illustration3 ,fig4 #af4 #a and #b ,it is evident that in fig4 #af four of the points determine a square1 which square we may take as the unit of measurement for areas4 ,but we can also measure areas in another way4 ,fig4 #af 7#a7 shows four points determining a square4 ,but four squares also meet in a point1 fig4 #af 7#b74 ,hence a point at the corner of a square belongs equally to four squares4 ,thus we may say that the point value of the square shown is one point1 for if we take the square in fig4 #af 7#a7 it has four points1 but each of these belong equally to four other squares4 ,hence one fourth of each of them belongs to the square 7#a7 in fig4 #af4 ,thus the point value of the square is one point4 ,the result of counting the points is the same as that arrived at by reckoning the square units enclosed4 ,hence1 if we wish to measure the area of any square we can take the number of points it encloses1 count these as one each1 and take one-fourth of the number of points at its corners4 ,illustration3 ,fig4 #ag4 ,now draw a diagonal square as shown in fig4 #ag4 ,it contains one point and the four corners count for one point more2 hence its point value is #b4 ,the value is the measure of its area--the size of this square is two of the unit squares4 ,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 points1 but have four corner points each1 which gives the point value of each as one point4 ,hence we see that the square on the diagonal is equal to the squares on the two sides2 or as it is generally expressed1 the square on the hypothenuse is equal to the sum of the squares on the sides4 ,illustration3 ,fig4 #ah4 ,noticing this fact we can proceed to ask if it is always true4 ,drawing the square shown in fig4 #ah1 we can count the number of its points4 ,there are five altogether4 ,there are four points inside the square on the diagonal1 and hence1 with the four points at its corners the point value is #e--that is1 the area is #e4 ,now the squares on the sides are respectively of the area #d and #a4 ,hence in this case also the square on the diagonal is equal to the sum of the square on the sides4 ,this property of matter is one of the first great discoveries of applied mathematics4 ,we shall prove afterwards that it is not a property of space4 ,for the present it is enough to remark that the positions in which the points are arranged is entirely experimental4 ,it is by means of equal pieces of some material1 or the same piece of material moved from one place to another1 that the points are arranged4 ,pythagoras next enquired what the relation must be so that a square drawn slanting-wise should be equal to one straight-wise4 ,he found that a square whose side is five can be placed either rectangularly along the lines of points1 or in a slanting position4 ,and this square is equivalent to two squares of sides #d and #c4 ,here he came upon a numerical relation embodied in a property of matter4 ,numbers immanent in the objects produced the equality so satisfactory for intellectual apprehension4 ,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 reason4 ,what wonder then that he ascribed an active power to number6 ,we must remember that1 sharing like ourselves the search for the permanent in changing phenomena1 the ,greeks had not that conception of the permanent in matter that we have4 ,to them material things were not permanent4 ,in fire solid things would vanish2 absolutely disappear4 ,rock and earth had a more stable existence1 but they too grew and decayed4 ,the permanence of matter1 the conservation of energy1 were unknown to them4 ,and that distinction which we draw so readily between the fleeting and permanent causes of sensation1 between a sound and a material object1 for instance1 had not the same meaning to them which it has for us4 ,let us but imagine for a moment that material things are fleeting1 disappearing1 and we shall enter with a far better appreciation into that search for the permanent which1 with the ,greeks1 as with us1 is the primary intellectual demand4 ,what is that which amid a thousand forms is ever the same1 which we can recognise under all its vicissitudes1 of which the diverse phenomena are the appearances8 ,to think that this is number is not so very wide of the mark4 ,with an intellectual apprehension which far outran the evidences for its application1 the atomists asserted that there were everlasting material particles1 which1 by their union1 produced all the varying forms and states of bodies4 ,but in view of the observed facts of nature as then known1 ,aristotle1 with perfect reason1 refused to accept this hypothesis4 ,he expressly states that there is a change of quality1 and that the change due to motion is only one of the possible modes of change4 ,with no permanent material world about us1 with the fleeting1 the unpermanent1 all around we should1 ,i think1 be ready to follow ,pythagoras in his identification of number with that principle which subsists amidst all changes1 which in multitudinous forms we apprehend immanent in the changing and disappearing substance of things4 ,and from the numerical idealism of ,pythagoras there is but a step to the more rich and full idealism of ,plato4 ,that which is apprehended by the sense of touch we put as primary and real1 and the other senses we say are merely concerned with appearances4 ,but ,plato took them all as valid1 as giving qualities of existence4 ,that the qualities were not permanent in the world as given to the senses forced him to attribute to them a different kind of permanence4 ,he formed the conception of a world of ideas1 in which all that really is1 all that affects us and gives the rich and wonderful wealth of our experience1 is not fleeting and transitory1 but eternal4 ,and of this real and eternal we see in the things about us the fleeting and transient images4 ,and this world of ideas was no exclusive one1 wherein was no place for the innermost convictions of the soul and its most authoritative assertions4 ,therein existed justice1 beauty--the one1 the good1 all that the soul demanded to be4 ,the world of ideas1 ,plato's wonderful creation preserved for man1 for his deliberate investigation and their sure development1 all that the rude incomprehensible changes of a harsh experience scatters and destroys4 ,plato believed in the reality of ideas4 ,he meets us fairly and squarely4 ,divide a line into two parts1 he says2 one to represent the real objects in the world1 the other to represent the transitory appearances1 such as the image in still water1 the glitter of the sun on a bright surface1 the shadows on the clouds4 ,a ,b ---------------------------------------- ---------------------------------------- ---------------------------------------- ----- ,real things3 ,appearances3 e4g41 the sun4 e4g41 the reflection of the sun4 ,take another line and divide it into two parts1 one representing our ideas1 the ordinary occupants of our minds1 such as whiteness1 equality1 and the other representing our true knowledge1 which is of eternal principles1 such as beauty1 goodness4 ,a#a ,b#a ---------------------------------------- ---------------------------------------- ---------------------------------------- ----- ,eternal principles1 ,appearances in the mind1 as beauty as whiteness1 equality ,then as ,a is to ,b1 so is ,a#a to ,b#a ,that is1 the soul can proceed1 going away from real things to a region of perfect certainty1 where it beholds what is1 not the scattered reflections2 beholds the sun1 not the glitter on the sands2 true being1 not chance opinion4 ,now1 this is to us1 as it was to ,aristotle1 absolutely inconceivable from a scientific point of view4 ,we can understand that a being is known in the fulness of his relations2 it is in his relations to his circumstances that a man's character is known2 it is in his acts under his conditions that his character exists4 ,we cannot grasp or conceive any principle of individuation apart from the fulness of the relations to the surroundings4 ,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 world4 ,do not his words begin to have a meaning8 ,such a being would have a consciousness of motion which is not as the motion he can see with the eyes of the body4 ,he1 in his own being1 knows a reality to which the outward matter of this too solid earth is flimsy superficiality4 ,he too knows a mode of being1 the fulness of relations1 in which can only be represented in the limited world of sense1 as the painter unsubstantially portrays the depths of woodland1 plains1 and air4 ,thinking of such a being in man1 was not ,plato's line well divided8 ,it is noteworthy that1 if ,plato omitted his doctrine of the independent origin of ideas1 he would present exactly the four-dimensional argument2 a real thing as we think it is an idea4 ,a plane being's idea of a square object is the idea of an abstraction1 namely1 a geometrical square4 ,similarly our idea of a solid thing is an abstraction1 for in our idea there is not the four-dimensional thickness which is necessary1 however slight1 to give reality4 ,the argument would then run1 as a shadow is to a solid object1 so is the solid object to the reality4 ,thus ,a and ,b would be identified4 ,in the allegory which ,i have already alluded to1 ,plato in almost as many words shows forth the relation between existence in a superficies and in solid space4 ,and he uses this relation to point to the conditions of a higher being4 ,he imagines a number of men prisoners1 chained so that they look at the wall of a cavern in which they are confined1 with their backs to the road and the light4 ,over the road pass men and women1 figures and processions1 but of all this pageant all that the prisoners behold is the shadow of it on the wall whereon they gaze4 ,their own shadows and the shadows of the things in the world are all that they see1 and identifying themselves with their shadows related as shadows to a world of shadows1 they live in a kind of dream4 ,plato imagines one of their number to pass out from amongst them into the real space world1 and then returning to tell them of their condition4 ,here he presents most plainly the relation between existence in a plane world and existence in a three-dimensional world4 ,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 know4 ,it must have hung upon the weight of a shadow which path he took6--whether the one we shall follow toward the higher solid and the four-dimensional existence1 or the one which makes ideas the higher realities1 and the direct perception of them the contact with the truer world4 ,passing on to ,aristotle1 we will touch on the points which most immediately concern our enquiry4 ,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 respectful1 asking of each and all wherein lay their relation to fact1 so ,aristotle1 in discussing the philosophy of ,greece as he found it1 asks1 above all other things3 8,does this represent the world8 ,in this system is there an adequate presentation of what is80 ,he finds them all defective1 some for the very reasons which we esteem them most highly1 as when he criticises the ,atomic theory for its reduction of all change to motion4 ,but in the lofty march of his reason he never loses sight of the whole2 and that wherein our views differ from his lies not so much in a superiority of our point of view1 as in the fact which he himself enunciates--that it is impossible for one principle to be valid in all branches of enquiry4 ,the conceptions of one method of investigation are not those of another2 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 ,aristotle4 ,he takes account of everything2 he does not separate matter and the manifestation of matter2 he fires all together in a conception of a vast world process in which everything takes part--the motion of a grain of dust1 the unfolding of a leaf1 the ordered motion of the spheres in heaven--all are parts of one whole which he will not separate into dead matter and adventitious modifications4 ,and just as our theories1 as representative of actuality1 fall before his unequalled grasp of fact1 so the doctrine of ideas fell4 ,it is not an adequate account of existence1 as ,plato himself shows in his 8,parmenides02 it only explains things by putting their doubles beside them4 ,for his own part ,aristotle invented a great marching definition which1 with a kind of power of its own1 cleaves its way through phenomena to limiting conceptions on either hand1 towards whose existence all experience points4 ,in ,aristotle's definition of matter and form as the constituent of reality1 as in ,plato's mystical vision of the kingdom of ideas1 the existence of the higher dimensionality is implicitly involved4 ,substance according to ,aristotle is relative1 not absolute4 ,in everything that is there is the matter of which it is composed1 the form which it exhibits2 but these are indissolubly connected1 and neither can be thought without the other4 ,the blocks of stone out of which a house is built are the material for the builder2 but1 as regards the quarrymen1 they are the matter of the rocks with the form he has imposed on them4 ,words are the final product of the grammarian1 but the mere matter of the orator or poet4 ,the atom is1 with us1 that out of which chemical substances are built up1 but looked at from another point of view is the result of complex processes4 ,nowhere do we find finality4 ,the matter in one sphere is the matter1 plus form1 of another sphere of thought4 ,making an obvious application to geometry1 plane figures exist as the limitation of different portions of the plane by one another4 ,in the bounding lines the separated matter of the plane shows its determination into form4 ,and as the plane is the matter relatively to determinations in the plane1 so the plane itself exists in virtue of the determination of space4 ,a plane is that wherein formless space has form superimposed on it1 and gives an actuality of real relations4 ,we cannot refuse to carry this process of reasoning a step farther back1 and say that space itself is that which gives form to higher space4 ,as a line is the determination of a plane1 and a plane of a solid1 so solid space itself is the determination of a higher space4 ,as a line by itself is inconceivable without that plane which it separates1 so the plane is inconceivable without the solids which it limits on either hand4 ,and so space itself cannot be positively defined4 ,it is the negation of the possibility of movement in more than three dimensions4 ,the conception of space demands that of a higher space4 ,as a surface is thin and unsubstantial without the substance of which it is the surface1 so matter itself is thin without the higher matter4 ,just as ,aristotle invented that algebraical method of representing unknown quantities by mere symbols1 not by lines necessarily determinate in length as was the habit of the ,greek geometers1 and so struck out the path towards those objectifications of thought which1 like independent machines for reasoning1 supply the mathematician with his analytical weapons1 so in the formulation of the doctrine of matter and form1 of potentiality and actuality1 of the relativity of substance1 he produced another kind of objectification of mind--a definition which had a vital force and an activity of its own4 ,in none of his writings1 as far as we know1 did he carry it to its legitimate conclusion on the side of matter1 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 matter4 ,the unmoved mover of all things is ,aristotle's highest principle4 ,towards it1 to partake of its perfection all things move4 ,the universe1 according to ,aristotle1 is an active process--he does not adopt the illogical conception that it was once set in motion and has kept on ever since4 ,there is room for activity1 will1 self-determination1 in ,aristotle's system1 and for the contingent and accidental as well4 ,we do not follow him1 because we are accustomed to find in nature infinite series1 and do not feel obliged to pass on to a belief in the ultimate limits to which they seem to point4 ,but apart from the pushing to the limit1 as a relative principle this doctrine of ,aristotle's as to the relativity of substance is irrefragible in its logic4 ,he was the first to show the necessity of that path of thought which when followed leads to a belief in a four-dimensional space4 ,antagonistic as he was to ,plato in his conception of the practical relation of reason to the world of phenomena1 yet in one point he coincided with him4 ,and in this he showed the candour of his intellect4 ,he was more anxious to lose nothing than to explain everything4 ,and that wherein so many have detected an inconsistency1 an inability to free himself from the school of ,plato1 appears to us in connection with our enquiry as an instance of the acuteness of his observation4 ,for beyond all knowledge given by the senses ,aristotle held that there is an active intelligence1 a mind not the passive recipient of impressions from without1 but an active and originative being1 capable of grasping knowledge at first hand4 ,in the active soul ,aristotle recognised something in man not produced by his physical surroundings1 something which creates1 whose activity is a knowledge underived from sense4 ,this1 he says1 is the immortal and undying being in man4 ,thus we see that ,aristotle was not far from the recognition of the four-dimensional existence1 both without and within man1 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 soul4 ,the next step in the unfolding of the drama of the recognition of the soul as connected with our scientific conception of the world1 and1 at the same time1 the recognition of that higher of which a three-dimensional world presents the superficial appearance1 took place many centuries later4 ,if we pass over the intervening time without a word it is because the soul was occupied with the assertion of itself in other ways than that of knowledge4 ,when it took up the task in earnest of knowing this material world in which it found itself1 and of directing the course of inanimate nature1 from that most objective aim came1 reflected back as from a mirror1 its knowledge of itself4 ,,chapter ,v ,,the ,,second ,,chapter ,,in ,,the ,,history ,,of ,,four ,,space ,,lobatchewsky1 ,,bolyai1 ,,and ,,gauss ,before entering on a description of the work of ,lobatchewsky and ,bolyai it will not be out of place to give a brief account of them1 the materials for which are to be found in an article by ,franz ,schmidt in the forty-second volume of the ,mathematische ,annalen1 and in ,engel's edition of ,lobatchewsky4 ,lobatchewsky was a man of the most complete and wonderful talents4 ,as a youth he was full of vivacity1 carrying his exuberance so far as to fall into serious trouble for hazing a professor1 and other freaks4 ,saved by the good offices of the mathematician ,bartels1 who appreciated his ability1 he managed to restrain himself within the bounds of prudence4 ,appointed professor at his own ,university1 ,kasan1 he entered on his duties under the regime of a pietistic reactionary1 who surrounded himself with sycophants and hypocrites4 ,esteeming probably the interests of his pupils as higher than any attempt at a vain resistance1 he made himself the tyrant's right-hand man1 doing an incredible amount of teaching and performing the most varied official duties4 ,amidst all his activities he found time to make important contributions to science4 ,his theory of parallels is most closely connected with his name1 but a study of his writings shows that he was a man capable of carrying on mathematics in its main lines of advance1 and of a judgment equal to discerning what these lines were4 ,appointed rector of his ,university1 he died at an advanced age1 surrounded by friends1 honoured1 with the results of his beneficent activity all around him4 ,to him no subject came amiss1 from the foundations of geometry to the improvement of the stoves by which the peasants warmed their houses4 ,he was born in #agic4 ,his scientific work was unnoticed till1 in #ahfg1 ,houel1 the ,french mathematician1 drew attention to its importance4 ,johann ,bolyai de ,bolyai was born in ,klausenburg1 a town in ,transylvania1 ,december #aeth1 #ahjb4 ,his father1 ,wolfgang ,bolyai1 a professor in the ,reformed ,college of ,maros ,vasarhely1 retained the ardour in mathematical studies which had made him a chosen companion of ,gauss in their early student days at ,gttingen4 ,he found an eager pupil in ,johann4 ,he relates that the boy sprang before him like a devil4 ,as soon as he had enunciated a problem the child would give the solution and command him to go on further4 ,as a thirteen-year-old boy his father sometimes sent him to fill his place when incapacitated from taking his classes4 ,the pupils listened to him with more attention than to his father for they found him clearer to understand4 ,in a letter to ,gauss ,wolfgang ,bolyai writes3-- 8,my boy is strongly built4 ,he has learned to recognise many constellations1 and the ordinary figures of geometry4 ,he makes apt applications of his notions1 drawing for instance the positions of the stars with their constellations4 ,last winter in the country1 seeing ,jupiter he asked3 #8,how is it that we can see him from here as well as from the town8 ,he must be far off4' ,and as to three different places to which he had been he asked me to tell him about them in one word4 ,i did not know what he meant1 and then he asked me if one was in a line with the other and all in a row1 or if they were in a triangle4 8,he enjoys cutting paper figures with a pair of scissors1 and without my ever having told him about triangles remarked that a right-angled triangle which he had cut out was half of an oblong4 ,i exercise his body with care1 he can dig well in the earth with his little hands4 ,the blossom can fall and no fruit left4 ,when he is fifteen ,i want to send him to you to be your pupil40 ,in ,johann's autobiography he says3-- 8,my father called my attention to the imperfections and gaps in the theory of parallels4 ,he told me he had gained more satisfactory results than his predecessors1 but had obtained no perfect and satisfying conclusion4 ,none of his assumptions had the necessary degree of geometrical certainty1 although they sufficed to prove the eleventh axiom and appeared acceptable on first sight4 8,he begged of me1 anxious not without a reason1 to hold myself aloof and to shun all investigation on this subject1 if ,i did not wish to live all my life in vain40 ,johann1 in the failure of his father to obtain any response from ,gauss1 in answer to a letter in which he asked the great mathematician to make of his son 8an apostle of truth in a far land10 entered the ,engineering ,school at ,vienna4 ,he writes from ,temesvar1 where he was appointed sub-lieutenant ,september1 #ahbc3-- 8,temesvar1 ,november #crd1 #ahbc4 8,,dear ,,good ,,father1 8,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 on4 ,i want an answer to my four-sheet letter4 8,i am unbroken in my determination to publish a work on ,parallels1 as soon as ,i have put my material in order and have the means4 8,at present ,i have not made any discovery1 but the way ,i have followed almost certainly promises me the attainment of my object if any possibility of it exists4 8,i have not got my object yet1 but ,i have produced such stupendous things that ,i was overwhelmed myself1 and it would be an eternal shame if they were lost4 ,when you see them you will find that it is so4 ,now ,i can only say that ,i have made a new world out of nothing4 ,everything that ,i have sent you before is a house of cards in comparison with a tower4 ,i am convinced that it will be no less to my honour than if ,i had already discovered it40 ,the discovery of which ,johann here speaks was published as an appendix to ,wolfgang ,bolyai's ,tentamen4 ,sending the book to ,gauss1 ,wolfgang writes1 after an interruption of eighteen years in his correspondence3-- 8,my son is first lieutenant of ,engineers and will soon be captain4 ,he is a fine youth1 a good violin player1 a skilful fencer1 and brave1 but has had many duels1 and is wild even for a soldier4 ,yet he is distinguished--light in darkness and darkness in light4 ,he is an impassioned mathematician with extraordinary capacities4444 ,he will think more of your judgment on his work than that of all ,europe40 ,wolfgang received no answer from ,gauss to this letter1 but sending a second copy of the book received the following reply3-- 8,you have rejoiced me1 my unforgotten friend1 by your letters4 ,i delayed answering the first because ,i wanted to wait for the arrival of the promised little book4 8,now something about your son's work4 8,if ,i begin with saying that #8,i ought not to praise it1' you will be staggered for a moment4 ,but ,i cannot say anything else4 ,to praise it is to praise myself1 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 reflections1 some of which date from thirty to thirty-five years ago4 8,in fact ,i am astonished to the uttermost4 ,my intention was to let nothing be known in my lifetime about my own work1 of which1 for the rest1 but little is committed to writing4 ,most people have but little perception of the problem1 and ,i have found very few who took any interest in the views ,i expressed to them4 ,to be able to do that one must first of all have had a real live feeling of what is wanting1 and as to that most men are completely in the dark4 8,still it was my intention to commit everything to writing in the course of time1 so that at least it should not perish with me4 8,i am deeply surprised that this task can be spared me1 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 me40 ,the impression which we receive from ,gauss's inexplicable silence towards his old friend is swept away by this letter4 ,hence we breathe the clear air of the mountain tops4 ,gauss would not have failed to perceive the vast significance of his thoughts1 sure to be all the greater in their effect on future ages from the want of comprehension of the present4 ,yet there is not a word or a sign in his writing to claim the thought for himself4 ,he published no single line on the subject4 ,by the measure of what he thus silently relinquishes1 by such a measure of a world-transforming thought1 we can appreciate his greatness4 ,it is a long step from ,gauss's serenity to the disturbed and passionate life of ,johann ,bolyai--he and ,galois1 the two most interesting figures in the history of mathematics4 ,for ,bolyai1 the wild soldier1 the duellist1 fell at odds with the world4 ,it is related of him that he was challenged by thirteen officers of his garrison1 a thing not unlikely to happen considering how differently he thought from every one else4 ,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 opponent4 ,he disarmed or wounded all his antagonists4 ,it can be easily imagined that a temperament such as his was one not congenial to his military superiors4 ,he was retired in #ahcc4 ,his epoch-making discovery awoke no attention4 ,he seems to have conceived the idea that his father had betrayed him in some inexplicable way by his communications with ,gauss1 and he challenged the excellent ,wolfgang to a duel4 ,he passed his life in poverty1 many a time1 says his biographer1 seeking to snatch himself from dissipation and apply himself again to mathematics4 ,but his efforts had no result4 ,he died ,january #bgth1 #ahfj1 fallen out with the world and with himself4 ,,metageometry ,the theories which are generally connected with the names of ,lobatchewsky and ,bolyai bear a singular and curious relation to the subject of higher space4 ,in order to show what this relation is1 ,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 figures4 ,no mathematical processes beyond this simple one of counting will be necessary4 ,illustration3 ,fig4 #ai4 ,let us suppose we have before us in fig4 #ai a plane covered with points at regular intervals1 so placed that every four determine a square4 ,now it is evident that as four points determine a square1 so four squares meet in a point4 ,illustration3 ,fig4 #bj4 ,thus1 considering a point inside a square as belonging to it1 we may say that a point on the corner of a square belongs to it and to three others equally3 belongs a quarter of it to each square4 ,illustration3 ,fig4 #ba4 ,illustration3 ,fig4 #bb4 ,thus the square ,,acde 7fig4 #ba7 contains one point1 and has four points at the four corners4 ,since one-fourth of each of these four belongs to the square1 the four together count as one point1 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 exclusively4 ,now the area of this square is two unit squares1 as can be seen by drawing two diagonals in fig4 #bb4 ,we also notice that the square in question is equal to the sum of the squares on the sides ,,ab1 ,,bc1 of the right-angled triangle ,,abc4 ,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 triangle4 ,now suppose we set ourselves the question of determining the whereabouts in the ordered system of points1 the end of a line would come when it turned about a point keeping one extremity fixed at the point4 ,we can solve this problem in a particular case4 ,if we can find a square lying slantwise amongst the dots which is equal to one which goes regularly1 we shall know that the two sides are equal1 and that the slanting side is equal to the straight-way side4 ,thus the volume and shape of a figure remaining unchanged will be the test of its having rotated about the point1 so that we can say that its side in its first position would turn into its side in the second position4 ,now1 such a square can be found in the one whose side is five units in length4 ,illustration3 ,fig4 #bc4 ,in fig4 #bc1 in the square on ,,ab1 there are-- #i points interior #i #d at the corners #a #d sides with #c on each side1 considered as #a on each side1 because belonging equally to two squares #f ,the total is #af4 ,there are #i points in the square on ,,bc4 ,in the square on ,,ac there are-- #bd points inside #bd #d at the corners #a or #be altogether4 ,hence we see again that the square on the hypothenuse is equal to the squares on the sides4 ,now take the square ,,afhg1 which is larger than the square on ,,ab4 ,it contains #be points4 #af inside #af #af on the sides1 counting as #h #d on the corners #a making #be altogether4 ,if two squares are equal we conclude the sides are equal4 ,hence1 the line ,,af turning round ,a would move so that it would after a certain turning coincide with ,,ac4 ,this is preliminary1 but it involves all the mathematical difficulties that will present themselves4 ,there are two alterations of a body by which its volume is not changed4 ,one is the one we have just considered1 rotation1 the other is what is called shear4 ,consider a book1 or heap of loose pages4 ,they can be slid so that each one slips over the preceding one1 and the whole assumes the shape b in fig4 #bd4 ,illustration3 ,fig4 #bd4 ,this deformation is not shear alone1 but shear accompanied by rotation4 ,shear can be considered as produced in another way4 ,take the square ,,abcd 7fig4 #be71 and suppose that it is pulled out from along one of its diagonals both ways1 and proportionately compressed along the other diagonal4 ,it will assume the shape in fig4 #bf4 ,this compression and expansion along two lines at right angles is what is called shear2 it is equivalent to the sliding illustrated above1 combined with a turning round4 ,illustration3 ,fig4 #be4 ,illustration3 ,fig4 #bf4 ,in pure shear a body is compressed and extended in two directions at right angles to each other1 so that its volume remains unchanged4 ,now we know that our material bodies resist shear--shear does violence to the internal arrangement of their particles1 but they turn as wholes without such internal resistance4 ,but there is an exception4 ,in a liquid shear and rotation take place equally easily1 there is no more resistance against a shear than there is against a rotation4 ,now1 suppose all bodies were to be reduced to the liquid state1 in which they yield to shear and to rotation equally easily1 and then were to be reconstructed as solids1 but in such a way that shear and rotation had interchanged places4 ,that is to say1 let us suppose that when they had become solids again they would shear without offering any internal resistance1 but a rotation would do violence to their internal arrangement4 ,that is1 we should have a world in which shear would have taken the place of rotation4 ,a shear does not alter the volume of a body3 thus an inhabitant living in such a world would look on a body sheared as we look on a body rotated4 ,he would say that it was of the same shape1 but had turned a bit round4 ,let us imagine a ,pythagoras in this world going to work to investigate1 as is his wont4 ,illustration3 ,fig4 #bg4 ,illustration3 ,fig4 #bh4 ,fig4 #bg represents a square unsheared4 ,fig4 #bh represents a square sheared4 ,it is not the figure into which the square in fig4 #bg would turn1 but the result of shear on some square not drawn4 ,it is a simple slanting placed figure1 taken now as we took a simple slanting placed square before4 ,now1 since bodies in this world of shear offer no internal resistance to shearing1 and keep their volume when sheared1 an inhabitant accustomed to them would not consider that they altered their shape under shear4 ,he would call ,,acde as much a square as the square in fig4 #bg4 ,we will call such figures shear squares4 ,counting the dots in ,,acde1 we find-- #b inside #b #d at corners #a or a total of #c4 ,now1 the square on the side ,,ab has #d points1 that on ,,bc has #a point4 ,here the shear square on the hypothenuse has not #e points but #c2 it is not the sum of the squares on the sides1 but the difference4 ,this relation always holds4 ,look at fig4 #bi4 ,illustration3 ,fig4 #bi4 ,shear square on hypothenuse-- #g internal #g #d at corners #a -- #h ,illustration3 ,fig4 #bi bis4 ,square on one side--which the reader can draw for himself-- #d internal #d #h on sides #d #d at corners #a -- #i and the square on the other side is #a4 ,hence in this case again the difference is equal to the shear square on the hypothenuse1 #i - #a #h4 ,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 triangle4 ,in fig4 #bi bis another shear square is drawn on which the above relation can be tested4 ,what now would be the position a line on turning by shear would take up8 ,we must settle this in the same way as previously with our turning4 ,since a body sheared remains the same1 we must find two equal bodies1 one in the straight way1 one in the slanting way1 which have the same volume4 ,then the side of one will by turning become the side of the other1 for the two figures are each what the other becomes by a shear turning4 ,we can solve the problem in a particular case-- ,illustration3 ,fig4 #cj4 ,in the figure ,,acde 7fig4 #cj7 there are-- #ae inside #ae #d at corners #a a total of #af4 ,now in the square ,,abgf1 there are #af-- #i inside #i #ab on sides #f #d at corners #a -- #af ,hence the square on ,,ab would1 by the shear turning1 become the shear square ,,acde4 ,and hence the inhabitant of this world would say that the line ,,ab turned into the line ,,ac4 ,these two lines would be to him two lines of equal length1 one turned a little way round from the other4 ,that is1 putting shear in place of rotation1 we get a different kind of figure1 as the result of the shear rotation1 from what we got with our ordinary rotation4 ,and as a consequence we get a position for the end of a line of invariable length when it turns by the shear rotation1 different from the position which it would assume on turning by our rotation4 ,a real material rod in the shear world would1 on turning about ,a1 pass from the position ,,ab to the position ,,ac4 ,we say that its length alters when it becomes ,,ac1 but this transformation of ,,ab would seem to an inhabitant of the shear world like a turning of ,,ab without altering in length4 ,if now we suppose a communication of ideas that takes place between one of ourselves and an inhabitant of the shear world1 there would evidently be a difference between his views of distance and ours4 ,we should say that his line ,,ab increased in length in turning to ,,ac4 ,he would say that our line ,,af 7fig4 #bc7 decreased in length in turning to ,,ac4 ,he would think that what we called an equal line was in reality a shorter one4 ,we should say that a rod turning round would have its extremities in the positions we call at equal distances4 ,so would he--but the positions would be different4 ,he could1 like us1 appeal to the properties of matter4 ,his rod to him alters as little as ours does to us4 ,now1 is there any standard to which we could appeal1 to say which of the two is right in this argument8 ,there is no standard4 ,we should say that1 with a change of position1 the configuration and shape of his objects altered4 ,he would say that the configuration and shape of our objects altered in what we called merely a change of position4 ,hence distance independent of position is inconceivable1 or practically distance is solely a property of matter4 ,there is no principle to which either party in this controversy could appeal4 ,there is nothing to connect the definition of distance with our ideas rather than with his1 except the behaviour of an actual piece of matter4 ,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 us4 ,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 world4 ,it was the geometry of such conceivable worlds that ,lobatchewsky and ,bolyai studied4 ,this kind of geometry has evidently nothing to do directly with four-dimensional space4 ,but a connection arises in this way4 ,it is evident that1 instead of taking a simple shear as ,i have done1 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 action1 ,i might take a complex motion1 composed of a shear and a rotation together1 or some other kind of deformation4 ,let us suppose such an alteration picked out and defined as the one which means simple rotation1 then the type1 according to which all bodies will alter by this rotation1 is fixed4 ,looking at the movements of this kind1 we should say that the objects were altering their shape as well as rotating4 ,but to the inhabitants of that world they would seem to be unaltered1 and our figures in their motions would seem to them to alter4 ,in such a world the features of geometry are different4 ,we have seen one such difference in the case of our illustration of the world of shear1 where the square on the hypothenuse was equal to the difference1 not the sum1 of the squares on the sides4 ,in our illustration we have the same laws of parallel lines as in our ordinary rotation world1 but in general the laws of parallel lines are different4 ,in one of these worlds of a different constitution of matter through one point there can be two parallels to a given line1 in another of them there can be none1 that is1 although a line be drawn parallel to another it will meet it after a time4 ,now it was precisely in this respect of parallels that ,lobatchewsky and ,bolyai discovered these different worlds4 ,they did not think of them as worlds of matter1 but they discovered that space did not necessarily mean that our law of parallels is true4 ,they made the distinction between laws of space and laws of matter1 although that is not the form in which they stated their results4 ,the way in which they were led to these results was the following4 ,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 drawn1 as a demand1 as something that must be assumed4 ,the words of his ninth postulate are these3 8,if a straight line meeting two other straight lines makes the interior angles on the same side of it equal to two right angles1 the two straight lines will never meet40 ,the mathematicians of later ages did not like this bald assumption1 and not being able to prove the proposition they called it an axiom--the eleventh axiom4 ,many attempts were made to prove the axiom2 no one doubted of its truth1 but no means could be found to demonstrate it4 ,at last an ,italian1 ,sacchieri1 unable to find a proof1 said3 8,let us suppose it not true40 ,he deduced the results of there being possibly two parallels to one given line through a given point1 but feeling the waters too deep for the human reason1 he devoted the latter half of his book to disproving what he had assumed in the first part4 ,then ,bolyai and ,lobatchewsky with firm step entered on the forbidden path4 ,there can be no greater evidence of the indomitable nature of the human spirit1 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 ,lobatchewsky4 ,c ,d ,a ,b ,take a line ,,ab and a point ,c4 ,we say and see and know that through ,c can only be drawn one line parallel to ,,ab4 ,but ,bolyai said3 8,i will draw two40 ,let ,,cd be parallel to ,,ab1 that is1 not meet ,,ab however far produced1 and let lines beyond ,,cd also not meet ,,ab2 let there be a certain region between ,,cd and ,,ce1 in which no line drawn meets ,,ab4 ,,ce and ,,cd produced backwards through ,c will give a similar region on the other side of ,c4 ,illustration3 ,fig4 #cb4 ,nothing so triumphantly1 one may almost say so insolently1 ignoring of sense had ever been written before4 ,men had struggled against the limitations of the body1 fought them1 despised them1 conquered them4 ,but no one had ever thought simply as if the body1 the bodily eyes1 the organs of vision1 all this vast experience of space1 had never existed4 ,the age-long contest of the soul with the body1 the struggle for mastery1 had come to a culmination4 ,bolyai and ,lobatchewsky simply thought as if the body was not4 ,the struggle for dominion1 the strife and combat of the soul were over2 they had mastered1 and the ,hungarian drew his line4 ,can we point out any connection1 as in the case of ,parmenides1 between these speculations and higher space8 ,can we suppose it was any inner perception by the soul of a motion not known to the senses1 which resulted in this theory so free from the bonds of sense8 ,no such supposition appears to be possible4 ,practically1 however1 metageometry had a great influence in bringing the higher space to the front as a working hypothesis4 ,this can be traced to the tendency the mind has to move in the direction of least resistance4 ,the results of the new geometry could not be neglected1 the problem of parallels had occupied a place too prominent in the development of mathematical thought for its final solution to be neglected4 ,but this utter independence of all mechanical considerations1 this perfect cutting loose from the familiar intuitions1 was so difficult that almost any other hypothesis was more easy of acceptance1 and when ,beltrami showed that the geometry of ,lobatchewsky and ,bolyai was the geometry of shortest lines drawn on certain curved surfaces1 the ordinary definitions of measurement being retained1 attention was drawn to the theory of a higher space4 ,an illustration of ,beltrami's theory is furnished by the simple consideration of hypothetical beings living on a spherical surface4 ,illustration3 ,fig4 #cc4 ,let ,,abcd be the equator of a globe1 and ,,ap1 ,,bp1 meridian lines drawn to the pole1 ,p4 ,the lines ,,ab1 ,,ap1 ,,bp would seem to be perfectly straight to a person moving on the surface of the sphere1 and unconscious of its curvature4 ,now ,,ap and ,,bp both make right angles with ,,ab4 ,hence they satisfy the definition of parallels4 ,yet they meet in ,p4 ,hence a being living on a spherical surface1 and unconscious of its curvature1 would find that parallel lines would meet4 ,he would also find that the angles in a triangle were greater than two right angles4 ,in the triangle ,,pab1 for instance1 the angles at ,a and ,b are right angles1 so the three angles of the triangle ,,pab are greater than two right angles4 ,now in one of the systems of metageometry 7for after ,lobatchewsky had shown the way it was found that other systems were possible besides his7 the angles of a triangle are greater than two right angles4 ,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 plane1 the matter in which had such properties as are presupposed by one of these systems of geometry4 ,beltrami also discovered a certain surface on which there could be drawn more than one 8straight0 line through a point which would not meet another given line4 ,i use the word straight as equivalent to the line having the property of giving the shortest path between any two points on it4 ,hence1 without giving up the ordinary methods of measurement1 it was possible to find conditions in which a plane being would necessarily have an experience corresponding to ,lobatchewsky's geometry4 ,and by the consideration of a higher space1 and a solid curved in such a higher space1 it was possible to account for a similar experience in a space of three dimensions4 ,now1 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 circle4 ,hence1 a logical conception having been found harder than that of a four dimensional space1 thought turned to the latter as a simple explanation of the possibilities to which ,lobatchewsky had awakened it4 ,thinkers became accustomed to deal with the geometry of higher space--it was ,kant1 says ,veronese1 who first used the expression of 8different spaces0--and with familiarity the inevitableness of the conception made itself felt4 ,from this point it is but a small step to adapt the ordinary mechanical conceptions to a higher spatial existence1 and then the recognition of its objective existence could be delayed no longer4 ,here1 too1 as in so many cases1 it turns out that the order and connection of our ideas is the order and connection of things4 ,what is the significance of ,lobatchewsky's and ,bolyai's work8 ,it must be recognised as something totally different from the conception of a higher space2 it is applicable to spaces of any number of dimensions4 ,by immersing the conception of distance in matter to which it properly belongs1 it promises to be of the greatest aid in analysis for the effective distance of any two particles is the product of complex material conditions and cannot be measured by hard and fast rules4 ,its ultimate significance is altogether unknown4 ,it is a cutting loose from the bonds of sense1 not coincident with the recognition of a higher dimensionality1 but indirectly contributory thereto4 ,thus1 finally1 we have come to accept what ,plato held in the hollow of his hand2 what ,aristotle's doctrine of the relativity of substance implies4 ,the vast universe1 too1 has its higher1 and in recognising it we find that the directing being within us no longer stands inevitably outside our systematic knowledge4 ,,chapter ,,vi ,,the ,,higher ,,world ,it is indeed strange1 the manner in which we must begin to think about the higher world4 ,those simplest objects analogous to those which are about us on every side in our daily experience such as a door1 a table1 a wheel are remote and uncognisable in the world of four dimensions1 while the abstract ideas of rotation1 stress and strain1 elasticity into which analysis resolves the familiar elements of our daily experience are transferable and applicable with no difficulty whatever4 ,thus we are in the unwonted position of being obliged to construct the daily and habitual experience of a four-dimensional being1 from a knowledge of the abstract theories of the space1 the matter1 the motion of it2 instead of1 as in our case1 passing to the abstract theories from the richness of sensible things4 ,what would a wheel be in four dimensions8 ,what the shafting for the transmission of power which a four-dimensional being would use4 ,the four-dimensional wheel1 and the four-dimensional shafting are what will occupy us for these few pages4 ,and it is no futile or insignificant enquiry4 ,for in the attempt to penetrate into the nature of the higher1 to grasp within our ken that which transcends all analogies1 because what we know are merely partial views of it1 the purely material and physical path affords a means of approach pursuing which we are in less likelihood of error than if we use the more frequently trodden path of framing conceptions which in their elevation and beauty seem to us ideally perfect4 ,for where we are concerned with our own thoughts1 the development of our own ideals1 we are as it were on a curve1 moving at any moment in a direction of tangency4 ,whither we go1 what we set up and exalt as perfect1 represents not the true trend of the curve1 but our own direction at the present--a tendency conditioned by the past1 and by a vital energy of motion essential but only true when perpetually modified4 ,that eternal corrector of our aspirations and ideals1 the material universe draws sublimely away from the simplest things we can touch or handle to the infinite depths of starry space1 in one and all uninfluenced by what we think or feel1 presenting unmoved fact to which1 think it good or think it evil1 we can but conform1 yet out of all that impassivity with a reference to something beyond our individual hopes and fears supporting us and giving us our being4 ,and to this great being we come with the question3 8,you1 too1 what is your higher80 ,or to put it in a form which will leave our conclusions in the shape of no barren formula1 and attacking the problem on its most assailable side3 8,what is the wheel and the shafting of the four-dimensional mechanic80 ,in entering on this enquiry we must make a plan of procedure4 ,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 rotation1 and then to apply an analogous process to the consideration of the higher movements4 ,the plane being must be imagined as no abstract figure1 but as a real body possessing all three dimensions4 ,his limitation to a plane must be the result of physical conditions4 ,we will therefore think of him as of a figure cut out of paper placed on a smooth plane4 ,sliding over this plane1 and coming into contact with other figures equally thin as he in the third dimension1 he will apprehend them only by their edges4 ,to him they will be completely bounded by lines4 ,a 8solid0 body will be to him a two-dimensional extent1 the interior of which can only be reached by penetrating through the bounding lines4 ,now such a plane being can think of our three-dimensional existence in two ways4 ,first1 he can think of it as a series of sections1 each like the solid he knows of extending in a direction unknown to him1 which stretches transverse to his tangible universe1 which lies in a direction at right angles to every motion which he made4 ,secondly1 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 sections1 each of them in itself exactly like the two-dimensional bodies he knows1 but extending away from his two-dimensional space4 ,a square lying in his space he regards as a solid bounded by four lines1 each of which lies in his space4 ,a square standing at right angles to his plane appears to him as simply a line in his plane1 for all of it except the line stretches in the third dimension4 ,he can think of a three-dimensional body as consisting of a number of such sections1 each of which starts from a line in his space4 ,now1 since in his world he can make any drawing or model which involves only two dimensions1 he can represent each such upright section as it actually is1 and can represent a turning from a known into the unknown dimension as a turning from one to another of his known dimensions4 ,to see the whole he must relinquish part of that which he has1 and take the whole portion by portion4 ,consider now a plane being in front of a square1 fig4 #cd4 ,the square can turn about any point in the plane--say the point ,a4 ,but it cannot turn about a line1 as ,,ab4 ,for1 in order to turn about the line ,,ab1 the square must leave the plane and move in the third dimension4 ,this motion is out of his range of observation1 and is therefore1 except for a process of reasoning1 inconceivable to him4 ,illustration3 ,fig4 #cd4 ,rotation will therefore be to him rotation about a point4 ,rotation about a line will be inconceivable to him4 ,the result of rotation about a line he can apprehend4 ,he can see the first and last positions occupied in a half-revolution about the line ,,ac4 ,the result of such a half revolution is to place the square ,,abcd on the left hand instead of on the right hand of the line ,,ac4 ,it would correspond to a pulling of the whole body ,,abcd through the line ,,ac1 or to the production of a solid body which was the exact reflection of it in the line ,,ac4 ,it would be as if the square ,,abcd turned into its image1 the line ,,ab acting as a mirror4 ,such a reversal of the positions of the parts of the square would be impossible in his space4 ,the occurrence of it would be a proof of the existence of a higher dimensionality4 ,let him now1 adopting the conception of a three-dimensional body as a series of sections lying1 each removed a little farther than the preceding one1 in direction at right angles to his plane1 regard a cube1 fig4 #cf1 as a series of sections1 each like the square which forms its base1 all rigidly connected together4 ,illustration3 ,fig4 #ce4 ,if now he turns the square about the point ,a in the plane of xy1 each parallel section turns with the square he moves4 ,in each of the sections there is a point at rest1 that vertically over ,a4 ,hence he would conclude that in the turning of a three-dimensional body there is one line which is at rest4 ,that is a three-dimensional turning in a turning about a line4 99 99 99 99 99 ,in a similar way let us regard ourselves as limited to a three-dimensional world by a physical condition4 ,let us imagine that there is a direction at right angles to every direction in which we can move1 and that we are prevented from passing in this direction by a vast solid1 that against which in every movement we make we slip as the plane being slips against his plane sheet4 ,we can then consider a four-dimensional body as consisting of a series of sections1 each parallel to our space1 and each a little farther off than the preceding on the unknown dimension4 ,take the simplest four-dimensional body--one which begins as a cube1 fig4 #cf1 in our space1 and consists of sections1 each a cube like fig4 #cf1 lying away from our space4 ,if we turn the cube which is its base in our space about a line1 if1 e4g41 in fig4 #cf we turn the cube about the line ,,ab1 not only it but each of the parallel cubes moves about a line4 ,the cube we see moves about the line ,,ab1 the cube beyond it about a line parallel to ,,ab and so on4 ,hence the whole four-dimensional body moves about a plane1 for the assemblage of these lines is our way of thinking about the plane which1 starting from the line ,,ab in our space1 runs off in the unknown direction4 ,illustration3 ,fig4 #cf4 ,in this case all that we see of the plane about which the turning takes place is the line ,,ab4 ,but it is obvious that the axis plane may lie in our space4 ,a point near the plane determines with it a three-dimensional space4 ,when it begins to rotate round the plane it does not move anywhere in this three-dimensional space1 but moves out of it4 ,a point can no more rotate round a plane in three-dimensional space than a point can move round a line in two-dimensional space4 ,we will now apply the second of the modes of representation to this case of turning about a plane1 building up our analogy step by step from the turning in a plane about a point and that in space about a line1 and so on4 ,in order to reduce our considerations to those of the greatest simplicity possible1 let us realise how the plane being would think of the motion by which a square is turned round a line4 ,let1 fig4 #cd1 ,,abcd be a square on his plane1 and represent the two dimensions of his space by the axes ,ax ,ay4 ,now the motion by which the square is turned over about the line ,,ac involves the third dimension4 ,he cannot represent the motion of the whole square in its turning1 but he can represent the motions of parts of it4 ,let the third axis perpendicular to the plane of the paper be called the axis of z4 ,of the three axes x1 y1 z1 the plane being can represent any two in his space4 ,let him then draw1 in fig4 #ce1 two axes1 x and z4 ,here he has in his plane a representation of what exists in the plane which goes off perpendicularly to his space4 ,in this representation the square would not be shown1 for in the plane of xz simply the line ,,ab of the square is contained4 ,the plane being then would have before him1 in fig4 #ce1 the representation of one line ,,ab of his square and two axes1 x and z1 at right angles4 ,now it would be obvious to him that1 by a turning such as he knows1 by a rotation about a point1 the line ,,ab can turn round ,a1 and occupying all the intermediate positions1 such as ,,ab#a1 come after half a revolution to lie as ,ax produced through ,a4 ,again1 just as he can represent the vertical plane through ,,ab1 so he can represent the vertical plane through ,a,b1 fig4 #cd1 and in a like manner can see that the line ,a,b can turn about the point ,a till it lies in the opposite direction from that which it ran in at first4 ,now these two turnings are not inconsistent4 ,in his plane1 if ,,ab turned about ,a1 and ,a,b about ,a1 the consistency of the square would be destroyed1 it would be an impossible motion for a rigid body to perform4 ,but in the turning which he studies portion by portion there is nothing inconsistent4 ,each line in the square can turn in this way1 hence he would realise the turning of the whole square as the sum of a number of turnings of isolated parts4 ,such turnings1 if they took place in his plane1 would be inconsistent1 but by virtue of a third dimension they are consistent1 and the result of them all is that the square turns about the line ,,ac and lies in a position in which it is the mirror image of what it was in its first position4 ,thus he can realise a turning about a line by relinquishing one of his axes1 and representing his body part by part4 ,let us apply this method to the turning of a cube so as to become the mirror image of itself4 ,in our space we can construct three independent axes1 x1 y1 z1 shown in fig4 #cf4 ,suppose that there is a fourth axis1 w1 at right angles to each and every one of them4 ,we cannot1 keeping all three axes1 x1 y1 z1 represent w in our space2 but if we relinquish one of our three axes we can let the fourth axis take its place1 and we can represent what lies in the space1 determined by the two axes we retain and the fourth axis4 ,illustration3 ,fig4 #cg4 ,let us suppose that we let the y axis drop1 and that we represent the w axis as occupying its direction4 ,we have in fig4 #cg a drawing of what we should then see of the cube4 ,the square ,,abcd1 remains unchanged1 for that is in the plane of xz1 and we still have that plane4 ,but from this plane the cube stretches out in the direction of the y axis4 ,now the y axis is gone1 and so we have no more of the cube than the face ,,abcd4 ,considering now this face ,,abcd1 we see that it is free to turn about the line ,,ab4 ,it can rotate in the x to w direction about this line4 ,in fig4 #ch it is shown on its way1 and it can evidently continue this rotation till it lies on the other side of the z axis in the plane of xz4 ,we can also take a section parallel to the face ,,abcd1 and then letting drop all of our space except the plane of that section1 introduce the w axis1 running in the old y direction4 ,this section can be represented by the same drawing1 fig4 #ch1 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 before4 ,these turnings of the different sections are not inconsistent1 and taken all together they will bring the cube from the position shown in fig4 #cf to that shown in fig4 #da4 ,illustration3 ,fig4 #ch4 ,since we have three axes at our disposal in our space1 we are not obliged to represent the w axis by any particular one4 ,we may let any axis we like disappear1 and let the fourth axis take its place4 ,illustration3 ,fig4 #ci4 ,illustration3 ,fig4 #dj4 ,illustration3 ,fig4 #da4 ,in fig4 #cf suppose the z axis to go4 ,we have then simply the plane of xy and the square base of the cube ,,aceg1 fig4 #ci1 is all that could be seen of it4 ,let now the w axis take the place of the z axis and we have1 in fig4 #ci again1 a representation of the space of xyw1 in which all that exists of the cube is its square base4 ,now1 by a turning of x to w1 this base can rotate around the line ,,ae1 it is shown on its way in fig4 #dj1 and finally it will1 after half a revolution1 lie on the other side of the y axis4 ,in a similar way we may rotate sections parallel to the base of the xw rotation1 and each of them comes to run in the opposite direction from that which they occupied at first4 ,thus again the cube comes from the position of fig4 #cf4 to that of fig4 #da4 ,in this x to w turning1 we see that it takes place by the rotations of sections parallel to the front face about lines parallel to ,,ab1 or else we may consider it as consisting of the rotation of sections parallel to the base about lines parallel to ,,ae4 ,it is a rotation of the whole cube about the plane ,,abef4 ,two separate sections could not rotate about two separate lines in our space without conflicting1 but their motion is consistent when we consider another dimension4 ,just1 then1 as a plane being can think of rotation about a line as a rotation about a number of points1 these rotations not interfering as they would if they took place in his two-dimensional space1 so we can think of a rotation about a plane as the rotation of a number of sections of a body about a number of lines in a plane1 these rotations not being inconsistent in a four-dimensional space as they are in three-dimensional space4 ,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 place4 ,let us draw the section of the cube1 fig4 #cf1 through ,a1 ,f1 ,c1 ,h1 forming a sloping plane4 ,now since the fourth dimension is at right angles to every line in our space it is at right angles to this section also4 ,we can represent our space by drawing an axis at right angles to the plane ,,aceg1 our space is then determined by the plane ,,aceg1 and the perpendicular axis4 ,if we let this axis drop and suppose the fourth axis1 w1 to take its place1 we have a representation of the space which runs off in the fourth dimension from the plane ,,aceg4 ,in this space we shall see simply the section ,,aceg of the cube1 and nothing else1 for one cube does not extend to any distance in the fourth dimension4 ,if1 keeping this plane1 we bring in the fourth dimension1 we shall have a space in which simply this section of the cube exists and nothing else4 ,the section can turn about the line ,,af1 and parallel sections can turn about parallel lines4 ,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 them4 ,illustration3 ,fig4 #db4 ,to bring out this point more clearly let us take two parallel lines1 ,a and ,b1 in the space of xyz1 and let ,,cd and ,,ef be two rods running above and below the plane of xy1 from these lines4 ,if we turn these rods in our space about the lines ,a and ,b1 as the upper end of one1 ,f1 is going down1 the lower end of the other1 ,c1 will be coming up4 ,they will meet and conflict4 ,but it is quite possible for these two rods each of them to turn about the two lines without altering their relative distances4 ,to see this suppose the y axis to go1 and let the w axis take its place4 ,we shall see the lines ,a and ,b no longer1 for they run in the y direction from the points ,g and ,h4 ,illustration3 ,fig4 #dc4 ,fig4 #dc is a picture of the two rods seen in the space of xzw4 ,if they rotate in the direction shown by the arrows--in the z to w direction--they move parallel to one another1 keeping their relative distances4 ,each will rotate about its own line1 but their rotation will not be inconsistent with their forming part of a rigid body4 ,now we have but to suppose a central plane with rods crossing it at every point1 like ,,cd and ,,ef cross the plane of xy1 to have an image of a mass of matter extending equal distances on each side of a diametral plane4 ,as two of these rods can rotate round1 so can all1 and the whole mass of matter can rotate round its diametral plane4 ,this rotation round a plane corresponds1 in four dimensions1 to the rotation round an axis in three dimensions4 ,rotation of a body round a plane is the analogue of rotation of a rod round an axis4 ,in a plane we have rotation round a point1 in three-space rotation round an axis line1 in four-space rotation round an axis plane4 ,the four-dimensional being's shaft by which he transmits power is a disk rotating round its central plane--the whole contour corresponds to the ends of an axis of rotation in our space4 ,he can impart the rotation at any point and take it off at any other point on the contour1 just as rotation round a line can in three-space be imparted at one end of a rod and taken off at the other end4 ,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 wheels4 ,suppose a wheel to move transverse to a plane1 so that the whole disk1 which ,i will consider to be solid and without spokes1 came at the same time into contact with the plane4 ,it would appear as a circular portion of plane matter completely enclosing another and smaller portion--the axle4 ,this appearance would last1 supposing the motion of the wheel to continue until it had traversed the plane by the extent of its thickness1 when there would remain in the plane only the small disk which is the section of the axle4 ,there would be no means obvious in the plane at first by which the axle could be reached1 except by going through the substance of the wheel4 ,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 disappeared4 ,in a similar way a four-dimensional wheel moving transverse to our space would appear first as a solid sphere1 completely surrounding a smaller solid sphere4 ,the outer sphere would represent the wheel1 and would last until the wheel has traversed our space by a distance equal to its thickness4 ,then the small sphere alone would remain1 representing the section of the axle4 ,the large sphere could move round the small one quite freely4 ,any line in space could be taken as an axis1 and round this line the outer sphere could rotate1 while the inner sphere remained still4 ,but in all these directions of revolution there would be in reality one line which remained unaltered1 that is the line which stretches away in the fourth direction1 forming the axis of the axle4 ,the four-dimensional wheel can rotate in any number of planes1 but all these planes are such that there is a line at right angles to them all unaffected by rotation in them4 ,an objection is sometimes experienced as to this mode of reasoning from a plane world to a higher dimensionality4 ,how artificial1 it is argued1 this conception of a plane world is4 ,if any real existence confined to a superficies could be shown to exist1 there would be an argument for one relative to which our three-dimensional existence is superficial4 ,but1 both on the one side and the other of the space we are familiar with1 spaces either with less or more than three dimensions are merely arbitrary conceptions4 ,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 motion1 while we have only one-fourth less than those of the higher space4 ,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 existence1 and that no material existence is possible with a more limited dimensionality than ours4 ,this is well seen if we try to construct the mechanics of a two-dimensional world4 ,no tube could exist1 for unless joined together completely at one end two parallel lines would be completely separate4 ,the possibility of an organic structure1 subject to conditions such as this1 is highly problematical2 yet1 possibly in the convolutions of the brain there may be a mode of existence to be described as two-dimensional4 ,we have but to suppose the increase in surface and the diminution in mass carried on to a certain extent to find a region which1 though without mobility of the constituents1 would have to be described as two-dimensional4 ,but1 however artificial the conception of a plane being may be1 it is none the less to be used in passing to the conception of a greater dimensionality than ours1 and hence the validity of the first part of this objection altogether disappears directly we find evidence for such a state of being4 ,the second part of the objection has more weight4 ,how is it possible to conceive that in a four-dimensional space any creatures should be confined to a three-dimensional existence8 ,in reply ,i would say that we know as a matter of fact that life is essentially a phenomenon of surface4 ,the amplitude of the movements which we can make is much greater along the surface of the earth than it is up or down4 ,now we have but to conceive the extent of a solid surface increased1 while the motions possible tranverse to it are diminished in the same proportion1 to obtain the image of a three-dimensional world in four-dimensional space4 ,and as our habitat is the meeting of air and earth on the world1 so we must think of the meeting place of two as affording the condition for our universe4 ,the meeting of what two8 ,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 curvature8 ,the perfection of the level suggests a liquid--a lake amidst what vast scenery6--whereon the matter of the universe floats speck-like4 ,but this aspect of the problem is like what are called in mathematics boundary conditions4 ,we can trace out all the consequences of four-dimensional movements down to their last detail4 ,then1 knowing the mode of action which would be characteristic of the minutest particles1 if they were free1 we can draw conclusions from what they actually do of what the constraint on them is4 ,of the two things1 the material conditions and the motion1 one is known1 and the other can be inferred4 ,if the place of this universe is a meeting of two1 there would be a one-sideness to space4 ,if it lies so that what stretches away in one direction in the unknown is unlike what stretches away in the other1 then1 as far as the movements which participate in that dimension are concerned1 there would be a difference as to which way the motion took place4 ,this would be shown in the dissimilarity of phenomena1 which1 so far as all three-space movements are concerned1 were perfectly symmetrical4 ,to take an instance1 merely1 for the sake of precising our ideas1 not for any inherent probability in it2 if it could be shown that the electric current in the positive direction were exactly like the electric current in the negative direction1 except for a reversal of the components of the motion in three-dimensional space1 then the dissimilarity of the discharge from the positive and negative poles would be an indication of a one-sideness to our space4 ,the only cause of difference in the two discharges would be due to a component in the fourth dimension1 which directed in one direction transverse to our space1 met with a different resistance to that which it met when directed in the opposite direction4 ,,chapter ,,vii ,,the ,,evidences ,,for ,a ,,fourth ,,dimension ,the method necessarily to be employed in the search for the evidences of a fourth dimension1 consists primarily in the formation of the conceptions of four-dimensional shapes and motions4 ,when we are in possession of these it is possible to call in the aid of observation1 without them we may have been all our lives in the familiar presence of a four-dimensional phenomenon without ever recognising its nature4 ,to take one of the conceptions we have already formed1 the turning of a real thing into its mirror image would be an occurrence which it would be hard to explain1 except on the assumption of a fourth dimension4 ,we know of no such turning4 ,but there exist a multitude of forms which show a certain relation to a plane1 a relation of symmetry1 which indicates more than an accidental juxtaposition of parts4 ,in organic life the universal type is of right- and left-handed symmetry1 there is a plane on each side of which the parts correspond4 ,now we have seen that in four dimensions a plane takes the place of a line in three dimensions4 ,in our space1 rotation about an axis is the type of rotation1 and the origin of bodies symmetrical about a line as the earth is symmetrical about an axis can easily be explained4 ,but where there is symmetry about a plane no simple physical motion1 such as we are accustomed to1 suffices to explain it4 ,in our space a symmetrical object must be built up by equal additions on each side of a central plane4 ,such additions about such a plane are as little likely as any other increments4 ,the probability against the existence of symmetrical form in inorganic nature is overwhelming in our space1 and in organic forms they would be as difficult of production as any other variety of configuration4 ,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 over4 ,the dots spread out on a symmetrical line1 and give the impression of a segmented form with antenn and legs4 ,now seeing a number of such figures we should naturally infer a folding over4 ,can1 then1 a folding over in four-dimensional space account for the symmetry of organic forms8 ,the folding cannot of course be of the bodies we see1 but it may be of those minute constituents1 the ultimate elements of living matter which1 turned in one way or the other1 become right- or left-handed1 and so produce a corresponding structure4 ,there is something in life not included in our conceptions of mechanical movement4 ,is this something a four-dimensional movement8 ,if we look at it from the broadest point of view1 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 world4 ,the interest and values of life as we know it in ourselves1 as we know it existing around us in subordinate forms1 is entirely and completely different to anything which inorganic nature shows4 ,and in living beings we have a kind of form1 a disposition of matter which is entirely different from that shown in inorganic matter4 ,right- and left-handed symmetry does not occur in the configurations of dead matter4 ,we have instances of symmetry about an axis1 but not about a plane4 ,it can be argued that the occurrence of symmetry in two dimensions involves the existence of a three-dimensional process1 as when a stone falls into water and makes rings of ripples1 or as when a mass of soft material rotates about an axis4 ,it can be argued that symmetry in any number of dimensions is the evidence of an action in a higher dimensionality4 ,thus considering living beings1 there is an evidence both in their structure1 and their different mode of activity1 of a something coming in from without into the inorganic world4 ,and the objections which will readily occur1 such as those derived from the forms of twin crystals and the theoretical structure of chemical molecules1 do not invalidate the argument2 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 mobility4 ,in another respect also the existence of symmetrical forms is noteworthy4 ,it is puzzling to conceive how two shapes exactly equal can exist which are not superposible4 ,such a pair of symmetrical figures as the two hands1 right and left1 show either a limitation in our power of movement1 by which we cannot superpose the one on the other1 or a definite influence and compulsion of space on matter1 inflicting limitations which are additional to those of the proportions of the parts4 ,we will1 however1 put aside the arguments to be drawn from the consideration of symmetry as inconclusive1 retaining one valuable indication which they afford4 ,if it is in virtue of a four-dimensional motion that symmetry exists1 it is only in the very minute particles of bodies that that motion is to be found1 for there is no such thing as a bending over in four dimensions of any object of a size which we can observe4 ,the region of the extremely minute is the one1 then1 which we shall have to investigate4 ,we must look for some phenomenon which1 occasioning movements of the kind we know1 still is itself inexplicable as any form of motion which we know4 ,now in the theories of the actions of the minute particles of bodies on one another1 and in the motions of the ether1 mathematicians have tacitly assumed that the mechanical principles are the same as those which prevail in the case of bodies which can be observed1 it has been assumed without proof that the conception of motion being three-dimensional1 holds beyond the region from observations in which it was formed4 ,hence it is not from any phenomenon explained by mathematics that we can derive a proof of four dimensions4 ,every phenomenon that has been explained is explained as three-dimensional4 ,and1 moreover1 since in the region of the very minute we do not find rigid bodies acting on each other at a distance1 but elastic substances and continuous fluids such as ether1 we shall have a double task4 ,we must form the conceptions of the possible movements of elastic and liquid four-dimensional matter1 before we can begin to observe4 ,let us1 therefore1 take the four-dimensional rotation about a plane1 and enquire what it becomes in the case of extensible fluid substances4 ,if four-dimensional movements exist1 this kind of rotation must exist1 and the finer portions of matter must exhibit it4 ,consider for a moment a rod of flexible and extensible material4 ,it can turn about an axis1 even if not straight2 a ring of india rubber can turn inside out4 ,what would this be in the case of four dimensions8 ,let us consider a sphere of our three-dimensional matter having a definite thickness4 ,to represent this thickness let us suppose that from every point of the sphere in fig4 #dd rods project both ways1 in and out1 like ,d and ,f4 ,we can only see the external portion1 because the internal parts are hidden by the sphere4 ,illustration3 ,fig4 #dd4 ,axis of x running towards the observer4 ,in this sphere the axis of x is supposed to come towards the observer1 the axis of z to run up1 the axis of y to go to the right4 ,illustration3 ,fig4 #de4 ,now take the section determined by the zy plane4 ,this will be a circle as shown in fig4 #de4 ,if we let drop the x axis1 this circle is all we have of the sphere4 ,letting the w axis now run in the place of the old x axis we have the space yzw1 and in this space all that we have of the sphere is the circle4 ,fig4 #de then represents all that there is of the sphere in the space of yzw4 ,in this space it is evident that the rods ,,cd and ,,ef can turn round the circumference as an axis4 ,if the matter of the spherical shell is sufficiently extensible to allow the particles ,c and ,e to become as widely separated as they would be in the positions ,d and ,f1 then the strip of matter represented by ,,cd and ,,ef and a multitude of rods like them can turn round the circular circumference4 ,thus this particular section of the sphere can turn inside out1 and what holds for any one section holds for all4 ,hence in four dimensions the whole sphere can1 if extensible turn inside out4 ,moreover1 any part of it--a bowl-shaped portion1 for instance--can turn inside out1 and so on round and round4 ,this is really no more than we had before in the rotation about a plane1 except that we see that the plane can1 in the case of extensible matter1 be curved1 and still play the part of an axis4 ,if we suppose the spherical shell to be of four-dimensional matter1 our representation will be a little different4 ,let us suppose there to be a small thickness to the matter in the fourth dimension4 ,this would make no difference in fig4 #dd1 for that merely shows the view in the xyz space4 ,but when the x axis is let drop1 and the w axis comes in1 then the rods ,,cd and ,,ef which represent the matter of the shell1 will have a certain thickness perpendicular to the plane of the paper on which they are drawn4 ,if they have a thickness in the fourth dimension they will show this thickness when looked at from the direction of the w axis4 ,supposing these rods1 then1 to be small slabs strung on the circumference of the circle in fig4 #de1 we see that there will not be in this case either any obstacle to their turning round the circumference4 ,we can have a shell of extensible material or of fluid material turning inside out in four dimensions4 ,and we must remember that in four dimensions there is no such thing as rotation round an axis4 ,if we want to investigate the motion of fluids in four dimensions we must take a movement about an axis in our space1 and find the corresponding movement about a plane in four space4 ,now1 of all the movements which take place in fluids1 the most important from a physical point of view is vortex motion4 ,a vortex is a whirl or eddy--it is shown in the gyrating wreaths of dust seen on a summer day2 it is exhibited on a larger scale in the destructive march of a cyclone4 ,a wheel whirling round will throw off the water on it4 ,but when this circling motion takes place in a liquid itself it is strangely persistent4 ,there is1 of course1 a certain cohesion between the particles of water by which they mutually impede their motions4 ,but in a liquid devoid of friction1 such that every particle is free from lateral cohesion on its path of motion1 it can be shown that a vortex or eddy separates from the mass of the fluid a certain portion1 which always remain in that vortex4 ,the shape of the vortex may alter1 but it always consists of the same particles of the fluid4 ,now1 a very remarkable fact about such a vortex is that the ends of the vortex cannot remain suspended and isolated in the fluid4 ,they must always run to the boundary of the fluid4 ,an eddy in water that remains half way down without coming to the top is impossible4 ,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 fluid1 terminating one end on each object1 the objects being internal boundaries of the fluid4 ,again1 a vortex may have its ends linked together1 so that it forms a ring4 ,circular vortex rings of this description are often seen in puffs of smoke1 and that the smoke travels on in the ring is a proof that the vortex always consists of the same particles of air4 ,let us now enquire what a vortex would be in a four-dimensional fluid4 ,we must replace the line axis by a plane axis4 ,we should have therefore a portion of fluid rotating round a plane4 ,we have seen that the contour of this plane corresponds with the ends of the axis line4 ,hence such a four-dimensional vortex must have its rim on a boundary of the fluid4 ,there would be a region of vorticity with a contour4 ,if such a rotation were started at one part of a circular boundary1 its edges would run round the boundary in both directions till the whole interior region was filled with the vortex sheet4 ,a vortex in a three-dimensional liquid may consist of a number of vortex filaments lying together producing a tube1 or rod of vorticity4 ,in the same way we can have in four dimensions a number of vortex sheets alongside each other1 each of which can be thought of as a bowl-shaped portion of a spherical shell turning inside out4 ,the rotation takes place at any point not in the space occupied by the shell1 but from that space to the fourth dimension and round back again4 ,is there anything analogous to this within the range of our observation8 ,an electric current answers this description in every respect4 ,electricity does not flow through a wire4 ,its effect travels both ways from the starting point along the wire4 ,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 it4 ,moreover1 it is known that the action of the current is not in the wire4 ,it is in the region enclosed by the wire1 this is the field of force1 the locus of the exhibition of the effects of the current4 ,and the necessity of a conducting circuit for a current is exactly that which we should expect if it were a four-dimensional vortex4 ,according to ,maxwell every current forms a closed circuit1 and this1 from the four-dimensional point of view1 is the same as saying a vortex must have its ends on a boundary of the fluid4 ,thus1 on the hypothesis of a fourth dimension1 the rotation of the fluid ether would give the phenomenon of an electric current4 ,we must suppose the ether to be full of movement1 for the more we examine into the conditions which prevail in the obscurity of the minute1 the more we find that an unceasing and perpetual motion reigns4 ,thus we may say that the conception of the fourth dimension means that there must be a phenomenon which presents the characteristics of electricity4 ,we know now that light is an electro-magnetic action1 and that so far from being a special and isolated phenomenon this electric action is universal in the realm of the minute4 ,hence1 may we not conclude that1 so far from the fourth dimension being remote and far away1 being a thing of symbolic import1 a term for the explanation of dubious facts by a more obscure theory1 it is really the most important fact within our knowledge4 ,our three-dimensional world is superficial4 ,these processes1 which really lie at the basis of all phenomena of matter1 escape our observation by their minuteness1 but reveal to our intellect an amplitude of motion surpassing any that we can see4 ,in such shapes and motions there is a realm of the utmost intellectual beauty1 and one to which our symbolic methods apply with a better grace than they do to those of three dimensions4 ,,chapter ,,viii ,,the ,,use ,,of ,,four ,,dimensions ,,in ,,thought ,having held before ourselves this outline of a conjecture of the world as four-dimensional1 having roughly thrown together those facts of movement which we can see apply to our actual experience1 let us pass to another branch of our subject4 ,the engineer uses drawings1 graphical constructions1 in a variety of manners4 ,he has1 for instance1 diagrams which represent the expansion of steam1 the efficiency of his valves4 ,these exist alongside the actual plans of his machines4 ,they are not the pictures of anything really existing1 but enable him to think about the relations which exist in his mechanisms4 ,and so1 besides showing us the actual existence of that world which lies beneath the one of visible movements1 four-dimensional space enables us to make ideal constructions which serve to represent the relations of things1 and throw what would otherwise be obscure into a definite and suggestive form4 ,from amidst the great variety of instances which lies before me ,i will select two1 one dealing with a subject of slight intrinsic interest1 which however gives within a limited field a striking example of the method of drawing conclusions and the use of higher space figures4#a #a ,it is suggestive also in another respect1 because it shows very clearly that in our processes of thought there are in play faculties other than logical2 in it the origin of the idea which proves to be justified is drawn from the consideration of symmetry1 a branch of the beautiful4 ,the other instance is chosen on account of the bearing it has on our fundamental conceptions4 ,in it ,i try to discover the real meaning of ,kant's theory of experience4 ,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--e4g41 #ab1 and #c are both numbers1 and the relation between them is #d1 another number4 ,the way is thus opened for a process of constructive theory1 without there being any necessity for a recourse to another class of concepts besides that which is given in the phenomena to be studied4 ,the discipline of number thus created is of great and varied applicability1 but it is not solely as quantitative that we learn to understand the phenomena of nature4 ,it is not possible to explain the properties of matter by number simply1 but all the activities of matter are energies in space4 ,they are numerically definite and also1 we may say1 directedly definite1 i4e4 definite in direction4 ,is there1 then1 a body of doctrine about space which1 like that of number1 is available in science8 ,it is needless to answer3 ,yes2 geometry4 ,but there is a method lying alongside the ordinary methods of geometry1 which tacitly used and presenting an analogy to the method of numerical thought deserves to be brought into greater prominence than it usually occupies4 ,the relation of numbers is a number4 ,can we say in the same way that the relation of shapes is a shape8 ,we can4 ,to take an instance chosen on account of its ready availability4 ,let us take two right-angled triangles of a given hypothenuse1 but having sides of different lengths 7fig4 #df74 ,these triangles are shapes which have a certain relation to each other4 ,let us exhibit their relation as a figure4 ,illustration3 ,fig4 #df4 ,draw two straight lines at right angles to each other1 the one ,,hl a horizontal level1 the other ,,vl a vertical level 7fig4 #dg74 ,by means of these two co-ordinating lines we can represent a double set of magnitudes2 one set as distances to the right of the vertical level1 the other as distances above the horizontal level1 a suitable unit being chosen4 ,illustration3 ,fig4 #dg4 ,thus the line marked #g will pick out the assemblage of points whose distance from the vertical level is #g1 and the line marked #a will pick out the points whose distance above the horizontal level is #a4 ,the meeting point of these two lines1 #g and #a1 will define a point which with regard to the one set of magnitudes is #g1 with regard to the other is #a4 ,let us take the sides of our triangles as the two sets of magnitudes in question4 ,then the point #g1 #a1 will represent the triangle whose sides are #g and #a4 ,similarly the point #e1 #e--e1 that is1 to the right of the vertical level and #e above the horizontal level--will represent the triangle whose sides are #e and #e 7fig4 #dh74 ,illustration3 ,fig4 #dh4 ,thus we have obtained a figure consisting of the two points #g1 #a1 and #e1 #e1 representative of our two triangles4 ,but we can go further1 and1 drawing an arc of a circle about ,o1 the meeting point of the horizontal and vertical levels1 which passes through #g1 #a1 and #e1 #e1 assert that all the triangles which are right-angled and have a hypothenuse whose square is #ej are represented by the points on this arc4 ,thus1 each individual of a class being represented by a point1 the whole class is represented by an assemblage of points forming a figure4 ,accepting this representation we can attach a definite and calculable significance to the expression1 resemblance1 or similarity between two individuals of the class represented1 the difference being measured by the length of the line between two representative points4 ,it is needless to multiply examples1 or to show how1 corresponding to different classes of triangles1 we obtain different curves4 ,a representation of this kind in which an object1 a thing in space1 is represented as a point1 and all its properties are left out1 their effect remaining only in the relative position which the representative point bears to the representative points of the other objects1 may be called1 after the analogy of ,sir ,william ,r4 ,hamilton's hodograph1 a 8,poiograph40 ,representations thus made have the character of natural objects2 they have a determinate and definite character of their own4 ,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 construction4 ,every system of classification is a poiograph4 ,in ,mendeleff's scheme of the elements1 for instance1 each element is represented by a point1 and the relations between the elements are represented by the relations between the points4 ,so far ,i have simply brought into prominence processes and considerations with which we are all familiar4 ,but it is worth while to bring into the full light of our attention our habitual assumptions and processes4 ,it often happens that we find there are two of them which have a bearing on each other1 which1 without this dragging into the light1 we should have allowed to remain without mutual influence4 ,there is a fact which it concerns us to take into account in discussing the theory of the poiograph4 ,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 object1 if he knew the co-ordinates of its particles in space1 and their velocity at any particular moment4 ,on the contrary1 in the presence of any natural object1 we have a great complexity of conditions before us1 which we cannot reduce to position in space and date in time4 ,there is mass1 attraction apparently spontaneous1 electrical and magnetic properties which must be superadded to spatial configuration4 ,to cut the list short we must say that practically the phenomena of the world present us problems involving many variables1 which we must take as independent4 ,from this it follows that in making poiographs we must be prepared to use space of more than three dimensions4 ,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 dimensions4 ,it is impossible to give an example of such a poiograph which will not be merely trivial1 without going into details of some kind irrelevant to our subject4 ,i prefer to introduce the irrelevant details rather than treat this part of the subject perfunctorily4 ,to take an instance of a poiograph which does not lead us into the complexities incident on its application in classificatory science1 let us follow ,mrs4 ,alicia ,boole ,stott in her representation of the syllogism by its means4 ,she will be interested to find that the curious gap she detected has a significance4 ,a syllogism consists of two statements1 the major and the minor premiss1 with the conclusion that can be drawn from them4 ,thus1 to take an instance1 fig4 #di4 ,it is evident1 from looking at the successive figures that1 if we know that the region ,m lies altogether within the region ,p1 and also know that the region ,s lies altogether within the region ,m1 we can conclude that the region ,s lies altogether within the region ,p4 ,m is ,p1 major premiss2 ,s is ,m1 minor premiss2 ,s is ,p1 conclusion4 ,given the first two data we must conclude that ,s lies in ,p4 ,the conclusion ,s is ,p involves two terms1 ,s and ,p1 which are respectively called the subject and the predicate1 the letters ,s and ,p being chosen with reference to the parts the notions they designate play in the conclusion4 ,s is the subject of the conclusion1 ,p is the predicate of the conclusion4 ,the major premiss we take to be1 that which does not involve ,s1 and here we always write it first4 ,illustration3 ,fig4 #di4 ,there are several varieties of statement possessing different degrees of universality and manners of assertiveness4 ,these different forms of statement are called the moods4 ,we will take the major premiss as one variable1 as a thing capable of different modifications of the same kind1 the minor premiss as another1 and the different moods we will consider as defining the variations which these variables undergo4 ,there are four moods3-- #a4 ,the universal affirmative2 all ,m is ,p1 called mood ,a4 #b4 ,the universal negative2 no ,m is ,p1 mood ,e4 #c4 ,the particular affirmative2 some ,m is ,p1 mood ,i4 #d4 ,the particular negative2 some ,m is not ,p1 mood ,o4 ,illustration3 #a4 #b4 #c4 #d4 ,mood ,a4 ,mood ,e4 ,mood ,i4 ,mood ,o4 ,fig4 #ej4 ,the dotted lines in #c and #d1 fig4 #ej1 denote that it is not known whether or no any objects exist1 corresponding to the space of which the dotted line forms one delimiting boundary2 thus1 in mood ,i we do not know if there are any ,m',s which are not ,p1 we only know some ,m',s are ,p4 ,illustration3 ,fig4 #ea4 ,representing the first premiss in its various moods by regions marked by vertical lines to the right of ,,pq1 we have in fig4 #ea1 running up from the four letters ,,aeio1 four columns1 each of which indicates that the major premiss is in the mood denoted by the respective letter4 ,in the first column to the right of ,,pq is the mood ,a4 ,now above the line ,,rs let there be marked off four regions corresponding to the four moods of the minor premiss4 ,thus1 in the first row above ,,rs all the region between ,,rs and the first horizontal line above it denotes that the minor premiss is in the mood ,a4 ,the letters ,e1 ,i1 ,o1 in the same way show the mood characterising the minor premiss in the rows opposite these letters4 ,we have still to exhibit the conclusion4 ,to do this we must consider the conclusion as a third variable1 characterised in its different varieties by four moods--this being the syllogistic classification4 ,the introduction of a third variable involves a change in our system of representation4 ,before we started with the regions to the right of a certain line as representing successively the major premiss in its moods2 now we must start with the regions to the right of a certain plane4 ,let ,,lmnr be the plane face of a cube1 fig4 #eb1 and let the cube be divided into four parts by vertical sections parallel to ,,lmnr4 ,the variable1 the major premiss1 is represented by the successive regions which occur to the right of the plane ,,lmnr--that region to which ,a stands opposite1 that slice of the cube1 is significative of the mood ,a4 ,this whole quarter-part of the cube represents that for every part of it the major premiss is in the mood ,a4 ,illustration3 ,fig4 #eb4 ,in a similar manner the next section1 the second with the letter ,e opposite it1 represents that for every one of the sixteen small cubic spaces in it1 the major premiss is in the mood ,e4 ,the third and fourth compartments made by the vertical sections denote the major premiss in the moods ,i and ,o4 ,but the cube can be divided in other ways by other planes4 ,let the divisions1 of which four stretch from the front face1 correspond to the minor premiss4 ,the first wall of sixteen cubes1 facing the observer1 has as its characteristic that in each of the small cubes1 whatever else may be the case1 the minor premiss is in the mood ,a4 ,the variable--the minor premiss--varies through the phases ,a1 ,e1 ,i1 ,o1 away from the front face of the cube1 or the front plane of which the front face is a part4 ,and now we can represent the third variable in a precisely similar way4 ,we can take the conclusion as the third variable1 going through its four phases from the ground plane upwards4 ,each of the small cubes at the base of the whole cube has this true about it1 whatever else may be the case1 that the conclusion is1 in it1 in the mood ,a4 ,thus1 to recapitulate1 the first wall of sixteen small cubes1 the first of the four walls which1 proceeding from left to right1 build up the whole cube1 is characterised in each part of it by this1 that the major premiss is in the mood ,a4 ,the next wall denotes that the major premiss is in the mood ,e1 and so on4 ,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 ,a4 ,the second wall is a region throughout which the minor premiss is in the mood ,e1 and so on4 ,in the layers1 from the bottom upwards1 the conclusion goes through its various moods beginning with ,a in the lowest1 ,e in the second1 ,i in the third1 ,o in the fourth4 ,in the general case1 in which the variables represented in the poiograph pass through a wide range of values1 the planes from which we measure their degrees of variation in our representation are taken to be indefinitely extended4 ,in this case1 however1 all we are concerned with is the finite region4 ,we have now to represent1 by some limitation of the complex we have obtained1 the fact that not every combination of premisses justifies any kind of conclusion4 ,this can be simply effected by marking the regions in which the premisses1 being such as are defined by the positions1 a conclusion which is valid is found4 ,taking the conjunction of the major premiss1 all ,m is ,p1 and the minor1 all ,s is ,m1 we conclude that all ,s is ,p4 ,hence1 that region must be marked in which we have the conjunction of major premiss in mood ,a2 minor premiss1 mood ,a2 conclusion1 mood ,a4 ,this is the cube occupying the lowest left-hand corner of the large cube4 ,illustration3 ,fig4 #ec4 ,proceeding in this way1 we find that the regions which must be marked are those shown in fig4 #ec4 ,to discuss the case shown in the marked cube which appears at the top of fig4 #ec4 ,here the major premiss is in the second wall to the right--it is in the mood ,e and is of the type no ,m is ,p4 ,the minor premiss is in the mood characterised by the third wall from the front4 ,it is of the type some ,s is ,m4 ,from these premisses we draw the conclusion that some ,s is not ,p1 a conclusion in the mood ,o4 ,now the mood ,o of the conclusion is represented in the top layer4 ,hence we see that the marking is correct in this respect4 ,illustration3 ,fig4 #ed4 ,it would1 of course1 be possible to represent the cube on a plane by means of four squares1 as in fig4 #ed1 if we consider each square to represent merely the beginning of the region it stands for4 ,thus the whole cube can be represented by four vertical squares1 each standing for a kind of vertical tray1 and the markings would be as shown4 ,in ,no4 #a the major premiss is in mood ,a for the whole of the region indicated by the vertical square of sixteen divisions2 in ,no4 #b it is in the mood ,e1 and so on4 ,a creature confined to a plane would have to adopt some such disjunctive way of representing the whole cube4 ,he would be obliged to represent that which we see as a whole in separate parts1 and each part would merely represent1 would not be1 that solid content which we see4 ,the view of these four squares which the plane creature would have would not be such as ours4 ,he would not see the interior of the four squares represented above1 but each would be entirely contained within its outline1 the internal boundaries of the separate small squares he could not see except by removing the outer squares4 ,we are now ready to introduce the fourth variable involved in the syllogism4 ,in assigning letters to denote the terms of the syllogism we have taken ,s and ,p to represent the subject and predicate in the conclusion1 and thus in the conclusion their order is invariable4 ,but in the premisses we have taken arbitrarily the order all ,m is ,p1 and all ,s is ,m4 ,there is no reason why ,m instead of ,p should not be the predicate of the major premiss1 and so on4 ,accordingly we take the order of the terms in the premisses as the fourth variable4 ,of this order there are four varieties1 and these varieties are called figures4 ,using the order in which the letters are written to denote that the letter first written is subject1 the one written second is predicate1 we have the following possibilities3-- #ast ,figure4 #bnd ,figure4 #crd ,figure4 #dth ,figure4 ,major ,m ,p ,p ,m ,m ,p ,p ,m ,minor ,s ,m ,s ,m ,m ,s ,m ,s ,there are therefore four possibilities with regard to this fourth variable as with regard to the premisses4 ,we have used up our dimensions of space in representing the phases of the premisses and the conclusion in respect of mood1 and to represent in an analogous manner the variations in figure we require a fourth dimension4 ,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 solid4 ,this fourth dimension is supposed to run at right angles to any of the three space dimensions1 as the third space dimension runs at right angles to the two dimensions of a plane1 and thus it gives us the opportunity of generating a new kind of volume4 ,if the whole cube moves in this dimension1 the solid itself traces out a path1 each section of which1 made at right angles to the direction in which it moves1 is a solid1 an exact repetition of the cube itself4 ,the cube as we see it is the beginning of a solid of such a kind4 ,it represents a kind of tray1 as the square face of the cube is a kind of tray against which the cube rests4 ,suppose the cube to move in this fourth dimension in four stages1 and let the hyper-solid region traced out in the first stage of its progress be characterised by this1 that the terms of the syllogism are in the first figure1 then we can represent in each of the three subsequent stages the remaining three figures4 ,thus the whole cube forms the basis from which we measure the variation in figure4 ,the first figure holds good for the cube as we see it1 and for that hyper-solid which lies within the first stage2 the second figure holds good in the second stage1 and so on4 ,thus we measure from the whole cube as far as figures are concerned4 ,but we saw that when we measured in the cube itself having three variables1 namely1 the two premisses and the conclusion1 we measured from three planes4 ,the base from which we measured was in every case the same4 ,hence1 in measuring in this higher space we should have bases of the same kind to measure from1 we should have solid bases4 ,the first solid base is easily seen1 it is the cube itself4 ,the other can be found from this consideration4 ,that solid from which we measure figure is that in which the remaining variables run through their full range of varieties4 ,now1 if we want to measure in respect of the moods of the major premiss1 we must let the minor premiss1 the conclusion1 run through their range1 and also the order of the terms4 ,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 minor1 the conclusion and the variation of the figures4 ,now the variation of the moods of the minor and of the conclusion are represented in the square face on the left of the cube4 ,here are all varieties of the minor premiss and the conclusion4 ,the varieties of the figures are represented by stages in a motion proceeding at right angles to all space directions1 at right angles consequently to the face in question1 the left-hand face of the cube4 ,consequently letting the left-hand face move in this direction we get a cube1 and in this cube all the varieties of the minor premiss1 the conclusion1 and the figure are represented4 ,thus another cubic base of measurement is given to the cube1 generated by movement of the left-hand square in the fourth dimension4 ,we find the other bases in a similar manner1 one is the cube generated by the front square moved in the fourth dimension so as to generate a cube4 ,from this cube variations in the mood of the minor are measured4 ,the fourth base is that found by moving the bottom square of the cube in the fourth dimension4 ,in this cube the variations of the major1 the minor1 and the figure are given4 ,considering this as a basis in the four stages proceeding from it1 the variation in the moods of the conclusion are given4 ,any one of these cubic bases can be represented in space1 and then the higher solid generated from them lies out of our space4 ,it can only be represented by a device analogous to that by which the plane being represents a cube4 ,he represents the cube shown above1 by taking four square sections and placing them arbitrarily at convenient distances the one from the other4 ,so we must represent this higher solid by four cubes3 each cube represents only the beginning of the corresponding higher volume4 ,it is sufficient for us1 then1 if we draw four cubes1 the first representing that region in which the figure is of the first kind1 the second that region in which the figure is of the second kind1 and so on4 ,these cubes are the beginnings merely of the respective regions--they are the trays1 as it were1 against which the real solids must be conceived as resting1 from which they start4 ,the first one1 as it is the beginning of the region of the first figure1 is characterised by the order of the terms in the premisses being that of the first figure4 ,the second similarly has the terms of the premisses in the order of the second figure1 and so on4 ,these cubes are shown below4 ,for the sake of showing the properties of the method of representation1 not for the logical problem1 ,i will make a digression4 ,i will represent in space the moods of the minor and of the conclusion and the different figures1 keeping the major always in mood ,a4 ,here we have three variables in different stages1 the minor1 the conclusion1 and the figure4 ,let the square of the left-hand side of the original cube be imagined to be standing by itself1 without the solid part of the cube1 represented by 7#b7 fig4 #ee4 ,the ,a1 ,e1 ,i1 ,o1 which run away represent the moods of the minor1 the ,a1 ,e1 ,i1 ,o1 which run up represent the moods of the conclusion4 ,the whole square1 since it is the beginning of the region in the major premiss1 mood ,a1 is to be considered as in major premiss1 mood ,a4 ,from this square1 let it be supposed that that direction in which the figures are represented runs to the left hand4 ,thus we have a cube 7#a7 running from the square above1 in which the square itself is hidden1 but the letters ,a1 ,e1 ,i1 ,o1 of the conclusion are seen4 ,in this cube we have the minor premiss and the conclusion in all their moods1 and all the figures represented4 ,with regard to the major premiss1 since the face 7#b7 belongs to the first wall from the left in the original arrangement1 and in this arrangement was characterised by the major premiss in the mood ,a1 we may say that the whole of the cube we now have put up represents the mood ,a of the major premiss4 ,illustration3 ,fig4 #ee4 ,hence the small cube at the bottom to the right in #a1 nearest to the spectator1 is major premiss1 mood ,a2 minor premiss1 mood ,a2 conclusion1 mood ,a2 and figure the first4 ,the cube next to it1 running to the left1 is major premiss1 mood ,a2 minor premiss1 mood ,a2 conclusion1 mood ,a2 figure #b4 ,so in this cube we have the representations of all the combinations which can occur when the major premiss1 remaining in the mood ,a1 the minor premiss1 the conclusion1 and the figures pass through their varieties4 ,in this case there is no room in space for a natural representation of the moods of the major premiss4 ,to represent them we must suppose as before that there is a fourth dimension1 and starting from this cube as base in the fourth direction in four equal stages1 all the first volume corresponds to major premiss ,a1 the second to major premiss1 mood ,e1 the next to the mood ,i1 and the last to mood ,o4 ,the cube we see is as it were merely a tray against which the four-dimensional figure rests4 ,its section at any stage is a cube4 ,but a transition in this direction being transverse to the whole of our space is represented by no space motion4 ,we can exhibit successive stages of the result of transference of the cube in that direction1 but cannot exhibit the product of a transference1 however small1 in that direction4 ,illustration3 ,fig4 #ef4 ,to return to the original method of representing our variables1 consider fig4 #ef4 ,these four cubes represent four sections of the figure derived from the first of them by moving it in the fourth dimension4 ,the first portion of the motion1 which begins with #a1 traces out a more than solid body1 which is all in the first figure4 ,the beginning of this body is shown in #a4 ,the next portion of the motion traces out a more than solid body1 all of which is in the second figure2 the beginning of this body is shown in #b2 #c and #d follow on in like manner4 ,here1 then1 in one four-dimensional figure we have all the combinations of the four variables1 major premiss1 minor premiss1 figure1 conclusion1 represented1 each variable going through its four varieties4 ,the disconnected cubes drawn are our representation in space by means of disconnected sections of this higher body4 ,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 accompany4 ,the total figure thus represented may be called the universe of thought in respect to these four constituents1 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 faculties4 ,we can go over each of the premisses in each of the moods1 and find out what conclusion logically follows4 ,but this is done in the works on logic2 most simply and clearly ,i believe in 8,jevon's ,logic40 ,as we are only concerned with a formal presentation of the results we will make use of the mnemonic lines printed below1 in which the words enclosed in brackets refer to the figures1 and are not significative3-- ,barbara celarent ,darii ferioque prioris4 ,caesare ,camestris ,festino ,baroko secundae4 ,tertia darapti disamis datisi felapton4 ,bokardo ferisson habet ,quarta insuper addit4 ,bramantip camenes dimaris ferapton fresison4 ,in these lines each significative word has three vowels1 the first vowel refers to the major premiss1 and gives the mood of that premiss1 8a0 signifying1 for instance1 that the major mood is in mood a4 ,the second vowel refers to the minor premiss1 and gives its mood4 ,the third vowel refers to the conclusion1 and gives its mood4 ,thus 7prioris7--of the first figure--the first mnemonic word is 8barbara10 and this gives major premiss1 mood ,a2 minor premiss1 mood ,a2 conclusion1 mood ,a4 ,accordingly in the first of our four cubes we mark the lowest left-hand front cube4 ,to take another instance in the third figure 8,tertia10 the word 8ferisson0 gives us major premiss mood ,e--e4g41 no ,m is ,p1 minor premiss mood ,i2 some ,m is ,s1 conclusion1 mood ,o2 some ,s is not ,p4 ,the region to be marked then in the third representative cube is the one in the second wall to the right for the major premiss1 the third wall from the front for the minor premiss1 and the top layer for the conclusion4 ,it is easily seen that in the diagram this cube is marked1 and so with all the valid conclusions4 ,the regions marked in the total region show which combinations of the four variables1 major premiss1 minor premiss1 figure1 and conclusion exist4 ,that is to say1 we objectify all possible conclusions1 and build up an ideal manifold1 containing all possible combinations of them with the premisses1 and then out of this we eliminate all that do not satisfy the laws of logic4 ,the residue is the syllogism1 considered as a canon of reasoning4 ,looking at the shape which represents the totality of the valid conclusions1 it does not present any obvious symmetry1 or easily characterisable nature4 ,a striking configuration1 however1 is obtained1 if we project the four-dimensional figure obtained into a three-dimensional one2 that is1 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 cube4 ,this corresponds to making abstraction of the figures1 giving all the conclusions which are valid whatever the figure may be4 ,illustration3 ,fig4 #eg4 ,proceeding in this way we obtain the arrangement of marked cubes shown in fig4 #eg4 ,we see that the valid conclusions are arranged almost symmetrically round one cube--the one on the top of the column starting from ,,aaa4 ,there is one breach of continuity however in this scheme4 ,one cube is unmarked1 which if marked would give symmetry4 ,it is the one which would be denoted by the letters ,i1 ,e1 ,o1 in the third wall to the right1 the second wall away1 the topmost layer4 ,now this combination of premisses in the mood ,,ie1 with a conclusion in the mood ,o1 is not noticed in any book on logic with which ,i am familiar4 ,let us look at it for ourselves1 as it seems that there must be something curious in connection with this break of continuity in the poiograph4 ,illustration3 ,fig4 #eh4 ,the propositions ,i1 ,e1 in the various figures are the following1 as shown in the accompanying scheme1 fig4 #eh3--;,first figure3 some ,m is ,p2 no ,s is ,m4 ,second figure3 some ,p is ,m2 no ,s is ,m4 ,third figure3 some ,m is ,p2 no ,m is ,s4 ,fourth figure3 some ,p is ,m2 no ,m is ,s4 ,examining these figures1 we see1 taking the first1 that if some ,m is ,p and no ,s is ,m1 we have no conclusion of the form ,s is ,p in the various moods4 ,it is quite indeterminate how the circle representing ,s lies with regard to the circle representing ,p4 ,it may lie inside1 outside1 or partly inside ,p4 ,the same is true in the other figures #b and #c4 ,but when we come to the fourth figure1 since ,m and ,s lie completely outside each other1 there cannot lie inside ,s that part of ,p which lies inside ,m4 ,now we know by the major premiss that some of ,p does lie in ,m4 ,hence ,s cannot contain the whole of ,p4 ,in words1 some ,p is ,m1 no ,m is ,s1 therefore ,s does not contain the whole of ,p4 ,if we take ,p as the subject1 this gives us a conclusion in the mood ,o about ,p4 ,some ,p is not ,s4 ,but it does not give us conclusion about ,s in any one of the four forms recognised in the syllogism and called its moods4 ,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 syllogism4 ,to take an instance3--,some ,americans 7,p7 are of ,african stock 7,m72 ,no ,aryans 7,s7 are of ,african stock 7,m72 ,aryans 7,s7 do not include all of ,americans 7,p74 ,in order to draw a conclusion about ,s we have to admit the statement1 8,s does not contain the whole of ,p10 as a valid logical form--it is a statement about ,s which can be made4 ,the logic which gives us the form1 8some ,p is not ,s10 and which does not allow us to give the exactly equivalent and equally primary form1 8,s does not contain the whole of ,p10 is artificial4 ,and ,i wish to point out that this artificiality leads to an error4 ,if one trusted to the mnemonic lines given above1 one would conclude that no logical conclusion about ,s can be drawn from the statement1 8some ,p are ,m1 no ,m are ,s40 ,but a conclusion can be drawn3 ,s does not contain the whole of ,p4 ,it is not that the result is given expressed in another form4 ,the mnemonic lines deny that any conclusion can be drawn from premisses in the moods ,i1 ,e1 respectively4 ,thus a simple four-dimensional poiograph has enabled us to detect a mistake in the mnemonic lines which have been handed down unchallenged from medival times4 ,to discuss the subject of these lines more fully a logician defending them would probably say that a particular statement cannot be a major premiss2 and so deny the existence of the fourth figure in the combination of moods4 ,to take our instance3 some ,americans are of ,african stock2 no ,aryans are of ,african stock4 ,he would say that the conclusion is some ,americans are not ,aryans2 and that the second statement is the major4 ,he would refuse to say anything about ,aryans1 condemning us to an eternal silence about them1 as far as these premisses are concerned6 ,but1 if there is a statement involving the relation of two classes1 it must be expressible as a statement about either of them4 ,to bar the conclusion1 8,aryans do not include the whole of ,americans10 is purely a makeshift in favour of a false classification4 ,and the argument drawn from the universality of the major premiss cannot be consistently maintained4 ,it would preclude such combinations as major ,o1 minor ,a1 conclusion ,o--i4e41 such as some mountains 7,m7 are not permanent 7,p72 all mountains 7,m7 are scenery 7,s72 some scenery 7,s7 is not permanent 7,p74 ,this is allowed in 8,jevon's ,logic10 and his omission to discuss ,i1 ,e1 ,o1 in the fourth figure1 is inexplicable4 ,a satisfactory poiograph of the logical scheme can be made by admitting the use of the words some1 none1 or all1 about the predicate as well as about the subject4 ,then we can express the statement1 8,aryans do not include the whole of ,americans10 clumsily1 but1 when its obscurity is fathomed1 correctly1 as 8,some ,aryans are not all ,americans40 ,and this method is what is called the 8quantification of the predicate40 ,the laws of formal logic are coincident with the conclusions which can be drawn about regions of space1 which overlap one another in the various possible ways4 ,it is not difficult so to state the relations or to obtain a symmetrical poiograph4 ,but to enter into this branch of geometry is beside our present purpose1 which is to show the application of the poiograph in a finite and limited region1 without any of those complexities which attend its use in regard to natural objects4 ,if we take the latter--plants1 for instance--and1 without assuming fixed directions in space as representative of definite variations1 arrange the representative points in such a manner as to correspond to the similarities of the objects1 we obtain configuration of singular interest2 and perhaps in this way1 in the making of shapes of shapes1 bodies with bodies omitted1 some insight into the structure of the species and genera might be obtained4 ,,chapter ,,ix ,,application ,,to ,,kant',s ,,theory ,,of ,,experience ,when we observe the heavenly bodies we become aware that they all participate in one universal motion--a diurnal revolution round the polar axis4 ,in the case of fixed stars this is most unqualifiedly true1 but in the case of the sun1 and the planets also1 the single motion of revolution can be discerned1 modified1 and slightly altered by other and secondary motions4 ,hence the universal characteristic of the celestial bodies is that they move in a diurnal circle4 ,but we know that this one great fact which is true of them all has in reality nothing to do with them4 ,the diurnal revolution which they visibly perform is the result of the condition of the observer4 ,it is because the observer is on a rotating earth that a universal statement can be made about all the celestial bodies4 ,the universal statement which is valid about every one of the celestial bodies is that which does not concern them at all1 and is but a statement of the condition of the observer4 ,now there are universal statements of other kinds which we can make4 ,we can say that all objects of experience are in space and subject to the laws of geometry4 ,does this mean that space and all that it means is due to a condition of the observer8 ,if a universal law in one case means nothing affecting the objects themselves1 but only a condition of observation1 is this true in every case8 ,there is shown us in astronomy a vera causa for the assertion of a universal4 ,is the same cause to be traced everywhere8 ,such is a first approximation to the doctrine of ,kant's critique4 ,it is the apprehension of a relation into which1 on the one side and the other1 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 unknown4 ,if spatiality is due to a condition of the observer1 the observer cannot be this bodily self of ours--the body1 like the objects around it1 are equally in space4 ,this conception ,kant applied1 not only to the intuitions of sense1 but to the concepts of reason--wherever a universal statement is made there is afforded him an opportunity for the application of his principle4 ,he constructed a system in which one hardly knows which the most to admire1 the architectonic skill1 or the reticence with regard to things in themselves1 and the observer in himself4 ,his system can be compared to a garden1 somewhat formal perhaps1 but with the charm of a quality more than intellectual1 a besonnenheit1 an exquisite moderation over all4 ,and from the ground he so carefully prepared with that buried in obscurity1 which it is fitting should be obscure1 science blossoms and the tree of real knowledge grows4 ,the critique is a storehouse of ideas of profound interest4 ,the one of which ,i have given a partial statement leads1 as we shall see on studying it in detail1 to a theory of mathematics suggestive of enquiries in many directions4 ,the justification for my treatment will be found amongst other passages in that part of the transcendental analytic1 in which ,kant speaks of objects of experience subject to the forms of sensibility1 not subject to the concepts of reason4 ,kant asserts that whenever we think we think of objects in space and time1 but he denies that the space and time exist as independent entities4 ,he goes about to explain them1 and their universality1 not by assuming them1 as most other philosophers do1 but by postulating their absence4 ,how then does it come to pass that the world is in space and time to us8 ,kant takes the same position with regard to what we call nature--a great system subject to law and order4 8,how do you explain the law and order in nature80 we ask the philosophers4 ,all except ,kant reply by assuming law and order somewhere1 and then showing how we can recognise it4 ,in explaining our notions1 philosophers from other than the ,kantian standpoint1 assume the notions as existing outside us1 and then it is no difficult task to show how they come to us1 either by inspiration or by observation4 ,we ask 8,why do we have an idea of law in nature80 8,because natural processes go according to law10 we are answered1 8and experience inherited or acquired1 gives us this notion40 ,but when we speak about the law in nature we are speaking about a notion of our own4 ,so all that these expositors do is to explain our notion by an assumption of it4 ,kant is very different4 ,he supposes nothing4 ,an experience such as ours is very different from experience in the abstract4 ,imagine just simply experience1 succession of states1 of consciousness6 ,why1 there would be no connecting any two together1 there would be no personal identity1 no memory4 ,it is out of a general experience such as this1 which1 in respect to anything we call real1 is less than a dream1 that ,kant shows the genesis of an experience such as ours4 ,kant takes up the problem of the explanation of space1 time1 order1 and so quite logically does not presuppose them4 ,but how1 when every act of thought is of things in space1 and time1 and ordered1 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 ordered4 ,that is our problem1 to represent that which ,kant assumes not subject to any of our forms of thought1 and then show some function which working on that makes it into a 8nature0 subject to law and order1 in space and time4 ,such a function ,kant calls the 8,unity of ,apperception02 i4e41 that which makes our state of consciousness capable of being woven into a system with a self1 an outer world1 memory1 law1 cause1 and order4 ,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 space1 and in time an existence not in time8 ,this difficulty is still more evident when we come to construct a poiograph1 for a poiograph is essentially a space structure4 ,but because more evident the difficulty is nearer a solution4 ,if we always think in space1 i4e4 using space concepts1 the first condition requisite for adapting them to the representation of non-spatial existence1 is to be aware of the limitation of our thought1 and so be able to take the proper steps to overcome it4 ,the problem before us1 then1 is to represent in space an existence not in space4 ,the solution is an easy one4 ,it is provided by the conception of alternativity4 ,to get our ideas clear let us go right back behind the distinctions of an inner and an outer world4 ,both of these1 ,kant says1 are products4 ,let us take merely states of consciousness1 and not ask the question whether they are produced or superinduced--to ask such a question is to have got too far on1 to have assumed something of which we have not traced the origin4 ,of these states let us simply say that they occur4 ,let us now use the word a 8posit0 for a phase of consciousness reduced to its last possible stage of evanescence2 let a posit be that phase of consciousness of which all that can be said is that it occurs4 ,let a1 b1 c1 be three such posits4 ,we cannot represent them in space without placing them in a certain order1 as a1 b1 c4 ,but ,kant distinguishes between the forms of sensibility and the concepts of reason4 ,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 reason4 ,it is partially subject to the concepts of reason because1 although there is no order of sequence1 still at any given time there is order4 ,perception of a thing as in space is a form of sensibility1 the perception of an order is a concept of reason4 ,we must1 therefore1 in order to get at that process which ,kant supposes to be constitutive of an ordered experience imagine the posits as in space without order4 ,as we know them they must be in some order1 abc1 bca1 cab1 acb1 cba1 bac1 one or another4 ,to represent them as having no order conceive all these different orders as equally existing4 ,introduce the conception of alternativity--let us suppose that the order abc1 and bac1 for example1 exist equally1 so that we cannot say about a that it comes before or after b4 ,this would correspond to a sudden and arbitrary change of a into b and b into a1 so that1 to use ,kant's words1 it would be possible to call one thing by one name at one time and at another time by another name4 ,in an experience of this kind we have a kind of chaos1 in which no order exists2 it is a manifold not subject to the concepts of reason4 ,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 arise8 ,in the precise condition in which the posits are1 as described above1 it does not seem to be possible4 ,but if we imagine a duality to exist in the manifold1 a function of consciousness can be easily discovered which will produce order out of no order4 ,let us imagine each posit1 then1 as having1 a dual aspect4 ,let a be #a;a in which the dual aspect is represented by the combination of symbols4 ,and similarly let b be #b;b1 c be #c;c1 in which #b and b represent the dual aspects of b1 #c and c those of c4 ,since a can arbitrarily change into b1 or into c1 and so on1 the particular combinations written above cannot be kept4 ,we have to assume the equally possible occurrence of form such as #b;a1 #b;b1 and so on2 and in order to get a representation of all those combinations out of which any set is alternatively possible1 we must take every aspect with every aspect4 ,we must1 that is1 have every letter with every number4 ,let us now apply the method of space representation4 ,note4--,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 passages4 ,they are there carried on to a greater multiplicity of dimensions1 and the significance of the process here briefly explained becomes more apparent4 ,illustration3 ,fig4 #ei4 ,take three mutually rectangular axes in space #a1 #b1 #c 7fig4 #ei71 and on each mark three points1 the common meeting point being the first on each axis4 ,then by means of these three points on each axis we define #bg positions1 #bg points in a cubical cluster1 shown in fig4 #fj1 the same method of co-ordination being used as has been described before4 ,each of these positions can be named by means of the axes and the points combined4 ,illustration3 ,fig4 #fj4 ,thus1 for instance1 the one marked by an asterisk can be called #a;c1 #b;b1 #c;c1 because it is opposite to c on #a1 to b on #b1 to c on #c4 ,let us now treat of the states of consciousness corresponding to these positions4 ,each point represents a composite of posits1 and the manifold of consciousness corresponding to them is of a certain complexity4 ,suppose now the constituents1 the points on the axes1 to interchange arbitrarily1 any one to become any other1 and also the axes #a1 #b1 and #c1 to interchange amongst themselves1 any one to become any other1 and to be subject to no system or law1 that is to say1 that order does not exist1 and that the points which run abc on each axis may run bac1 and so on4 ,then any one of the states of consciousness represented by the points in the cluster can become any other4 ,we have a representation of a random consciousness of a certain degree of complexity4 ,now let us examine carefully one particular case of arbitrary interchange of the points1 a1 b1 c2 as one such case1 carefully considered1 makes the whole clear4 ,illustration3 ,fig4 #fa4 ,consider the points named in the figure #a;c1 #b;a1 #c;c2 #a;c1 #b;c1 #c;a2 #a;a1 #b;c1 #c;c1 and examine the effect on them when a change of order takes place4 ,let us suppose1 for instance1 that a changes into b1 and let us call the two sets of points we get1 the one before and the one after1 their change conjugates4 ,before the change #a;c #b;a #c;c #a;c #b;c #c;a #a;a #b;c #c;c,conjug- ,after the change #a;c #b;b #c;c #a;c #b;c #c;b #a;b #b;c #c;c ates4 ,the points surrounded by rings represent the conjugate points4 ,it is evident that as consciousness1 represented first by the first set of points and afterwards by the second set of points1 would have nothing in common in its two phases4 ,it would not be capable of giving an account of itself4 ,there would be no identity4 ,if1 however1 we can find any set of points in the cubical cluster1 which1 when any arbitrary change takes place in the points on the axes1 or in the axes themselves1 repeats itself1 is reproduced1 then a consciousness represented by those points would have a permanence4 ,it would have a principle of identity4 ,despite the no law1 the no order1 of the ultimate constituents1 it would have an order1 it would form a system1 the condition of a personal identity would be fulfilled4 ,the question comes to this1 then4 ,can we find a system of points which is self-conjugate which is such that when any posit on the axes becomes any other1 or when any axis becomes any other1 such a set is transformed into itself1 its identity is not submerged1 but rises superior to the chaos of its constituents8 ,illustration3 ,fig4 #fb4 ,such a set can be found4 ,consider the set represented in fig4 #fb1 and written down in the first of the two lines-- ,self- #a;a #b;b #c;c #a;b #b;a #c;c #a;c #b;a #c;b conjugate4 #a;c #b;b #c;a #a;b #b;c #c;a #a;a #b;c #c;b ,self- #a;c #b;b #c;a #a;b #b;c #c;a #a;a #b;c #c;b conjugate4 #a;a #b;b #c;c #a;b #b;a #c;c #a;c #b;a #c;b ,if now a change into c and c into a1 we get the set in the second line1 which has the same members as are in the upper line4 ,looking at the diagram we see that it would correspond simply to the turning of the figures as a whole4#b ,any arbitrary change of the points on the axes1 or of the axes themselves1 reproduces the same set4 #b ,these figures are described more fully1 and extended1 in the next chapter4 ,thus1 a function1 by which a random1 an unordered1 consciousness could give an ordered and systematic one1 can be represented4 ,it is noteworthy that it is a system of selection4 ,if out of all the alternative forms that only is attended to which is self-conjugate1 an ordered consciousness is formed4 ,a selection gives a feature of permanence4 ,can we say that the permanent consciousness is this selection8 ,an analogy between ,kant and ,darwin comes into light4 ,that which is swings clear of the fleeting1 in virtue of its presenting a feature of permanence4 ,there is no need to suppose any function of 8attending to40 ,a consciousness capable of giving an account of itself is one which is characterised by this combination4 ,all combinations exist--of this kind is the consciousness which can give an account of itself4 ,and the very duality which we have presupposed may be regarded as originated by a process of selection4 ,darwin set himself to explain the origin of the fauna and flora of the world4 ,he denied specific tendencies4 ,he assumed an indefinite variability--that is1 chance--but a chance confined within narrow limits as regards the magnitude of any consecutive variations4 ,he showed that organisms possessing features of permanence1 if they occurred would be preserved4 ,so his account of any structure or organised being was that it possessed features of permanence4 ,kant1 undertaking not the explanation of any particular phenomena but of that which we call nature as a whole1 had an origin of species of his own1 an account of the flora and fauna of consciousness4 ,he denied any specific tendency of the elements of consciousness1 but taking our own consciousness1 pointed out that in which it resembled any consciousness which could survive1 which could give an account of itself4 ,he assumes a chance or random world1 and as great and small were not to him any given notions of which he could make use1 he did not limit the chance1 the randomness1 in any way4 ,but any consciousness which is permanent must possess certain features--those attributes namely which give it permanence4 ,any consciousness like our own is simply a consciousness which possesses those attributes4 ,the main thing is that which he calls the unity of apperception1 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-conjugate1 and so permanent4 ,as with ,darwin so with ,kant1 the reason for existence of any feature comes to this--show that it tends to the permanence of that which possesses it4 ,we can thus regard ,kant as the creator of the first of the modern evolution theories4 ,and1 as is so often the case1 the first effort was the most stupendous in its scope4 ,kant does not investigate the origin of any special part of the world1 such as its organisms1 its chemical elements1 its social communities of men4 ,he simply investigates the origin of the whole--of all that is included in consciousness1 the origin of that 8thought thing0 whose progressive realisation is the knowable universe4 ,this point of view is very different from the ordinary one1 in which a man is supposed to be placed in a world like that which he has come to think of it1 and then to learn what he has found out from this model which he himself has placed on the scene4 ,we all know that there are a number of questions in attempting an answer to which such an assumption is not allowable4 ,mill1 for instance1 explains our notion of 8law0 by an invariable sequence in nature4 ,but what we call nature is something given in thought4 ,so he explains a thought of law and order by a thought of an invariable sequence4 ,he leaves the problem where he found it4 ,kant's theory is not unique and alone4 ,it is one of a number of evolution theories4 ,a notion of its import and significance can be obtained by a comparison of it with other theories4 ,thus in ,darwin's theoretical world of natural selection a certain assumption is made1 the assumption of indefinite variability--slight variability it is true1 over any appreciable lapse of time1 but indefinite in the postulated epochs of transformation--and a whole chain of results is shown to follow4 ,this element of chance variation is not1 however1 an ultimate resting place4 ,it is a preliminary stage4 ,this supposing the all is a preliminary step towards finding out what is4 ,if every kind of organism can come into being1 those that do survive will present such and such characteristics4 ,this is the necessary beginning for ascertaining what kinds of organisms do come into existence4 ,and so ,kant's hypothesis of a random consciousness is the necessary beginning for the rational investigation of consciousness as it is4 ,his assumption supplies1 as it were1 the space in which we can observe the phenomena4 ,it gives the general laws constitutive of any experience4 ,if1 on the assumption of absolute randomness in the constituents1 such and such would be characteristic of the experience1 then1 whatever the constituents1 these characteristics must be universally valid4 ,we will now proceed to examine more carefully the poiograph1 constructed for the purpose of exhibiting an illustration of ,kant's unity of apperception4 ,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 elements1 each non-ordered1 and it is in the reciprocal relation of them that the order1 the definite system1 originates4 ,is there anything in our experience of the nature of a duality8 ,there certainly are objects in our experience which have order and those which are incapable of order4 ,the two roots of a quadratic equation have no order4 ,no one can tell which comes first4 ,if a body rises vertically and then goes at right angles to its former course1 no one can assign any priority to the direction of the north or to the east4 ,there is no priority in directions of turning4 ,we associate turnings with no order progressions in a line with order4 ,but in the axes and points we have assumed above there is no such distinction4 ,it is the same1 whether we assume an order among the turnings1 and no order among the points on the axes1 or1 vice versa1 an order in the points and no order in the turnings4 ,a being with an infinite number of axes mutually at right angles1 with a definite sequence between them and no sequence between the points on the axes1 would be in a condition formally indistinguishable from that of a creature who1 according to an assumption more natural to us1 had on each axis an infinite number of ordered points and no order of priority amongst the axes4 ,a being in such a constituted world would not be able to tell which was turning and which was length along an axis1 in order to distinguish between them4 ,thus to take a pertinent illustration1 we may be in a world of an infinite number of dimensions1 with three arbitrary points on each--three points whose order is indifferent1 or in a world of three axes of arbitrary sequence with an infinite number of ordered points on each4 ,we can't tell which is which1 to distinguish it from the other4 ,thus it appears the mode of illustration which we have used is not an artificial one4 ,there really exists in nature a duality of the kind which is necessary to explain the origin of order out of no order--the duality1 namely1 of dimension and position4 ,let us use the term group for that system of points which remains unchanged1 whatever arbitrary change of its constituents takes place4 ,we notice that a group involves a duality1 is inconceivable without a duality4 ,thus1 according to ,kant1 the primary element of experience is the group1 and the theory of groups would be the most fundamental branch of science4 ,owing to an expression in the critique the authority of ,kant is sometimes adduced against the assumption of more than three dimensions to space4 ,it seems to me1 however1 that the whole tendency of his theory lies in the opposite direction1 and points to a perfect duality between dimension and position in a dimension4 ,if the order and the law we see is due to the conditions of conscious experience1 we must conceive nature as spontaneous1 free1 subject to no predication that we can devise1 but1 however apprehended1 subject to our logic4 ,and our logic is simply spatiality in the general sense--that resultant of a selection of the permanent from the unpermanent1 the ordered from the unordered1 by the means of the group and its underlying duality4 ,we can predicate nothing about nature1 only about the way in which we can apprehend nature4 ,all that we can say is that all that which experience gives us will be conditioned as spatial1 subject to our logic4 ,thus1 in exploring the facts of geometry from the simplest logical relations to the properties of space of any number of dimensions1 we are merely observing ourselves1 becoming aware of the conditions under which we must perceive4 ,do any phenomena present themselves incapable of explanation under the assumption of the space we are dealing with1 then we must habituate ourselves to the conception of a higher space1 in order that our logic may be equal to the task before us4 ,we gain a repetition of the thought that came before1 experimentally suggested4 ,if the laws of the intellectual comprehension of nature are those derived from considering her as absolute chance1 subject to no law save that derived from a process of selection1 then1 perhaps1 the order of nature requires different faculties from the intellectual to apprehend it4 ,the source and origin of ideas may have to be sought elsewhere than in reasoning4 ,the total outcome of the critique is to leave the ordinary man just where he is1 justified in his practical attitude towards nature1 liberated from the fetters of his own mental representations4 ,the truth of a picture lies in its total effect4 ,it is vain to seek information about the landscape from an examination of the pigments4 ,and in any method of thought it is the complexity of the whole that brings us to a knowledge of nature4 ,dimensions are artificial enough1 but in the multiplicity of them we catch some breath of nature4 ,we must therefore1 and this seems to me the practical conclusion of the whole matter1 proceed to form means of intellectual apprehension of a greater and greater degree of complexity1 both dimensionally and in extent in any dimension4 ,such means of representation must always be artificial1 but in the multiplicity of the elements with which we deal1 however incipiently arbitrary1 lies our chance of apprehending nature4 ,and as a concluding chapter to this part of the book1 ,i will extend the figures1 which have been used to represent ,kant's theory1 two steps1 so that the reader may have the opportunity of looking at a four-dimensional figure which can be delineated without any of the special apparatus1 to the consideration of which ,i shall subsequently pass on4 ,,chapter ,x ,a ,,four-,,dimensional ,,figure ,the method used in the preceding chapter to illustrate the problem of ,kant's critique1 gives a singularly easy and direct mode of constructing a series of important figures in any number of dimensions4 ,we have seen that to represent our space a plane being must give up one of his axes1 and similarly to represent the higher shapes we must give up one amongst our three axes4 ,but there is another kind of giving up which reduces the construction of higher shapes to a matter of the utmost simplicity4 ,ordinarily we have on a straight line any number of positions4 ,the wealth of space in position is illimitable1 while there are only three dimensions4 ,i propose to give up this wealth of positions1 and to consider the figures obtained by taking just as many positions as dimensions4 ,in this way ,i consider dimensions and positions as two 8kinds10 and applying the simple rule of selecting every one of one kind with every other of every other kind1 get a series of figures which are noteworthy because they exactly fill space of any number of dimensions 7as the hexagon fills a plane7 by equal repetitions of themselves4 ,the rule will be made more evident by a simple application4 ,let us consider one dimension and one position4 ,i will call the axis i1 and the position o4 -------------------------------i o ,here the figure is the position o on the line i4 ,take now two dimensions and two positions on each4 ,illustration3 ,fig4 #fc4 ,we have the two positions o2 #a on i1 and the two positions o1 #a on j1 fig4 #fc4 ,these give rise to a certain complexity4 ,i will let the two lines i and j meet in the position ,i call o on each1 and ,i will consider i as a direction starting equally from every position on j1 and j as starting equally from every position on i4 ,we thus obtain the following figure3--,a is both oi and oj1 ,b is #a;i and oj1 and so on as shown in fig4 #fc;b4 ,the positions on ,,ac are all oi positions4 ,they are1 if we like to consider it in that way1 points at no distance in the i direction from the line ,,ac4 ,we can call the line ,,ac the oi line4 ,similarly the points on ,,ab are those no distance from ,,ab in the j direction1 and we can call them oj points and the line ,,ab the oj line4 ,again1 the line ,,cd can be called the #a;j line because the points on it are at a distance1 #a in the j direction4 ,illustration3 ,fig4 #fc;b4 ,we have then four positions or points named as shown1 and1 considering directions and positions as 8kinds10 we have the combination of two kinds with two kinds4 ,now1 selecting every one of one kind with every other of every other kind will mean that we take #a of the kind i and with it o of the kind j2 and then1 that we take o of the kind i and with it #a of the kind j4 ,thus we get a pair of positions lying in the straight line ,,bc1 fig4 #fd4 ,we can call this pair #aj and #ja if we adopt the plan of mentally1 adding an i to the first and a j to the second of the symbols written thus--#ja is a short expression for ,oi1 #a;j4 ,illustration3 ,fig4 #fd4 ,coming now to our space1 we have three dimensions1 so we take three positions on each4 ,these positions ,i will suppose to be at equal distances along each axis4 ,the three axes and the three positions on each are shown in the accompanying diagrams1 fig4 #fe1 of which the first represents a cube with the front faces visible1 the second the rear faces of the same cube2 the positions ,i will call #j1 #a1 #b2 the axes1 i1 j1 k4 ,i take the base ,,abc as the starting place1 from which to determine distances in the k direction1 and hence every point in the base ,,abc will be an ok position1 and the base ,,abc can be called an ok plane4 ,illustration3 ,fig4 #fe4 ,in the same way1 measuring the distances from the face ,,adc1 we see that every position in the face ,,adc is an oi position1 and the whole plane of the face may be called an oi plane4 ,thus we see that with the introduction of a new dimension the signification of a compound symbol1 such as 8oi10 alters4 ,in the plane it meant the line ,,ac4 ,in space it means the whole plane ,,acd4 ,now1 it is evident that we have twenty-seven positions1 each of them named4 ,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 positions4 ,a is oi1 oj1 ok4 ,it is at the distance #j along i1 #j along j1 #j along k1 and io can be written in short #jjj1 where the ijk symbols are omitted4 ,the point immediately above is #jja1 for it is no distance in the i direction1 and a distance of #a in the k direction4 ,again1 looking at ,b1 it is at a distance of #b from ,a1 or from the plane ,,adc1 in the i direction1 #j in the j direction from the plane ,,abd1 and #j in the k direction1 measured from the plane ,,abc4 ,hence it is #bjj written for #b;i1 #j;j1 #jk4 ,now1 out of these twenty-seven 8things0 or compounds of position and dimension1 select those which are given by the rule1 every one of one kind with every other of every other kind4 ,take #b of the i kind4 ,with this we must have a #a of the j kind1 and then by the rule we can only have a #j of the k kind1 for if we had any other of the k kind we should repeat one of the kinds we already had4 ,in #b;i1 #a;j1 #ak1 for instance1 #a is repeated4 ,the point we obtain is that marked #baj1 fig4 #ff4 ,illustration3 ,fig4 #ff4 ,proceeding in this way1 we pick out the following cluster of points1 fig4 #fg4 ,they are joined by lines1 dotted where they are hidden by the body of the cube1 and we see that they form a figure--a hexagon which could be taken out of the cube and placed on a plane4 ,it is a figure which will fill a plane by equal repetitions of itself4 ,the plane being representing this construction in his plane would take three squares to represent the cube4 ,let us suppose that he takes the ij axes in his space and k represents the axis running out of his space1 fig4 #fh4 ,in each of the three squares shown here as drawn separately he could select the points given by the rule1 and he would then have to try to discover the figure determined by the three lines drawn4 ,the line from #baj to #abj is given in the figure1 but the line from #bja to #ajb or ,,gk is not given4 ,he can determine ,,gk by making another set of drawings and discovering in them what the relation between these two extremities is4 ,illustration3 ,fig4 #fg4 ,illustration3 ,fig4 #fh4 ,illustration3 ,fig4 #fi4 ,let him draw the i and k axes in his plane1 fig4 #fi4 ,the j axis then runs out and he has the accompanying figure4 ,in the first of these three squares1 fig4 #fi1 he can pick out by the rule the two points #bja1 #ajb--;,g1 and ,k4 ,here they occur in one plane and he can measure the distance between them4 ,in his first representation they occur at ,g and ,k in separate figures4 ,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 position4 ,joining them he would have the figure of a hexagon4 ,illustration3 ,fig4 #gj4 ,we may also notice that the plane being could make a representation of the whole cube simultaneously4 ,the three squares1 shown in perspective in fig4 #gj1 all lie in one plane1 and on these the plane being could pick out any selection of points just as well as on three separate squares4 ,he would obtain a hexagon by joining the points marked4 ,this hexagon1 as drawn1 is of the right shape1 but it would not be so if actual squares were used instead of perspective1 because the relation between the separate squares as they lie in the plane figure is not their real relation4 ,the figure1 however1 as thus constructed1 would give him an idea of the correct figure1 and he could determine it accurately by remembering that distances in each square were correct1 but in passing from one square to another their distance in the third dimension had to be taken into account4 ,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 each1 we must first draw a catalogue figure in which the whole assemblage is shown4 ,we can represent this assemblage of points by four solid figures4 ,the first giving all those positions which are at a distance ,o from our space in the fourth dimension1 the second showing all those that are at a distance #a1 and so on4 ,these figures will each be cubes4 ,the first two are drawn showing the front faces1 the second two the rear faces4 ,we will mark the points #j1 #a1 #b1 #c1 putting points at those distances along each of these axes1 and suppose all the points thus determined to be contained in solid models of which our drawings in fig4 #ga are representatives4 ,here we notice that as on the plane #j;i meant the whole line from which the distances in the i direction was measured1 and as in space #j;i means the whole plane from which distances in the i direction are measured1 so now #j;h means the whole space in which the first cube stands--measuring away from that space by a distance of one we come to the second cube represented4 ,illustration3 ,fig4 #ga4 ,now selecting according to the rule every one of one kind with every other of every other kind1 we must take1 for instance1 #c;i1 #b;j1 #ak1 #j;h4 ,this point is marked #cbaj at the lower star in the figure4 ,it is #c in the i direction1 #b in the j direction1 #a in the k direction1 #j in the h direction4 ,with #c;i we must also take #a;j1 #bk1 #j;h4 ,this point is shown by the second star in the cube #j;h4 ,illustration3 ,fig4 #gb4 ,in the first cube1 since all the points are #j;h points1 we can only have varieties in which i1 j1 k1 are accompanied by #c1 #b1 #a4 ,the points determined are marked off in the diagram fig4 #gb1 and lines are drawn joining the adjacent pairs in each figure1 the lines being dotted when they pass within the substance of the cube in the first two diagrams4 ,opposite each point1 on one side or the other of each cube1 is written its name4 ,it will be noticed that the figures are symmetrical right and left2 and right and left the first two numbers are simply interchanged4 ,now this being our selection of points1 what figure do they make when all are put together in their proper relative positions8 ,to determine this we must find the distance between corresponding corners of the separate hexagons4 ,illustration3 ,fig4 #gc4 ,to do this let us keep the axes i1 j1 in our space1 and draw h instead of k1 letting k run out in the fourth dimension1 fig4 #gc4 ,here we have four cubes again1 in the first of which all the points are #jk points2 that is1 points at a distance zero in the k direction from the space of the three dimensions ijh4 ,we have all the points selected before1 and some of the distances1 which in the last diagram led from figure to figure are shown here in the same figure1 and so capable of measurement4 ,take for instance the points #cabj to #cjba1 which in the first diagram 7fig4 #gb7 lie in the first and second figures4 ,their actual relation is shown in fig4 #gc in the cube marked #b,k1 where the points in question are marked with a 994 ,we see that the distance in question is the diagonal of a unit square4 ,in like manner we find that the distance between corresponding points of any two hexagonal figures is the diagonal of a unit square4 ,the total figure is now easily constructed4 ,an idea of it may be gained by drawing all the four cubes in the catalogue figure in one 7fig4 #gd74 ,these cubes are exact repetitions of one another1 so one drawing will serve as a representation of the whole series1 if we take care to remember where we are1 whether in a #j;h1 a #a;h1 a #b;h1 or a #c;h figure1 when we pick out the points required4 ,fig4 #gd is a representation of all the catalogue cubes put in one4 ,for the sake of clearness the front faces and the back faces of this cube are represented separately4 ,illustration3 ,fig4 #gd4 ,the figure determined by the selected points is shown below4 ,in putting the sections together some of the outlines in them disappear4 ,the line ,,tw for instance is not wanted4 ,we notice that ,,pqtw and ,,twrs are each the half of a hexagon4 ,now ,,qv and ,,vr lie in one straight line4 ,hence these two hexagons fit together1 forming one hexagon1 and the line ,,tw is only wanted when we consider a section of the whole figure1 we thus obtain the solid represented in the lower part of fig4 #gd4 ,equal repetitions of this figure1 called a tetrakaidecagon1 will fill up three-dimensional space4 ,to make the corresponding four-dimensional figure we have to take five axes mutually at right angles with five points on each4 ,a catalogue of the positions determined in five-dimensional space can be found thus4 ,take a cube with five points on each of its axes1 the fifth point is at a distance of four units of length from the first on any one of the axes4 ,and since the fourth dimension also stretches to a distance of four we shall need to represent the successive sets of points at distances #j1 #a1 #b1 #c1 #d1 in the fourth dimensions1 five cubes4 ,now all of these extend to no distance at all in the fifth dimension4 ,to represent what lies in the fifth dimension we shall have to draw1 starting from each of our cubes1 five similar cubes to represent the four steps on in the fifth dimension4 ,by this assemblage we get a catalogue of all the points shown in fig4 #ge1 in which ,l represents the fifth dimension4 ,illustration3 ,fig4 #ge4 ,now1 as we saw before1 there is nothing to prevent us from putting all the cubes representing the different stages in the fourth dimension in one figure1 if we take note when we look at it1 whether we consider it as a #j;h1 a #a;h1 a #b;h1 etc41 cube4 ,putting then the #j;h1 #a;h1 #b;h1 #c;h1 #d;h cubes of each row in one1 we have five cubes with the sides of each containing five positions1 the first of these five cubes represents the #jl points1 and has in it the i points from #j to #d1 the j points from #j to #d1 the k points from #j to #d1 while we have to specify with regard to any selection we make from it1 whether we regard it as a #j;h1 a #a;h1 a #b;h1 a #c;h1 or a #d;h figure4 ,in fig4 #gf each cube is represented by two drawings1 one of the front part1 the other of the rear part4 ,let then our five cubes be arranged before us and our selection be made according to the rule4 ,take the first figure in which all points are #jl points4 ,we cannot have #j with any other letter4 ,then1 keeping in the first figure1 which is that of the #jl positions1 take first of all that selection which always contains #a;h4 ,we suppose1 therefore1 that the cube is a #a;h cube1 and in it we take i1 j1 k in combination with #d1 #c1 #b according to the rule4 ,the figure we obtain is a hexagon1 as shown1 the one in front4 ,the points on the right hand have the same figures as those on the left1 with the first two numerals interchanged4 ,next keeping still to the #jl figure let us suppose that the cube before us represents a section at a distance of #b in the h direction4 ,let all the points in it be considered as #b;h points4 ,we then have a #jl1 #b;h region1 and have the sets ijk and #dca left over4 ,we must then pick out in accordance with our rule all such points as #d;i1 #c;j1 #ak4 ,these are shown in the figure and we find that we can draw them without confusion1 forming the second hexagon from the front4 ,going on in this way it will be seen that in each of the five figures a set of hexagons is picked out1 which put together form a three-space figure something like the tetrakaidecagon4 ,illustration3 ,fig4 #gf4 ,these separate figures are the successive stages in which the whole four-dimensional figure in which they cohere can be apprehended4 ,the first figure and the last are tetrakaidecagons4 ,these are two of the solid boundaries of the figure4 ,the other solid boundaries can be traced easily4 ,some of them are complete from one face in the figure to the corresponding face in the next1 as for instance the solid which extends from the hexagonal base of the first figure to the equal hexagonal base of the second figure4 ,this kind of boundary is a hexagonal prism4 ,the hexagonal prism also occurs in another sectional series1 as for instance1 in the square at the bottom of the first figure1 the oblong at the base of the second and the square at the base of the third figure4 ,other solid boundaries can be traced through four of the five sectional figures4 ,thus taking the hexagon at the top of the first figure we find in the next a hexagon also1 of which some alternate sides are elongated4 ,the top of the third figure is also a hexagon with the other set of alternate rules elongated1 and finally we come in the fourth figure to a regular hexagon4 ,these four sections are the sections of a tetrakaidecagon as can be recognised from the sections of this figure which we have had previously4 ,hence the boundaries are of two kinds1 hexagonal prisms and tetrakaidecagons4 ,these four-dimensional figures exactly fill four-dimensional space by equal repetitions of themselves4 ,,chapter ,,xi ,,nomenclature ,,and ,,analogies ,,preliminary ,,to ,,the ,,study ,,of ,,four-,,dimensional ,,figures ,in the following pages a method of designating different regions of space by a systematic colour scheme has been adopted4 ,the explanations have been given in such a manner as to involve no reference to models1 the diagrams will be found sufficient4 ,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 obtain4 ,if models are used the diagrams in ,chapters ,,xi4 and ,,xii4 will form a guide sufficient to indicate their use4 ,cubes of the colours designated by the diagrams should be picked out and used to reinforce the diagrams4 ,the reader1 in the following description1 should suppose that a board or wall stretches away from him1 against which the figures are placed4 ,illustration3 ,fig4 #gg4 ,take a square1 one of those shown in ,fig4 #gg and give it a neutral colour1 let this colour be called 8null10 and be such that it makes no appreciable difference to any colour with which it mixed4 ,if there is no such real colour let us imagine such a colour1 and assign to it the properties of the number zero1 which makes no difference in any number to which it is added4 ,above this square place a red square4 ,thus we symbolise the going up by adding red to null4 ,away from this null square place a yellow square1 and represent going away by adding yellow to null4 ,to complete the figure we need a fourth square4 ,colour this orange1 which is a mixture of red and yellow1 and so appropriately represents a going in a direction compounded of up and away4 ,we have thus a colour scheme which will serve to name the set of squares drawn4 ,we have two axes of colours--red and yellow--and they may occupy as in the figure the direction up and away1 or they may be turned about2 in any case they enable us to name the four squares drawn in their relation to one another4 ,now take1 in ,fig4 #gh1 nine squares1 and suppose that at the end of the going in any direction the colour started with repeats itself4 ,illustration3 ,fig4 #gh4 ,we obtain a square named as shown4 ,let us now1 in fig4 #gi1 suppose the number of squares to be increased1 keeping still to the principle of colouring already used4 ,here the nulls remain four in number4 ,there are three reds between the first null and the null above it1 three yellows between the first null and the null beyond it1 while the oranges increase in a double way4 ,illustration3 ,fig4 #gi4 ,suppose this process of enlarging the number of the squares to be indefinitely pursued and the total figure obtained to be reduced in size1 we should obtain a square of which the interior was all orange1 while the lines round it were red and yellow1 and merely the points null colour1 as in fig4 #hj4 ,thus all the points1 lines1 and the area would have a colour4 ,illustration3 ,fig4 #hj4 ,we can consider this scheme to originate thus3--,let a null point move in a yellow direction and trace out a yellow line and end in a null point4 ,then let the whole line thus traced move in a red direction4 ,the null points at the ends of the line will produce red lines1 and end in null points4 ,the yellow line will trace out a yellow and red1 or orange square4 ,now1 turning back to fig4 #gh1 we see that these two ways of naming1 the one we started with and the one we arrived at1 can be combined4 ,by its position in the group of four squares1 in fig4 #gg1 the null square has a relation to the yellow and to the red directions4 ,we can speak therefore of the red line of the null square without confusion1 meaning thereby the line ,,ab1 fig4 #ha1 which runs up from the initial null point ,a in the figure as drawn4 ,the yellow line of the null square is its lower horizontal line ,,ac as it is situated in the figure4 ,illustration3 ,fig4 #ha4 ,if we wish to denote the upper yellow line ,,bd1 fig4 #ha1 we can speak of it as the yellow line1 meaning the yellow line which is separated from the primary yellow line by the red movement4 ,in a similar way each of the other squares has null points1 red and yellow lines4 ,although the yellow square is all yellow1 its line ,,cd1 for instance1 can be referred to as its red line4 ,this nomenclature can be extended4 ,if the eight cubes drawn1 in fig4 #hb1 are put close together1 as on the right hand of the diagram1 they form a cube1 and in them1 as thus arranged1 a going up is represented by adding red to the zero1 or null colour1 a going away by adding yellow1 a going to the right by adding white4 ,white is used as a colour1 as a pigment1 which produces a colour change in the pigments with which it is mixed4 ,from whatever cube of the lower set we start1 a motion up brings us to a cube showing a change to red1 thus light yellow becomes light yellow red1 or light orange1 which is called ochre4 ,and going to the right from the null on the left we have a change involving the introduction of white1 while the yellow change runs from front to back4 ,there are three colour axes--the red1 the white1 the yellow--and these run in the position the cubes occupy in the drawing--up1 to the right1 away--but they could be turned about to occupy any positions in space4 ,illustration3 ,fig4 #hb4 ,illustration3 ,fig4 #hc4 ,the three layers4 ,we can conveniently represent a block of cubes by three sets of squares1 representing each the base of a cube4 ,thus the block1 fig4 #hc1 can be represented by the layers on the right4 ,here1 as in the case of the plane1 the initial colours repeat themselves at the end of the series4 ,proceeding now to increase the number of the cubes we obtain fig4 #hd1 in which the initial letters of the colours are given instead of their full names4 ,here we see that there are four null cubes as before1 but the series which spring from the initial corner will tend to become lines of cubes1 as also the sets of cubes parallel to them1 starting from other corners4 ,thus1 from the initial null springs a line of red cubes1 a line of white cubes1 and a line of yellow cubes4 ,if the number of the cubes is largely increased1 and the size of the whole cube is diminished1 we get a cube with null points1 and the edges coloured with these three colours4 ,illustration3 ,fig4 #hd4 ,the light yellow cubes increase in two ways1 forming ultimately a sheet of cubes1 and the same is true of the orange and pink sets4 ,hence1 ultimately the cube thus formed would have red1 white1 and yellow lines surrounding pink1 orange1 and light yellow faces4 ,the ochre cubes increase in three ways1 and hence ultimately the whole interior of the cube would be coloured ochre4 ,we have thus a nomenclature for the points1 lines1 faces1 and solid content of a cube1 and it can be named as exhibited in fig4 #he4 ,illustration3 ,fig4 #he4 ,we can consider the cube to be produced in the following way4 ,a null point moves in a direction to which we attach the colour indication yellow2 it generates a yellow line and ends in a null point4 ,the yellow line thus generated moves in a direction to which we give the colour indication red4 ,this lies up in the figure4 ,the yellow line traces out a yellow1 red1 or orange square1 and each of its null points trace out a red line1 and ends in a null point4 ,this orange square moves in a direction to which we attribute the colour indication white1 in this case the direction is the right4 ,the square traces out a cube coloured orange1 red1 or ochre1 the red lines trace out red to white or pink squares1 and the yellow lines trace out light yellow squares1 each line ending in a line of its own colour4 ,while the points each trace out a null white1 or white line to end in a null point4 ,now returning to the first block of eight cubes we can name each point1 line1 and square in them by reference to the colour scheme1 which they determine by their relation to each other4 ,thus1 in fig4 #hf1 the null cube touches the red cube by a light yellow square2 it touches the yellow cube by a pink square1 and touches the white cube by an orange square4 ,there are three axes to which the colours red1 yellow1 and white are assigned1 the faces of each cube are designated by taking these colours in pairs4 ,taking all the colours together we get a colour name for the solidity of a cube4 ,illustration3 ,fig4 #hf4 ,let us now ask ourselves how the cube could be presented to the plane being4 ,without going into the question of how he could have a real experience of it1 let us see how1 if we could turn it about and show it to him1 he1 under his limitations1 could get information about it4 ,if the cube were placed with its red and yellow axes against a plane1 that is resting against it by its orange face1 the plane being would observe a square surrounded by red and yellow lines1 and having null points4 ,see the dotted square1 fig4 #hg4 ,illustration3 ,fig4 #hg4 ,we could turn the cube about the red line so that a different face comes into juxtaposition with the plane4 ,suppose the cube turned about the red line4 ,as it is turning from its first position all of it except the red line leaves the plane--goes absolutely out of the range of the plane being's apprehension4 ,but when the yellow line points straight out from the plane then the pink face comes into contact with it4 ,thus the same red line remaining as he saw it at first1 now towards him comes a face surrounded by white and red lines4 ,if we call the direction to the right the unknown direction1 then the line he saw before1 the yellow line1 goes out into this unknown direction1 and the line which before went into the unknown direction1 comes in4 ,it comes in in the opposite direction to that in which the yellow line ran before2 the interior of the face now against the plane is pink4 ,it is a property of two lines at right angles that1 if one turns out of a given direction and stands at right angles to it1 then the other of the two lines comes in1 but runs the opposite way in that given direction1 as in fig4 #hh4 ,illustration3 ,fig4 #hh4 ,now these two presentations of the cube would seem1 to the plane creature like perfectly different material bodies1 with only that line in common in which they both meet4 ,again our cube can be turned about the yellow line4 ,in this case the yellow square would disappear as before1 but a new square would come into the plane after the cube had rotated by an angle of #ij about this line4 ,the bottom square of the cube would come in thus in figure #hi4 ,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 plane4 ,here1 as before1 the red line going out into the unknown dimension1 the white line which before ran in the unknown dimension would come in downwards in the opposite sense to that in which the red line ran before4 ,illustration3 ,fig4 #hi4 ,now if we use i1 j1 k1 for the three space directions1 i left to right1 j from near away1 k from below up2 then1 using the colour names for the axes1 we have that first of all white runs i1 yellow runs j1 red runs k2 then after the first turning round the k axis1 white runs negative j1 yellow runs i1 red runs k2 thus we have the table3-- i j k #ast position white yellow red #bnd position yellow white-- red #crd position red yellow white-- ,here white with a negative sign after it in the column under j means that white runs in the negative sense of the j direction4 ,we may express the fact in the following way3-- ,in the plane there is room for two axes while the body has three4 ,therefore in the plane we can represent any two4 ,if we want to keep the axis that goes in the unknown dimension always running in the positive sense1 then the axis which originally ran in the unknown dimension 7the white axis7 must come in in the negative sense of that axis which goes out of the plane into the unknown dimension4 ,it is obvious that the unknown direction1 the direction in which the white line runs at first1 is quite distinct from any direction which the plane creature knows4 ,the white line may come in towards him1 or running down4 ,if he is looking at a square1 which is the face of a cube 7looking at it by a line71 then any one of the bounding lines remaining unmoved1 another face of the cube may come in1 any one of the faces1 namely1 which have the white line in them4 ,and the white line comes sometimes in one of the space directions he knows1 sometimes in another4 ,now this turning which leaves a line unchanged is something quite unlike any turning he knows in the plane4 ,in the plane a figure turns round a point4 ,the square can turn round the null point in his plane1 and the red and yellow lines change places1 only of course1 as with every rotation of lines at right angles1 if red goes where yellow went1 yellow comes in negative of red's old direction4 ,this turning1 as the plane creature conceives it1 we should call turning about an axis perpendicular to the plane4 ,what he calls turning about the null point we call turning about the white line as it stands out from his plane4 ,there is no such thing as turning about a point1 there is always an axis1 and really much more turns than the plane being is aware of4 ,taking now a different point of view1 let us suppose the cubes to be presented to the plane being by being passed transverse to his plane4 ,let us suppose the sheet of matter over which the plane being and all objects in his world slide1 to be of such a nature that objects can pass through it without breaking it4 ,let us suppose it to be of the same nature as the film of a soap bubble1 so that it closes around objects pushed through it1 and1 however the object alters its shape as it passes through it1 let us suppose this film to run up to the contour of the object in every part1 maintaining its plane surface unbroken4 ,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 it4 ,illustration3 ,fig4 #ij4 ,fig4 #ij represents a cube passing through a plane film4 ,the plane being now comes into contact with a very thin slice of the cube somewhere between the left and right hand faces4 ,this very thin slice he thinks of as having no thickness1 and consequently his idea of it is what we call a section4 ,it is bounded by him by pink lines front and back1 coming from the part of the pink face he is in contact with1 and above and below1 by light yellow lines4 ,its corners are not null-coloured points1 but white points1 and its interior is ochre1 the colour of the interior of the cube4 ,if now we suppose the cube to be an inch in each dimension1 and to pass across1 from right to left1 through the plane1 then we should explain the appearances presented to the plane being by saying3 ,first of all you have the face of a cube1 this lasts only a moment2 then you have a figure of the same shape but differently coloured4 ,this1 which appears not to move to you in any direction which you know of1 is really moving transverse to your plane world4 ,its appearance is unaltered1 but each moment it is something different--a section further on1 in the white1 the unknown dimension4 ,finally1 at the end of the minute1 a face comes in exactly like the face you first saw4 ,this finishes up the cube--it is the further face in the unknown dimension4 ,the white line1 which extends in length just like the red or the yellow1 you do not see as extensive2 you apprehend it simply as an enduring white point4 ,the null point1 under the condition of movement of the cube1 vanishes in a moment1 the lasting white point is really your apprehension of a white line1 running in the unknown dimension4 ,in the same way the red line of the face by which the cube is first in contact with the plane lasts only a moment1 it is succeeded by the pink line1 and this pink line lasts for the inside of a minute4 ,this lasting pink line in your apprehension of a surface1 which extends in two dimensions just like the orange surface extends1 as you know it1 when the cube is at rest4 ,but the plane creature might answer1 8,this orange object is substance1 solid substance1 bounded completely and on every side40 ,here1 of course1 the difficulty comes in4 ,his solid is our surface--his notion of a solid is our notion of an abstract surface with no thickness at all4 ,we should have to explain to him that1 from every point of what he called a solid1 a new dimension runs away4 ,from every point a line can be drawn in a direction unknown to him1 and there is a solidity of a kind greater than that which he knows4 ,this solidity can only be realised by him by his supposing an unknown direction1 by motion in which what he conceives to be solid matter instantly disappears4 ,the higher solid1 however1 which extends in this dimension as well as in those which he knows1 lasts when a motion of that kind takes place1 different sections of it come consecutively in the plane of his apprehension1 and take the place of the solid which he at first conceives to be all4 ,thus1 the higher solid--our solid in contradistinction to his area solid1 his two-dimensional solid1 must be conceived by him as something which has duration in it1 under circumstances in which his matter disappears out of his world4 ,we may put the matter thus1 using the conception of motion4 ,a null point moving in a direction away generates a yellow line1 and the yellow line ends in a null point4 ,we suppose1 that is1 a point to move and mark out the products of this motion in such a manner4 ,now suppose this whole line as thus produced to move in an upward direction2 it traces out the two-dimensional solid1 and the plane being gets an orange square4 ,the null point moves in a red line and ends in a null point1 the yellow line moves and generates an orange square and ends in a yellow line1 the farther null point generates a red line and ends in a null point4 ,thus1 by movement in two successive directions known to him1 he can imagine his two-dimensional solid produced with all its boundaries4 ,now we tell him3 8,this whole two-dimensional solid can move in a third or unknown dimension to you4 ,the null point moving in this dimension out of your world generates a white line and ends in a null point4 ,the yellow line moving generates a light yellow two-dimensional solid and ends in a yellow line1 and this two-dimensional solid1 lying end on to your plane world1 is bounded on the far side by the other yellow line4 ,in the same way each of the lines surrounding your square traces out an area1 just like the orange area you know4 ,but there is something new produced1 something which you had no idea of before2 it is that which is produced by the movement of the orange square4 ,that1 than which you can imagine nothing more solid1 itself moves in a direction open to it and produces a three-dimensional solid4 ,using the addition of white to symbolise the products of this motion this new kind of solid will be light orange or ochre1 and it will be bounded on the far side by the final position of the orange square which traced it out1 and this final position we suppose to be coloured like the square in its first position1 orange with yellow and red boundaries and null corners40 ,this product of movement1 which it is so easy for us to describe1 would be difficult for him to conceive4 ,but this difficulty is connected rather with its totality than with any particular part of it4 ,any line1 or plane of this1 to him higher1 solid we could show to him1 and put in his sensible world4 ,we have already seen how the pink square could be put in his world by a turning of the cube about the red line4 ,and any section which we can conceive made of the cube could be exhibited to him4 ,you have simply to turn the cube and push it through1 so that the plane of his existence is the plane which cuts out the given section of the cube1 then the section would appear to him as a solid4 ,in his world he would see the contour1 get to any part of it by digging down into it4 ,,the ,,process ,,by ,,which ,a ,,plane ,,being ,,would ,,gain ,a ,,notion ,,of ,a ,,solid4 ,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 method1 the discipline1 by which he would acquire a working familiarity with our space existence4 ,the process begins with an adequate realisation of a simple solid figure4 ,for this purpose we will suppose eight cubes forming a larger cube1 and first we will suppose each cube to be coloured throughout uniformly4 ,let the cubes in fig4 #ia be the eight making a larger cube4 ,illustration3 ,fig4 #ia4 ,now1 although each cube is supposed to be coloured entirely through with the colour1 the name of which is written on it1 still we can speak of the faces1 edges1 and corners of each cube as if the colour scheme we have investigated held for it4 ,thus1 on the null cube we can speak of a null point1 a red line1 a white line1 a pink face1 and so on4 ,these colour designations are shown on ,no4 #a of the views of the tesseract in the plate4 ,here these colour names are used simply in their geometrical significance4 ,they denote what the particular line1 etc41 referred to would have as its colour1 if in reference to the particular cube the colour scheme described previously were carried out4 ,if such a block of cubes were put against the plane and then passed through it from right to left1 at the rate of an inch a minute1 each cube being an inch each way1 the plane being would have the following appearances3-- ,first of all1 four squares null1 yellow1 red1 orange1 lasting each a minute2 and secondly1 taking the exact places of these four squares1 four others1 coloured white1 light yellow1 pink1 ochre4 ,thus1 to make a catalogue of the solid body1 he would have to put side by side in his world two sets of four squares each1 as in fig4 #ib4 ,the first are supposed to last a minute1 and then the others to come in in place of them1 and also last a minute4 ,illustration3 ,fig4 #ib4 ,in speaking of them he would have to denote what part of the respective cube each square represents4 ,thus1 at the beginning he would have null cube orange face1 and after the motion had begun he would have null cube ochre section4 ,as he could get the same coloured section whichever way the cube passed through1 it would be best for him to call this section white section1 meaning that it is transverse to the white axis4 ,these colour-names1 of course1 are merely used as names1 and do not imply in this case that the object is really coloured4 ,finally1 after a minute1 as the first cube was passing beyond his plane he would have null cube orange face again4 ,the same names will hold for each of the other cubes1 describing what face or section of them the plane being has before him2 and the second wall of cubes will come on1 continue1 and go out in the same manner4 ,in the area he thus has he can represent any movement which we carry out in the cubes1 as long as it does not involve a motion in the direction of the white axis4 ,the relation of parts that succeed one another in the direction of the white axis is realised by him as a consecution of states4 ,now1 his means of developing his space apprehension lies in this1 that that which is represented as a time sequence in one position of the cubes1 can become a real co-existence1 if something that has a real co-existence becomes a time sequence4 ,we must suppose the cubes turned round each of the axes1 the red line1 and the yellow line1 then something1 which was given as time before1 will now be given as the plane creature's space2 something1 which was given as space before1 will now be given as a time series as the cube is passed through the plane4 ,the three positions in which the cubes must be studied are the one given above and the two following ones4 ,in each case the original null point which was nearest to us at first is marked by an asterisk4 ,in figs4 #ic and #id the point marked with a star is the same in the cubes and in the plane view4 ,illustration3 ,fig4 #ic4 ,the cube swung round the red line1 so that the white line points towards us4 ,in fig4 #ic the cube is swung round the red line so as to point towards us1 and consequently the pink face comes next to the plane4 ,as it passes through there are two varieties of appearance designated by the figures #a and #b in the plane4 ,these appearances are named in the figure1 and are determined by the order in which the cubes come in the motion of the whole block through the plane4 ,with regard to these squares severally1 however1 different names must be used1 determined by their relations in the block4 ,thus1 in fig4 #ic1 when the cube first rests against the plane the null cube is in contact by its pink face2 as the block passes through we get an ochre section of the null cube1 but this is better called a yellow section1 as it is made by a plane perpendicular to the yellow line4 ,when the null cube has passed through the plane1 as it is leaving it1 we get again a pink face4 ,illustration3 ,fig4 #id4 ,the cube swung round yellow line1 with red line running from left to right1 and white line running down4 ,the same series of changes take place with the cube appearances which follow on those of the null cube4 ,in this motion the yellow cube follows on the null cube1 and the square marked yellow in #b in the plane will be first 8yellow pink face10 then 8yellow yellow section10 then 8yellow pink face40 ,in fig4 #id1 in which the cube is turned about the yellow line1 we have a certain difficulty1 for the plane being will find that the position his squares are to be placed in will lie below that which they first occupied4 ,they will come where the support was on which he stood his first set of squares4 ,he will get over this difficulty by moving his support4 ,then1 since the cubes come upon his plane by the light yellow face1 he will have1 taking the null cube as before for an example1 null1 light yellow face2 null1 red section1 because the section is perpendicular to the red line2 and finally1 as the null cube leaves the plane1 null1 light yellow face4 ,then1 in this case red following on null1 he will have the same series of views of the red as he had of the null cube4 ,illustration3 ,fig4 #ie4 ,there is another set of considerations which we will briefly allude to4 ,suppose there is a hollow cube1 and a string is stretched across it from null to null1 r1 y1 wh1 as we may call the far diagonal point1 how will this string appear to the plane being as the cube moves transverse to his plane8 ,let us represent the cube as a number of sections1 say #e1 corresponding to #d equal divisions made along the white line perpendicular to it4 ,we number these sections #j1 #a1 #b1 #c1 #d1 corresponding to the distances along the white line at which they are taken1 and imagine each section to come in successively1 taking the place of the preceding one4 ,these sections appear to the plane being1 counting from the first1 to exactly coincide each with the preceding one4 ,but the section of the string occupies a different place in each to that which it does in the preceding section4 ,the section of the string appears in the position marked by the dots4 ,hence the slant of the string appears as a motion in the frame work marked out by the cube sides4 ,if we suppose the motion of the cube not to be recognised1 then the string appears to the plane being as a moving point4 ,hence extension on the unknown dimension appears as duration4 ,extension sloping in the unknown direction appears as continuous movement4 ,,chapter ,,xii ,,the ,,simplest ,,four-,,dimensional ,,solid ,a plane being1 in learning to apprehend solid existence1 must first of all realise that there is a sense of direction altogether wanting to him4 ,that which we call right and left does not exist in his perception4 ,he must assume a movement in a direction1 and a distinction of positive and negative in that direction1 which has no reality corresponding to it in the movements he can make4 ,this direction1 this new dimension1 he can only make sensible to himself by bringing in time1 and supposing that changes1 which take place in time1 are due to objects of a definite configuration in three dimensions passing transverse to his plane1 and the different sections of it being apprehended as changes of one and the same plane figure4 ,he must also acquire a distinct notion about his plane world1 he must no longer believe that it is the all of space1 but that space extends on both sides of it4 ,in order1 then1 to prevent his moving off in this unknown direction1 he must assume a sheet1 an extended solid sheet1 in two dimensions1 against which1 in contact with which1 all his movements take place4 ,when we come to think of a four-dimensional solid1 what are the corresponding assumptions which we must make8 ,we must suppose a sense which we have not1 a sense of direction wanting in us1 something which a being in a four-dimensional world has1 and which we have not4 ,it is a sense corresponding to a new space direction1 a direction which extends positively and negatively from every point of our space1 and which goes right away from any space direction we know of4 ,the perpendicular to a plane is perpendicular1 not only to two lines in it1 but to every line1 and so we must conceive this fourth dimension as running perpendicularly to each and every line we can draw in our space4 ,and as the plane being had to suppose something which prevented his moving off in the third1 the unknown dimension to him1 so we have to suppose something which prevents us moving off in the direction unknown to us4 ,this something1 since we must be in contact with it in every one of our movements1 must not be a plane surface1 but a solid2 it must be a solid1 which in every one of our movements we are against1 not in4 ,it must be supposed as stretching out in every space dimension that we know2 but we are not in it1 we are against it1 we are next to it1 in the fourth dimension4 ,that is1 as the plane being conceives himself as having a very small thickness in the third dimension1 of which he is not aware in his sense experience1 so we must suppose ourselves as having a very small thickness in the fourth dimension1 and1 being thus four-dimensional beings1 to be prevented from realising that we are such beings by a constraint which keeps us always in contact with a vast solid sheet1 which stretches on in every direction4 ,we are against that sheet1 so that1 if we had the power of four-dimensional movement1 we should either go away from it or through it2 all our space movements as we know them being such that1 performing them1 we keep in contact with this solid sheet4 ,now consider the exposition a plane being would make for himself as to the question of the enclosure of a square1 and of a cube4 ,he would say the square ,a1 in ,fig4 #if1 is completely enclosed by the four squares1 ,a far1 ,a near1 ,a above1 ,a below1 or as they are written ,an1 ,af1 ,aa1 ,ab4 ,illustration3 ,fig4 #if4 ,if now he conceives the square ,a to move in the1 to him1 unknown dimension it will trace out a cube1 and the bounding squares will form cubes4 ,will these completely surround the cube generated by ,a8 ,no2 there will be two faces of the cube made by ,a left uncovered2 the first1 that face which coincides with the square ,a in its first position2 the next1 that which coincides with the square ,a in its final position4 ,against these two faces cubes must be placed in order to completely enclose the cube ,a4 ,these may be called the cubes left and right or ,al and ,ar4 ,thus each of the enclosing squares of the square ,a becomes a cube and two more cubes are wanted to enclose the cube formed by the movement of ,a in the third dimension4 ,illustration3 ,fig4 #ig4 ,the plane being could not see the square ,a with the squares ,an1 ,af1 etc41 placed about it1 because they completely hide it from view2 and so we1 in the analogous case in our three-dimensional world1 cannot see a cube ,a surrounded by six other cubes4 ,these cubes we will call ,a near ,an1 ,a far ,af1 ,a above ,aa1 ,a below ,ab1 ,a left ,al1 ,a right ,ar1 shown in fig4 #ig4 ,if now the cube ,a moves in the fourth dimension right out of space1 it traces out a higher cube--a tesseract1 as it may be called4 ,each of the six surrounding cubes carried on in the same motion will make a tesseract also1 and these will be grouped around the tesseract formed by ,a4 ,but will they enclose it completely8 ,all the cubes ,an1 ,af1 etc41 lie in our space4 ,but there is nothing between the cube ,a and that solid sheet in contact with which every particle of matter is4 ,when the cube ,a moves in the fourth direction it starts from its position1 say ,ak1 and ends in a final position ,an 7using the words 8ana0 and 8kata0 for up and down in the fourth dimension74 ,now the movement in this fourth dimension is not bounded by any of the cubes ,an1 ,af1 nor by what they form when thus moved4 ,the tesseract which ,a becomes is bounded in the positive and negative ways in this new direction by the first position of ,a and the last position of ,a4 ,or1 if we ask how many tesseracts lie around the tesseract which ,a forms1 there are eight1 of which one meets it by the cube ,a1 and another meets it by a cube like ,a at the end of its motion4 ,we come here to a very curious thing4 ,the whole solid cube ,a is to be looked on merely as a boundary of the tesseract4 ,yet this is exactly analogous to what the plane being would come to in his study of the solid world4 ,the square ,a 7fig4 #if71 which the plane being looks on as a solid existence in his plane world1 is merely the boundary of the cube which he supposes generated by its motion4 ,the fact is that we have to recognise that1 if there is another dimension of space1 our present idea of a solid body1 as one which has three dimensions only1 does not correspond to anything real1 but is the abstract idea of a three-dimensional boundary limiting a four-dimensional solid1 which a four-dimensional being would form4 ,the plane being's thought of a square is not the thought of what we should call a possibly existing real square1 but the thought of an abstract boundary1 the face of a cube4 ,let us now take our eight coloured cubes1 which form a cube in space1 and ask what additions we must make to them to represent the simplest collection of four-dimensional bodies--namely1 a group of them of the same extent in every direction4 ,in plane space we have four squares4 ,in solid space we have eight cubes4 ,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 space1 and these bodies we call tesseracts4 ,given then the null1 white1 red1 yellow cubes1 and those which make up the block1 we notice that we represent perfectly well the extension in three directions 7fig4 #ih74 ,from the null point of the null cube1 travelling one inch1 we come to the white cube2 travelling one inch away we come to the yellow cube2 travelling one inch up we come to the red cube4 ,now1 if there is a fourth dimension1 then travelling from the same null point for one inch in that direction1 we must come to the body lying beyond the null region4 ,illustration3 ,fig4 #ih4 ,i say null region1 not cube2 for with the introduction of the fourth dimension each of our cubes must become something different from cubes4 ,if they are to have existence in the fourth dimension1 they must be 8filled up from0 in this fourth dimension4 ,now we will assume that as we get a transference from null to white going in one way1 from null to yellow going in another1 so going from null in the fourth direction we have a transference from null to blue1 using thus the colours white1 yellow1 red1 blue1 to denote transferences in each of the four directions--right1 away1 up1 unknown or fourth dimension4 ,illustration3 ,fig4 #ii4 ,a plane being's representation of a block of eight cubes by two sets of four squares4 ,hence1 as the plane being must represent the solid regions1 he would come to by going right1 as four squares lying in some position in his plane1 arbitrarily chosen1 side by side with his original four squares1 so we must represent those eight four-dimensional regions1 which we should come to by going in the fourth dimension from each of our eight cubes1 by eight cubes placed in some arbitrary position relative to our first eight cubes4 ,illustration3 ,fig4 #ajj4 ,our representation of a block of sixteen tesseracts by two blocks of eight cubes4#c #c ,the eight cubes used here in #b can be found in the second of the model blocks4 ,they can be taken out and used4 ,hence1 of the two sets of eight cubes1 each one will serve us as a representation of one of the sixteen tesseracts which form one single block in four-dimensional space4 ,each cube1 as we have it1 is a tray1 as it were1 against which the real four-dimensional figure rests--just as each of the squares which the plane being has is a tray1 so to speak1 against which the cube it represents could rest4 ,if we suppose the cubes to be one inch each way1 then the original eight cubes will give eight tesseracts of the same colours1 or the cubes1 extending each one inch in the fourth dimension4 ,but after these there come1 going on in the fourth dimension1 eight other bodies1 eight other tesseracts4 ,these must be there1 if we suppose the four-dimensional body we make up to have two divisions1 one inch each in each of four directions4 ,the colour we choose to designate the transference to this second region in the fourth dimension is blue4 ,thus1 starting from the null cube and going in the fourth dimension1 we first go through one inch of the null tesseract1 then we come to a blue cube1 which is the beginning of a blue tesseract4 ,this blue tesseract stretches one inch farther on in the fourth dimension4 ,thus1 beyond each of the eight tesseracts1 which are of the same colour as the cubes which are their bases1 lie eight tesseracts whose colours are derived from the colours of the first eight by adding blue4 ,thus-- ,null gives blue ,yellow 0 green ,red 0 purple ,orange 0 brown ,white 0 light blue ,pink 0 light purple ,light yellow 0 light green ,ochre 0 light brown ,the addition of blue to yellow gives green--this is a natural supposition to make4 ,it is also natural to suppose that blue added to red makes purple4 ,orange and blue can be made to give a brown1 by using certain shades and proportions4 ,and ochre and blue can be made to give a light brown4 ,but the scheme of colours is merely used for getting a definite and realisable set of names and distinctions visible to the eye4 ,their naturalness is apparent to any one in the habit of using colours1 and may be assumed to be justifiable1 as the sole purpose is to devise a set of names which are easy to remember1 and which will give us a set of colours by which diagrams may be made easy of comprehension4 ,no scientific classification of colours has been attempted4 ,starting1 then1 with these sixteen colour names1 we have a catalogue of the sixteen tesseracts1 which form a four-dimensional block analogous to the cubic block4 ,but the cube which we can put in space and look at is not one of the constituent tesseracts2 it is merely the beginning1 the solid face1 the side1 the aspect1 of a tesseract4 ,we will now proceed to derive a name for each region1 point1 edge1 plane face1 solid and a face of the tesseract4 ,the system will be clear1 if we look at a representation in the plane of a tesseract with three1 and one with four divisions in its side4 ,the tesseract made up of three tesseracts each way corresponds to the cube made up of three cubes each way1 and will give us a complete nomenclature4 ,in this diagram1 fig4 #aja1 #a represents a cube of #bg cubes1 each of which is the beginning of a tesseract4 ,these cubes are represented simply by their lowest squares1 the solid content must be understood4 #b represents the #bg cubes which are the beginnings of the #bg tesseracts one inch on in the fourth dimension4 ,these tesseracts are represented as a block of cubes put side by side with the first block1 but in their proper positions they could not be in space with the first set4 #c represents #bg cubes 7forming a larger cube7 which are the beginnings of the tesseracts1 which begin two inches in the fourth direction from our space and continue another inch4 ,illustration3 ,fig4 #aja4 ,illustration3 ,fig4 #ajbd #d ,the coloured plate1 figs4 #a1 #b1 #c1 shows these relations more conspicuously4 ,in fig4 #ajb1 we have the representation of a block of #d #d #d #d or #bef tesseracts4 ,they are given in four consecutive sections1 each supposed to be taken one inch apart in the fourth dimension1 and so giving four blocks of cubes1 #fd in each block4 ,here we see1 comparing it with the figure of #ha tesseracts1 that the number of the different regions show a different tendency of increase4 ,by taking five blocks of five divisions each way this would become even more clear4 ,we see1 fig4 #ajb1 that starting from the point at any corner1 the white coloured regions only extend out in a line4 ,the same is true for the yellow1 red1 and blue4 ,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 drawing1 but in portions which come in in different cubes4 ,the portions which lie next to one another in the fourth dimension must always be represented so1 when we have a three-dimensional representation4 ,again1 those regions such as the pink one1 go on increasing in two dimensions4 ,about the pink region this is seen without going out of the cube itself1 the pink regions increase in length and height1 but in no other dimension4 ,in examining these regions it is sufficient to take one as a sample4 ,the purple increases in the same manner1 for it comes in in a succession from below to above in block #b1 and in a succession from block to block in #b and #c4 ,now1 a succession from below to above represents a continuous extension upwards1 and a succession from block to block represents a continuous extension in the fourth dimension4 ,thus the purple regions increase in two dimensions1 the upward and the fourth1 so when we take a very great many divisions1 and let each become very small1 the purple region forms a two-dimensional extension4 ,in the same way1 looking at the regions marked l4 b4 or light blue1 which starts nearest a corner1 we see that the tesseracts occupying it increase in length from left to right1 forming a line1 and that there are as many lines of light blue tesseracts as there are sections between the first and last section4 ,hence the light blue tesseracts increase in number in two ways--in the right and left1 and in the fourth dimension4 ,they ultimately form what we may call a plane surface4 ,now all those regions which contain a mixture of two simple colours1 white1 yellow1 red1 blue1 increase in two ways4 ,on the other hand1 those which contain a mixture of three colours increase in three ways4 ,take1 for instance1 the ochre region2 this has three colours1 white1 yellow1 red2 and in the cube itself it increases in three ways4 ,now regard the orange region2 if we add blue to this we get a brown4 ,the region of the brown tesseracts extends in two ways on the left of the second block1 ,no4 #b in the figure4 ,it extends also from left to right in succession from one section to another1 from section #b to section #c in our figure4 ,hence the brown tesseracts increase in number in three dimensions upwards1 to and fro1 fourth dimension4 ,hence they form a cubic1 a three-dimensional region2 this region extends up and down1 near and far1 and in the fourth direction1 but is thin in the direction from left to right4 ,it is a cube which1 when the complete tesseract is represented in our space1 appears as a series of faces on the successive cubic sections of the tesseract4 ,compare fig4 #ajc in which the middle block1 #b1 stands as representing a great number of sections intermediate between #a and #c4 ,in a similar way from the pink region by addition of blue we have the light purple region1 which can be seen to increase in three ways as the number of divisions becomes greater4 ,the three ways in which this region of tesseracts extends is up and down1 right and left1 fourth dimension4 ,finally1 therefore1 it forms a cubic mass of very small tesseracts1 and when the tesseract is given in space sections it appears on the faces containing the upward and the right and left dimensions4 ,we get then altogether1 as three-dimensional regions1 ochre1 brown1 light purple1 light green4 ,finally1 there is the region which corresponds to a mixture of all the colours2 there is only one region such as this4 ,it is the one that springs from ochre by the addition of blue--this colour we call light brown4 ,looking at the light brown region we see that it increases in four ways4 ,hence1 the tesseracts of which it is composed increase in number in each of four dimensions1 and the shape they form does not remain thin in any of the four dimensions4 ,consequently this region becomes the solid content of the block of tesseracts1 itself2 it is the real four-dimensional solid4 ,all the other regions are then boundaries of this light brown region4 ,if we suppose the process of increasing the number of tesseracts and diminishing their size carried on indefinitely1 then the light brown coloured tesseracts become the whole interior mass1 the three-coloured tesseracts become three-dimensional boundaries1 thin in one dimension1 and form the ochre1 the brown1 the light purple1 the light green4 ,the two-coloured tesseracts become two-dimensional boundaries1 thin in two dimensions1 e4g41 the pink1 the green1 the purple1 the orange1 the light blue1 the light yellow4 ,the one-coloured tesseracts become bounding lines1 thin in three dimensions1 and the null points become bounding corners1 thin in four dimensions4 ,from these thin real boundaries we can pass in thought to the abstractions--points1 lines1 faces1 solids--bounding the four-dimensional solid1 which in this case is light brown coloured1 and under this supposition the light brown coloured region is the only real one1 is the only one which is not an abstraction4 ,it should be observed that1 in taking a square as the representation of a cube on a plane1 we only represent one face1 or the section between two faces4 ,the squares1 as drawn by a plane being1 are not the cubes themselves1 but represent the faces or the sections of a cube4 ,thus in the plane being's diagram a cube of twenty-seven cubes 8null0 represents a cube1 but is really1 in the normal position1 the orange square of a null cube1 and may be called null1 orange square4 ,a plane being would save himself confusion if he named his representative squares1 not by using the names of the cubes simply1 but by adding to the names of the cubes a word to show what part of a cube his representative square was4 ,thus a cube null standing against his plane touches it by null orange face1 passing through his plane it has in the plane a square as trace1 which is null white section1 if we use the phrase white section to mean a section drawn perpendicular to the white line4 ,in the same way the cubes which we take as representative of the tesseract are not the tesseract itself1 but definite faces or sections of it4 ,in the preceding figures we should say then1 not null1 but 8null tesseract ochre cube10 because the cube we actually have is the one determined by the three axes1 white1 red1 yellow4 ,there is another way in which we can regard the colour nomenclature of the boundaries of a tesseract4 ,consider a null point to move tracing out a white line one inch in length1 and terminating in a null point1 see fig4 #ajc or in the coloured plate4 ,then consider this white line with its terminal points itself to move in a second dimension1 each of the points traces out a line1 the line itself traces out an area1 and gives two lines as well1 its initial and its final position4 ,thus1 if we call 8a region0 any element of the figure1 such as a point1 or a line1 etc41 every 8region0 in moving traces out a new kind of region1 8a higher region10 and gives two regions of its own kind1 an initial and a final position4 ,the 8higher region0 means a region with another dimension in it4 ,now the square can move and generate a cube4 ,the square light yellow moves and traces out the mass of the cube4 ,letting the addition of red denote the region made by the motion in the upward direction we get an ochre solid4 ,the light yellow face in its initial and terminal positions give the two square boundaries of the cube above and below4 ,then each of the four lines of the light yellow square--white1 yellow1 and the white1 yellow opposite them--trace out a bounding square4 ,so there are in all six bounding squares1 four of these squares being designated in colour by adding red to the colour of the generating lines4 ,finally1 each point moving in the up direction gives rise to a line coloured null red1 or red1 and then there are the initial and terminal positions of the points giving eight points4 ,the number of the lines is evidently twelve1 for the four lines of this light yellow square give four lines in their initial1 four lines in their final position1 while the four points trace out four lines1 that is altogether twelve lines4 ,now the squares are each of them separate boundaries of the cube1 while the lines belong1 each of them1 to two squares1 thus the red line is that which is common to the orange and pink squares4 ,now suppose that there is a direction1 the fourth dimension1 which is perpendicular alike to every one of the space dimensions already used--a dimension perpendicular1 for instance1 to up and to right hand1 so that the pink square moving in this direction traces out a cube4 ,a dimension1 moreover1 perpendicular to the up and away directions1 so that the orange square moving in this direction also traces out a cube1 and the light yellow square1 too1 moving in this direction traces out a cube4 ,under this supposition1 the whole cube moving in the unknown dimension1 traces out something new--a new kind of volume1 a higher volume4 ,this higher volume is a four-dimensional volume1 and we designate it in colour by adding blue to the colour of that which by moving generates it4 ,it is generated by the motion of the ochre solid1 and hence it is of the colour we call light brown 7white1 yellow1 red1 blue1 mixed together74 ,it is represented by a number of sections like #b in fig4 #ajc4 ,now this light brown higher solid has for boundaries3 first1 the ochre cube in its initial position1 second1 the same cube in its final position1 #a and #c1 fig4 #ajc4 ,each of the squares which bound the cube1 moreover1 by movement in this new direction traces out a cube1 so we have from the front pink faces of the cube1 third1 a pink blue or light purple cube1 shown as a light purple face on cube #b in fig4 #ajc1 this cube standing for any number of intermediate sections2 fourth1 a similar cube from the opposite pink face2 fifth1 a cube traced out by the orange face--this is coloured brown and is represented by the brown face of the section cube in fig4 #ajc2 sixth1 a corresponding brown cube on the right hand2 seventh1 a cube starting from the light yellow square below2 the unknown dimension is at right angles to this also4 ,this cube is coloured light yellow and blue or light green2 and1 finally1 eighth1 a corresponding cube from the upper light yellow face1 shown as the light green square at the top of the section cube4 ,the tesseract has thus eight cubic boundaries4 ,these completely enclose it1 so that it would be invisible to a four-dimensional being4 ,now1 as to the other boundaries1 just as the cube has squares1 lines1 points1 as boundaries1 so the tesseract has cubes1 squares1 lines1 points1 as boundaries4 ,the number of squares is found thus--round the cube are six squares1 these will give six squares in their initial and six in their final positions4 ,then each of the twelve lines of the cube trace out a square in the motion in the fourth dimension4 ,hence there will be altogether #ab #ab #bd squares4 ,if we look at any one of these squares we see that it is the meeting surface of two of the cubic sides4 ,thus1 the red line by its movement in the fourth dimension1 traces out a purple square--this is common to two cubes1 one of which is traced out by the pink square moving in the fourth dimension1 and the other is traced out by the orange square moving in the same way4 ,to take another square1 the light yellow one1 this is common to the ochre cube and the light green cube4 ,the ochre cube comes from the light yellow square by moving it in the up direction1 the light green cube is made from the light yellow square by moving it in the fourth dimension4 ,the number of lines is thirty-two1 for the twelve lines of the cube give twelve lines of the tesseract in their initial position1 and twelve in their final position1 making twenty-four1 while each of the eight points traces out a line1 thus forming thirty-two lines altogether4 ,the lines are each of them common to three cubes1 or to three square faces2 take1 for instance1 the red line4 ,this is common to the orange face1 the pink face1 and that face which is formed by moving the red line in the sixth dimension1 namely1 the purple face4 ,it is also common to the ochre cube1 the pale purple cube1 and the brown cube4 ,the points are common to six square faces and to four cubes2 thus1 the null point from which we start is common to the three square faces--pink1 light yellow1 orange1 and to the three square faces made by moving the three lines white1 yellow1 red1 in the fourth dimension1 namely1 the light blue1 the light green1 the purple faces--that is1 to six faces in all4 ,the four cubes which meet in it are the ochre cube1 the light purple cube1 the brown cube1 and the light green cube4 ,illustration3 ,fig4 #ajc4 ,the tesseract1 red1 white1 yellow axes in space4 ,in the lower line the three rear faces are shown1 the interior being removed4 ,illustration3 ,fig4 #ajd4 ,the tesseract1 red1 yellow1 blue axes in space1 the blue axis running to the left1 opposite faces are coloured identically4 ,a complete view of the tesseract in its various space presentations is given in the following figures or catalogue cubes1 figs4 #ajc-ajf4 ,the first cube in each figure represents the view of a tesseract coloured as described as it begins to pass transverse to our space4 ,the intermediate figure represents a sectional view when it is partly through1 and the final figure represents the far end as it is just passing out4 ,these figures will be explained in detail in the next chapter4 ,illustration3 ,fig4 #aje4 ,the tesseract1 with red1 white1 blue axes in space4 ,opposite faces are coloured identically4 ,illustration3 ,fig4 #ajf4 ,the tesseract1 with blue1 white1 yellow axes in space4 ,the blue axis runs downward from the base of the ochre cube as it stands originally4 ,opposite faces are coloured identically4 ,we have thus obtained a nomenclature for each of the regions of a tesseract2 we can speak of any one of the eight bounding cubes1 the twenty square faces1 the thirty-two lines1 the sixteen points4 ,,chapter ,,xiii ,,remarks ,,on ,,the ,,figures ,an inspection of above figures will give an answer to many questions about the tesseract4 ,if we have a tesseract one inch each way1 then it can be represented as a cube--a cube having white1 yellow1 red axes1 and from this cube as a beginning1 a volume extending into the fourth dimension4 ,now suppose the tesseract to pass transverse to our space1 the cube of the red1 yellow1 white axes disappears at once1 it is indefinitely thin in the fourth dimension4 ,its place is occupied by those parts of the tesseract which lie further away from our space in the fourth dimension4 ,each one of these sections will last only for one moment1 but the whole of them will take up some appreciable time in passing4 ,if we take the rate of one inch a minute the sections will take the whole of the minute in their passage across our space1 they will take the whole of the minute except the moment which the beginning cube and the end cube occupy in their crossing our space4 ,in each one of the cubes1 the section cubes1 we can draw lines in all directions except in the direction occupied by the blue line1 the fourth dimension2 lines in that direction are represented by the transition from one section cube to another4 ,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 cube1 the ochre cube1 and the last bounding cube1 the other ochre cube4 ,practically three intermediate sectional cubes will be found sufficient for most purposes4 ,we will take then a series of five figures--two terminal cubes1 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 space4 ,in fig4 #ajg initial letters are used for the colours4 ,a reference to fig4 #ajc will show the complete nomenclature1 which is merely indicated here4 ,illustration3 ,fig4 #ajg4 ,in this figure the tesseract is shown in five stages distant from our space3 first1 zero2 second1 #a_/d in42 third1 #b_/d in42 fourth1 #c_/d in42 fifth1 #a in42 which are called b#j1 b#a1 b#b1 b#c1 b#d1 because they are sections taken at distances #j1 #a1 #b1 #c1 #d quarter inches along the blue line4 ,all the regions can be named from the first cube1 the b#j cube1 as before1 simply by remembering that transference along the b axis gives the addition of blue to the colour of the region in the ochre1 the b#j cube4 ,in the final cube b#d1 the colouring of the original b#j cube is repeated4 ,thus the red line moved along the blue axis gives a red and blue or purple square4 ,this purple square appears as the three purple lines in the sections b#a1 b#b1 b#c1 taken at #a_/d1 #b_/d1 #c_/d of an inch in the fourth dimension4 ,if the tesseract moves transverse to our space we have then in this particular region1 first of all a red line which lasts for a moment1 secondly a purple line which takes its place4 ,this purple line lasts for a minute--that is1 all of a minute1 except the moment taken by the crossing our space of the initial and final red line4 ,the purple line having lasted for this period is succeeded by a red line1 which lasts for a moment2 then this goes and the tesseract has passed across our space4 ,the final red line we call red bl41 because it is separated from the initial red line by a distance along the axis for which we use the colour blue4 ,thus a line that lasts represents an area duration2 is in this mode of presentation equivalent to a dimension of space4 ,in the same way the white line1 during the crossing our space by the tesseract1 is succeeded by a light blue line which lasts for the inside of a minute1 and as the tesseract leaves our space1 having crossed it1 the white bl4 line appears as the final termination4 ,take now the pink face4 ,moved in the blue direction it traces out a light purple cube4 ,this light purple cube is shown in sections in b#a1 b#b1 b#c1 and the farther face of this cube in the blue direction is shown in b#d--;a pink face1 called pink b because it is distant from the pink face we began with in the blue direction4 ,thus the cube which we colour light purple appears as a lasting square4 ,the square face itself1 the pink face1 vanishes instantly the tesseract begins to move1 but the light purple cube appears as a lasting square4 ,here also duration is the equivalent of a dimension of space--a lasting square is a cube4 ,it is useful to connect these diagrams with the views given in the coloured plate4 ,take again the orange face1 that determined by the red and yellow axes2 from it goes a brown cube in the blue direction1 for red and yellow and blue are supposed to make brown4 ,this brown cube is shown in three sections in the faces b#a1 b#b1 b#c4 ,in b#d is the opposite orange face of the brown cube1 the face called orange b1 for it is distant in the blue direction from the orange face4 ,as the tesseract passes transverse to our space1 we have then in this region an instantly vanishing orange square1 followed by a lasting brown square1 and finally an orange face which vanishes instantly4 ,now1 as any three axes will be in our space1 let us send the white axis out into the unknown1 the fourth dimension1 and take the blue axis into our known space dimension4 ,since the white and blue axes are perpendicular to each other1 if the white axis goes out into the fourth dimension in the positive sense1 the blue axis will come into the direction the white axis occupied1 in the negative sense4 ,illustration3 ,fig4 #ajh4 ,hence1 not to complicate matters by having to think of two senses in the unknown direction1 let us send the white line into the positive sense of the fourth dimension1 and take the blue one as running in the negative sense of that direction which the white line has left2 let the blue line1 that is1 run to the left4 ,we have now the row of figures in fig4 #ajh4 ,the dotted cube shows where we had a cube when the white line ran in our space--now it has turned out of our space1 and another solid boundary1 another cubic face of the tesseract comes into our space4 ,this cube has red and yellow axes as before2 but now1 instead of a white axis running to the right1 there is a blue axis running to the left4 ,here we can distinguish the regions by colours in a perfectly systematic way4 ,the red line traces out a purple square in the transference along the blue axis by which this cube is generated from the orange face4 ,this purple square made by the motion of the red line is the same purple face that we saw before as a series of lines in the sections b#a1 b#b1 b#c4 ,here1 since both red and blue axes are in our space1 we have no need of duration to represent the area they determine4 ,in the motion of the tesseract across space this purple face would instantly disappear4 ,from the orange face1 which is common to the initial cubes in fig4 #ajg and fig4 #ajh1 there goes in the blue direction a cube coloured brown4 ,this brown cube is now all in our space1 because each of its three axes run in space directions1 up1 away1 to the left4 ,it is the same brown cube which appeared as the successive faces on the sections b#a1 b#b1 b#c4 ,having all its three axes in our space1 it is given in extension2 no part of it needs to be represented as a succession4 ,the tesseract is now in a new position with regard to our space1 and when it moves across our space the brown cube instantly disappears4 ,in order to exhibit the other regions of the tesseract we must remember that now the white line runs in the unknown dimension4 ,where shall we put the sections at distances along the line8 ,any arbitrary position in our space will do3 there is no way by which we can represent their real position4 ,however1 as the brown cube comes off from the orange face to the left1 let us put these successive sections to the left4 ,we can call them wh#j1 wh#a1 wh#b1 wh#c1 wh#d1 because they are sections along the white axis1 which now runs in the unknown dimension4 ,running from the purple square in the white direction we find the light purple cube4 ,this is represented in the sections wh#a1 wh#b1 wh#c1 wh#d1 fig4 #ajh4 ,it is the same cube that is represented in the sections b#a1 b#b1 b#c3 in fig4 #ajg the red and white axes are in our space1 the blue out of it2 in the other case1 the red and blue are in our space1 the white out of it4 ,it is evident that the face pink y1 opposite the pink face in fig4 #ajg1 makes a cube shown in squares in b#a1 b#b1 b#c1 b#d1 on the opposite side to the l purple squares4 ,also the light yellow face at the base of the cube b#j1 makes a light green cube1 shown as a series of base squares4 ,the same light green cube can be found in fig4 #ajg4 ,the base square in wh#j is a green square1 for it is enclosed by blue and yellow axes4 ,from it goes a cube in the white direction1 this is then a light green cube and the same as the one just mentioned as existing in the sections b#j1 b#a1 b#b1 b#c1 b#d4 ,the case is1 however1 a little different with the brown cube4 ,this cube we have altogether in space in the section wh#j1 fig4 #ajh1 while it exists as a series of squares1 the left-hand ones1 in the sections b#j1 b#a1 b#b1 b#c1 b#d4 ,the brown cube exists as a solid in our space1 as shown in fig4 #ajh4 ,in the mode of representation of the tesseract exhibited in fig4 #ajg1 the same brown cube appears as a succession of squares4 ,that is1 as the tesseract moves across space1 the brown cube would actually be to us a square--it would be merely the lasting boundary of another solid4 ,it would have no thickness at all1 only extension in two dimensions1 and its duration would show its solidity in three dimensions4 ,it is obvious that1 if there is a four-dimensional space1 matter in three dimensions only is a mere abstraction2 all material objects must then have a slight four-dimensional thickness4 ,in this case the above statement will undergo modification4 ,the material cube which is used as the model of the boundary of a tesseract will have a slight thickness in the fourth dimension1 and when the cube is presented to us in another aspect1 it would not be a mere surface4 ,but it is most convenient to regard the cubes we use as having no extension at all in the fourth dimension4 ,this consideration serves to bring out a point alluded to before1 that1 if there is a fourth dimension1 our conception of a solid is the conception of a mere abstraction1 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 squares1 real triangles1 etc41 would seem to us4 ,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 space4 ,the brown faces in b#a1 b#b1 b#c1 are the very same brown sections that would be obtained by cutting the brown cube1 wh#j1 across at the right distances along the blue line1 as shown in fig4 #ajh4 ,but as these sections are placed in the brown cube1 wh#j1 they come behind one another in the blue direction4 ,now1 in the sections wh#a1 wh#b1 wh#c1 we are looking at these sections from the white direction--the blue direction does not exist in these figures4 ,so we see them in a direction at right angles to that in which they occur behind one another in wh#j4 ,there are intermediate views1 which would come in the rotation of a tesseract4 ,these brown squares can be looked at from directions intermediate between the white and blue axes4 ,it must be remembered that the fourth dimension is perpendicular equally to all three space axes4 ,hence we must take the combinations of the blue axis1 with each two of our three axes1 white1 red1 yellow1 in turn4 ,in fig4 #aji we take red1 white1 and blue axes in space1 sending yellow into the fourth dimension4 ,if it goes into the positive sense of the fourth dimension the blue line will come in the opposite direction to that in which the yellow line ran before4 ,hence1 the cube determined by the white1 red1 blue axes1 will start from the pink plane and run towards us4 ,the dotted cube shows where the ochre cube was4 ,when it is turned out of space1 the cube coming towards from its front face is the one which comes into our space in this turning4 ,since the yellow line now runs in the unknown dimension we call the sections y#j1 y#a1 y#b1 y#c1 y#d1 as they are made at distances #j1 #a1 #b1 #c1 #d1 quarter inches along the yellow line4 ,we suppose these cubes arranged in a line coming towards us--not that that is any more natural than any other arbitrary series of positions1 but it agrees with the plan previously adopted4 ,illustration3 ,fig4 #aji4 ,the interior of the first cube1 y#j1 is that derived from pink by adding blue1 or1 as we call it1 light purple4 ,the faces of the cube are light blue1 purple1 pink4 ,as drawn1 we can only see the face nearest to us1 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 us4 ,the successive sections of the series1 y#j1 y#a1 y#b1 etc41 can be considered as derived from sections of the b#j cube made at distances along the yellow axis4 ,what is distant a quarter inch from the pink face in the yellow direction8 ,this question is answered by taking a section from a point a quarter inch along the yellow axis in the cube b#j1 fig4 #ajg4 ,it is an ochre section with lines orange and light yellow4 ,this section will therefore take the place of the pink face in y#a when we go on in the yellow direction4 ,thus1 the first section1 y#a1 will begin from an ochre face with light yellow and orange lines4 ,the colour of the axis which lies in space towards us is blue1 hence the regions of this section-cube are determined in nomenclature1 they will be found in full in fig4 #aje4 ,there remains only one figure to be drawn1 and that is the one in which the red axis is replaced by the blue4 ,here1 as before1 if the red axis goes out into the positive sense of the fourth dimension1 the blue line must come into our space in the negative sense of the direction which the red line has left4 ,accordingly1 the first cube will come in beneath the position of our ochre cube1 the one we have been in the habit of starting with4 ,illustration3 ,fig4 #aaj4 ,to show these figures we must suppose the ochre cube to be on a movable stand4 ,when the red line swings out into the unknown dimension1 and the blue line comes in downwards1 a cube appears below the place occupied by the ochre cube4 ,the dotted cube shows where the ochre cube was4 ,that cube has gone and a different cube runs downwards from its base4 ,this cube has white1 yellow1 and blue axes4 ,its top is a light yellow square1 and hence its interior is light yellow blue or light green4 ,its front face is formed by the white line moving along the blue axis1 and is therefore light blue1 the left-hand side is formed by the yellow line moving along the blue axis1 and therefore green4 ,as the red line now runs in the fourth dimension1 the successive sections can he called r#j1 r#a1 r#b1 r#c1 r#d1 these letters indicating that at distances #j1 #a_/d1 #b_/d1 #c_/d1 #a inch along the red axis we take all of the tesseract that can be found in a three-dimensional space1 this three-dimensional space extending not at all in the fourth dimension1 but up and down1 right and left1 far and near4 ,we can see what should replace the light yellow face of r#j1 when the section r#a comes in1 by looking at the cube b#j1 fig4 #ajg4 ,what is distant in it one-quarter of an inch from the light yellow face in the red direction8 ,it is an ochre section with orange and pink lines and red points2 see also fig4 #ajc4 ,this square then forms the top square of r#a4 ,now we can determine the nomenclature of all the regions of r#a by considering what would be formed by the motion of this square along a blue axis4 ,but we can adopt another plan4 ,let us take a horizontal section of r#j1 and finding that section in the figures1 of fig4 #ajg or fig4 #ajc1 from them determine what will replace it1 going on in the red direction4 ,a section of the r#j cube has green1 light blue1 green1 light blue sides and blue points4 ,now this square occurs on the base of each of the section figures1 b#a1 b#b1 etc4 ,in them we see that #a_/d inch in the red direction from it lies a section with brown and light purple lines and purple corners1 the interior being of light brown4 ,hence this is the nomenclature of the section which in r#a replaces the section of r#j made from a point along the blue axis4 ,hence the colouring as given can be derived4 ,we have thus obtained a perfectly named group of tesseracts4 ,we can take a group of eighty-one of them #c #c #c #c1 in four dimensions1 and each tesseract will have its name null1 red1 white1 yellow1 blue1 etc41 and whatever cubic view we take of them we can say exactly what sides of the tesseracts we are handling1 and how they touch each other4#e #e ,at this point the reader will find it advantageous1 if he has the models1 to go through the manipulations described in the appendix4 ,thus1 for instance1 if we have the sixteen tesseracts shown below1 we can ask how does null touch blue4 ,illustration3 ,fig4 #aaa4 ,in the arrangement given in fig4 #aaa we have the axes white1 red1 yellow1 in space1 blue running in the fourth dimension4 ,hence we have the ochre cubes as bases4 ,imagine now the tesseractic group to pass transverse to our space--we have first of all null ochre cube1 white ochre cube1 etc42 these instantly vanish1 and we get the section shown in the middle cube in fig4 #ajc1 and finally1 just when the tesseract block has moved one inch transverse to our space1 we have null ochre cube1 and then immediately afterwards the ochre cube of blue comes in4 ,hence the tesseract null touches the tesseract blue by its ochre cube1 which is in contact1 each and every point of it1 with the ochre cube of blue4 ,how does null touch white1 we may ask8 ,looking at the beginning ,a1 fig4 #aaa1 where we have the ochre cubes1 we see that null ochre touches white ochre by an orange face4 ,now let us generate the null and white tesseracts by a motion in the blue direction of each of these cubes4 ,each of them generates the corresponding tesseract1 and the plane of contact of the cubes generates the cube by which the tesseracts are in contact4 ,now an orange plane carried along a blue axis generates a brown cube4 ,hence null touches white by a brown cube4 ,illustration3 ,fig4 #aab4 ,if we ask again how red touches light blue tesseract1 let us rearrange our group1 fig4 #aab1 or rather turn it about so that we have a different space view of it2 let the red axis and the white axis run up and right1 and let the blue axis come in space towards us1 then the yellow axis runs in the fourth dimension4 ,we have then two blocks in which the bounding cubes of the tesseracts are given1 differently arranged with regard to us--the arrangement is really the same1 but it appears different to us4 ,starting from the plane of the red and white axes we have the four squares of the null1 white1 red1 pink tesseracts as shown in ,a1 on the red1 white plane1 unaltered1 only from them now comes out towards us the blue axis4 ,hence we have null1 white1 red1 pink tesseracts in contact with our space by their cubes which have the red1 white1 blue axis in them1 that is by the light purple cubes4 ,following on these four tesseracts we have that which comes next to them in the blue direction1 that is the four blue1 light blue1 purple1 light purple4 ,these are likewise in contact with our space by their light purple cubes1 so we see a block as named in the figure1 of which each cube is the one determined by the red1 white1 blue1 axes4 ,the yellow line now runs out of space2 accordingly one inch on in the fourth dimension we come to the tesseracts which follow on the eight named in ,c1 fig4 #aab1 in the yellow direction4 ,these are shown in ,c4y#a1 fig4 #aab4 ,between figure ,c and ,c4y#a is that four-dimensional mass which is formed by moving each of the cubes in ,c one inch in the fourth dimension--that is1 along a yellow axis2 for the yellow axis now runs in the fourth dimension4 ,in the block ,c we observe that red 7light purple cube7 touches light blue 7light purple cube7 by a point4 ,now these two cubes moving together remain in contact during the period in which they trace out the tesseracts red and light blue4 ,this motion is along the yellow axis1 consequently red and light blue touch by a yellow line4 ,we have seen that the pink face moved in a yellow direction traces out a cube2 moved in the blue direction it also traces out a cube4 ,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 directions4 ,what section of the tesseract will it make8 ,we will first consider the red line alone4 ,let us take a cube with the red line in it and the yellow and blue axes4 ,the cube with the yellow1 red1 blue axes is shown in fig4 #aac4 ,if the red line is moved equally in the yellow and in the blue direction by four equal motions of inch each1 it takes the positions #aa1 #bb1 #cc1 and ends as a red line4 ,illustration3 ,fig4 #aac4 ,now1 the whole of this red1 yellow1 blue1 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 dimension4 ,hence the plane traced out by the red line appears as a series of lines in the successive sections1 in our ordinary way of representing the tesseract2 these lines are in different places in each successive section4 ,illustration3 ,fig4 #aad4 ,thus drawing our initial cube and the successive sections1 calling them b#j1 b#a1 b#b1 b#c1 b#d1 fig4 #aae1 we have the red line subject to this movement appearing in the positions indicated4 ,we will now investigate what positions in the tesseract another line in the pink face assumes when it is moved in a similar manner4 ,take a section of the original cube containing a vertical line1 #d1 in the pink plane1 fig4 #aae4 ,we have1 in the section1 the yellow direction1 but not the blue4 ,from this section a cube goes off in the fourth dimension1 which is formed by moving each point of the section in the blue direction4 ,illustration3 ,fig4 #aae4 ,illustration3 ,fig4 #aaf4 ,drawing this cube we have fig4 #aaf4 ,now this cube occurs as a series of sections in our original representation of the tesseract4 ,taking four steps as before this cube appears as the sections drawn in b#j1 b#a1 b#b1 b#c1 b#d1 fig4 #aag1 and if the line #d is subjected to a movement equal in the blue and yellow directions1 it will occupy the positions designated by #d1 #da1 #db1 #dc1 #dd4 ,illustration3 ,fig4 #aag4 ,hence1 reasoning in a similar manner about every line1 it is evident that1 moved equally in the blue and yellow directions1 the pink plane will trace out a space which is shown by the series of section planes represented in the diagram4 ,thus the space traced out by the pink face1 if it is moved equally in the yellow and blue directions1 is represented by the set of planes delineated in ,fig4 #aah1 pink face or #j1 then #a1 #b1 #c1 and finally pink face or #d4 ,this solid is a diagonal solid of the tesseract1 running from a pink face to a pink face4 ,its length is the length of the diagonal of a square1 its side is a square4 ,let us now consider the unlimited space which springs from the pink face extended4 ,this space1 if it goes off in the yellow direction1 gives us in it the ochre cube of the tesseract4 ,thus1 if we have the pink face given and a point in the ochre cube1 we have determined this particular space4 ,similarly going off from the pink face in the blue direction is another space1 which gives us the light purple cube of the tesseract in it4 ,and any point being taken in the light purple cube1 this space going off from the pink face is fixed4 ,illustration3 ,fig4 #aah4 ,the space we are speaking of can be conceived as swinging round the pink face1 and in each of its positions it cuts out a solid figure from the tesseract1 one of which we have seen represented in fig4 #aah4 ,each of these solid figures is given by one position of the swinging space1 and by one only4 ,hence in each of them1 if one point is taken1 the particular one of the slanting spaces is fixed4 ,thus we see that given a plane and a point out of it a space is determined4 ,now1 two points determine a line4 ,again1 think of a line and a point outside it4 ,imagine a plane rotating round the line4 ,at some time in its rotation it passes through the point4 ,thus a line and a point1 or three points1 determine a plane4 ,and finally four points determine a space4 ,we have seen that a plane and a point determine a space1 and that three points determine a plane2 so four points will determine a space4 ,these four points may be any points1 and we can take1 for instance1 the four points at the extremities of the red1 white1 yellow1 blue axes1 in the tesseract4 ,these will determine a space slanting with regard to the section spaces we have been previously considering4 ,this space will cut the tesseract in a certain figure4 ,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 point4 ,we see at once that this plane would cut the cube in a triangle1 but we will go through the process by which a plane being would most conveniently treat the problem of the determination of this shape1 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 #d points at unit distance from a corner4 ,we know that two points determine a line1 three points determine a plane1 and given any two points in a plane the line between them lies wholly in the plane4 ,illustration3 ,fig4 #aai4 ,let now the plane being study the section made by a plane passing through the null r1 null wh1 and null y points1 fig4 #aai4 ,looking at the orange square1 which1 as usual1 we suppose to be initially in his plane1 he sees that the line from null r to null y1 which is a line in the section plane1 the plane1 namely1 through the three extremities of the edges meeting in null1 cuts the orange face in an orange line with null points4 ,this then is one of the boundaries of the section figure4 ,let now the cube be so turned that the pink face comes in his plane4 ,the points null r and null wh are now visible4 ,the line between them is pink with null points1 and since this line is common to the surface of the cube and the cutting plane1 it is a boundary of the figure in which the plane cuts the cube4 ,again1 suppose the cube turned so that the light yellow face is in contact with the plane being's plane4 ,he sees two points1 the null wh and the null y4 ,the line between these lies in the cutting plane4 ,hence1 since the three cutting lines meet and enclose a portion of the cube between them1 he has determined the figure he sought4 ,it is a triangle with orange1 pink1 and light yellow sides1 all equal1 and enclosing an ochre area4 ,let us now determine in what figure the space1 determined by the four points1 null r1 null y1 null wh1 null b1 cuts the tesseract4 ,we can see three of these points in the primary position of the tesseract resting against our solid sheet by the ochre cube4 ,these three points determine a plane which lies in the space we are considering1 and this plane cuts the ochre cube in a triangle1 the interior of which is ochre 7fig4 #aai will serve for this view71 with pink1 light yellow and orange sides1 and null points4 ,going in the fourth direction1 in one sense1 from this plane we pass into the tesseract1 in the other sense we pass away from it4 ,the whole area inside the triangle is common to the cutting plane we see1 and a boundary of the tesseract4 ,hence we conclude that the triangle drawn is common to the tesseract and the cutting space4 ,now let the ochre cube turn out and the brown cube come in4 ,the dotted lines show the position the ochre cube has left 7fig4 #abj74 ,illustration3 ,fig4 #abj4 ,here we see three out of the four points through which the cutting plane passes1 null r1 null y1 and null b4 ,the plane they determine lies in the cutting space1 and this plane cuts out of the brown cube a triangle with orange1 purple and green sides1 and null points4 ,the orange line of this figure is the same as the orange line in the last figure4 ,now let the light purple cube swing into our space1 towards us1 fig4 #aba4 ,illustration3 ,fig4 #aba4 ,the cutting space which passes through the four points1 null r1 y1 wh1 b1 passes through the null r1 wh1 b1 and therefore the plane these determine lies in the cutting space4 ,this triangle lies before us4 ,it has a light purple interior and pink1 light blue1 and purple edges with null points4 ,this1 since it is all of the plane that is common to it1 and this bounding of the tesseract1 gives us one of the bounding faces of our sectional figure4 ,the pink line in it is the same as the pink line we found in the first figure--that of the ochre cube4 ,finally1 let the tesseract swing about the light yellow plane1 so that the light green cube comes into our space4 ,it will point downwards4 ,the three points1 n4y1 n4wh1 n4b1 are in the cutting space1 and the triangle they determine is common to the tesseract and the cutting space4 ,hence this boundary is a triangle having a light yellow line1 which is the same as the light yellow line of the first figure1 a light blue line and a green line4 ,illustration3 ,fig4 #abb4 ,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 tesseract1 and have got four faces which all join on to each other by lines4 ,illustration3 ,fig4 #abc4 ,the triangles are shown in fig4 #abc as they join on to the triangle in the ochre cube4 ,but they join on each to the other in an exactly similar manner2 their edges are all identical two and two4 ,they form a closed figure1 a tetrahedron1 enclosing a light brown portion which is the portion of the cutting space which lies inside the tesseract4 ,we cannot expect to see this light brown portion1 any more than a plane being could expect to see the inside of a cube if an angle of it were pushed through his plane4 ,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 plane4 ,thus in this solid section2 the whole interior lies perfectly open in the fourth dimension4 ,go round it as we may we are simply looking at the boundaries of the tesseract which penetrates through our solid sheet4 ,if the tesseract were not to pass across so far1 the triangle would be smaller2 if it were to pass farther1 we should have a different figure1 the outlines of which can be determined in a similar manner4 ,the preceding method is open to the objection that it depends rather on our inferring what must be1 than our seeing what is4 ,let us therefore consider our sectional space as consisting of a number of planes1 each very close to the last1 and observe what is to be found in each plane4 ,the corresponding method in the case of two dimensions is as follows3--,the plane being can see that line of the sectional plane through null y1 null wh1 null r1 which lies in the orange plane4 ,let him now suppose the cube and the section plane to pass half way through his plane4 ,replacing the red and yellow axes are lines parallel to them1 sections of the pink and light yellow faces4 ,illustration3 ,fig4 #abd4 ,where will the section plane cut these parallels to the red and yellow axes8 ,let him suppose the cube1 in the position of the drawing1 fig4 #abd1 turned so that the pink face lies against his plane4 ,he can see the line from the null r point to the null wh point1 and can see 7compare fig4 #aai7 that it cuts ,,ab a parallel to his red axis1 drawn at a point half way along the white line1 in a point ,b1 half way up4 ,i shall speak of the axis as having the length of an edge of the cube4 ,similarly1 by letting the cube turn so that the light yellow square swings against his plane1 he can see 7compare fig4 #aai7 that a parallel to his yellow axis drawn from a point half-way along the white axis1 is cut at half its length by the trace of the section plane in the light yellow face4 ,hence when the cube had passed half-way through he would have--instead of the orange line with null points1 which he had at first--an ochre line of half its length1 with pink and light yellow points4 ,thus1 as the cube passed slowly through his plane1 he would have a succession of lines gradually diminishing in length and forming an equilateral triangle4 ,the whole interior would be ochre1 the line from which it started would be orange4 ,the succession of points at the ends of the succeeding lines would form pink and light yellow lines and the final point would be null4 ,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 bit4 ,coming now to the section of the tesseract1 let us imagine that the tesseract and its cutting space pass slowly across our space2 we can examine portions of it1 and their relation to portions of the cutting space4 ,take the section space which passes through the four points1 null r1 wh1 y1 b2 we can see in the ochre cube 7fig4 #aai7 the plane belonging to this section space1 which passes through the three extremities of the red1 white1 yellow axes4 ,now let the tesseract pass half way through our space4 ,instead of our original axes we have parallels to them1 purple1 light blue1 and green1 each of the same length as the first axes1 for the section of the tesseract is of exactly the same shape as its ochre cube4 ,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 first4 ,to see where the sectional space would cut these parallels to the original axes let the tesseract swing so that1 the orange face remaining stationary1 the blue line comes in to the left4 ,here 7fig4 #abe7 we have the null r1 y1 b points1 and of the sectional space all we see is the plane through these three points in it4 ,illustration3 ,fig4 #abe4 ,in this figure we can draw the parallels to the red and yellow axes and see that1 if they started at a point half way along the blue axis1 they would each be cut at a point so as to be half of their previous length4 ,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 space4 ,hence in a section made when the tesseract had passed half across our space the parallels to the red1 white1 yellow axes1 which are now in our space1 are cut by the section space1 each of them half way along1 and for this stage of the traversing motion we should have fig4 #abf4 ,the section made of this cube by the plane in which the sectional space cuts it1 is an equilateral triangle with purple1 l4 blue1 green points1 and l4 purple1 brown1 l4 green lines4 ,illustration3 ,fig4 #abf4 ,thus the original ochre triangle1 with null points and pink1 orange1 light yellow lines1 would be succeeded by a triangle coloured in manner just described4 ,this triangle would initially be only a very little smaller than the original triangle1 it would gradually diminish1 until it ended in a point1 a null point4 ,each of its edges would be of the same length4 ,thus the successive sections of the successive planes into which we analyse the cutting space would be a tetrahedron of the description shown 7fig4 #abc71 and the whole interior of the tetrahedron would be light brown4 ,illustration3 ,fig4 #abg4 ,front view4 ,the rear faces4 ,in fig4 #abg the tetrahedron is represented by means of its faces as two triangles which meet in the p4 line1 and two rear triangles which join on to them1 the diagonal of the pink face being supposed to run vertically upward4 ,we have now reached a natural termination4 ,the reader may pursue the subject in further detail1 but will find no essential novelty4 ,i conclude with an indication as to the manner in which figures previously given may be used in determining sections by the method developed above4 ,applying this method to the tesseract1 as represented in ,chapter ,,ix41 sections made by a space cutting the axes equidistantly at any distance can be drawn1 and also the sections of tesseracts arranged in a block4 ,if we draw a plane1 cutting all four axes at a point six units distance from null1 we have a slanting space4 ,this space cuts the red1 white1 yellow axes in the points ,,lmn 7fig4 #abh71 and so in the region of our space before we go off into the fourth dimension1 we have the plane represented by ,,lmn extended4 ,this is what is common to the slanting space and our space4 ,illustration3 ,fig4 #abh4 ,this plane cuts the ochre cube in the triangle ,,efg4 ,comparing this with 7fig4 #gb7 oh1 we see that the hexagon there drawn is part of the triangle ,,efg4 ,let us now imagine the tesseract and the slanting space both together to pass transverse to our space1 a distance of one unit1 we have in #a;h a section of the tesseract1 whose axes are parallels to the previous axes4 ,the slanting space cuts them at a distance of five units along each4 ,drawing the plane through these points in #a;h it will be found to cut the cubical section of the tesseract in the hexagonal figure drawn4 ,in #b;h 7fig4 #gb7 the slanting space cuts the parallels to the axes at a distance of four along each1 and the hexagonal figure is the section of this section of the tesseract by it4 ,finally when #c;h comes in the slanting space cuts the axes at a distance of three along each1 and the section is a triangle1 of which the hexagon drawn is a truncated portion4 ,after this the tesseract1 which extends only three units in each of the four dimensions1 has completely passed transverse of our space1 and there is no more of it to be cut4 ,hence1 putting the plane sections together in the right relations1 we have the section determined by the particular slanting space3 namely an octahedron4 ,,chapter ,,xiv4#f ,a ,,recapitulation ,,and ,,extension ,,of ,,the ,,physical ,,argument ,there are two directions of inquiry in which the research for the physical reality of a fourth dimension can be prosecuted4 ,one is the investigation of the infinitely great1 the other is the investigation of the infinitely small4 #f ,the contents of this chapter are taken from a paper read before the ,philosophical ,society of ,washington4 ,the mathematical portion of the paper has appeared in part in the ,proceedings of the ,royal ,irish ,academy under the title1 8,cayley's formul of orthogonal transformation10 ,nov4 #bith1 #aijc4 ,by the measurement of the angles of vast triangles1 whose sides are the distances between the stars1 astronomers have sought to determine if there is any deviation from the values given by geometrical deduction4 ,if the angles of a celestial triangle do not together equal two right angles1 there would be an evidence for the physical reality of a fourth dimension4 ,this conclusion deserves a word of explanation4 ,if space is really four-dimensional1 certain conclusions follow which must be brought clearly into evidence if we are to frame the questions definitely which we put to ,nature4 ,to account for our limitation let us assume a solid material sheet against which we move4 ,this sheet must stretch alongside every object in every direction in which it visibly moves4 ,every material body must slip or slide along this sheet1 not deviating from contact with it in any motion which we can observe4 ,the necessity for this assumption is clearly apparent1 if we consider the analogous case of a suppositionary plane world4 ,if there were any creatures whose experiences were confined to a plane1 we must account for their limitation4 ,if they were free to move in every space direction1 they would have a three-dimensional motion2 hence they must be physically limited1 and the only way in which we can conceive such a limitation to exist is by means of a material surface against which they slide4 ,the existence of this surface could only be known to them indirectly4 ,it does not lie in any direction from them in which the kinds of motion they know of leads them4 ,if it were perfectly smooth and always in contact with every material object1 there would be no difference in their relations to it which would direct their attention to it4 ,but if this surface were curved--if it were1 say1 in the form of a vast sphere--the triangles they drew would really be triangles of a sphere1 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 plane4 ,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 world1 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 with4 ,now1 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 say1 they have tried astronomical measurements to find out if the vast solid sheet against which1 on the supposition of a fourth dimension1 everything slides is curved or not4 ,these results have been negative4 ,the solid sheet1 if it exists1 is not curved or1 being curved1 has not a sufficient curvature to cause any observable deviation from the theoretical value of the angles calculated4 ,hence the examination of the infinitely great leads to no decisive criterion4 ,if it did we should have to decide between the present theory and that of metageometry4 ,coming now to the prosecution of the inquiry in the direction of the infinitely small1 we have to state the question thus3 ,our laws of movement are derived from the examination of bodies which move in three-dimensional space4 ,all our conceptions are founded on the supposition of a space which is represented analytically by three independent axes and variations along them--that is1 it is a space in which there are three independent movements4 ,any motion possible in it can be compounded out of these three movements1 which we may call3 up1 right1 away4 ,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 space1 we must commence by clearly defining what the laws of mechanics would be on the supposition of a fourth dimension4 ,it is of no use asking if the phenomena of the smallest particles of matter are like--we do not know what4 ,we must have a definite conception of what the laws of motion would be on the supposition of the fourth dimension1 and then inquire if the phenomena of the activity of the smaller particles of matter resemble the conceptions which we have elaborated4 ,now1 the task of forming these conceptions is by no means one to be lightly dismissed4 ,movement in space has many features which differ entirely from movement on a plane2 and when we set about to form the conception of motion in four dimensions1 we find that there is at least as great a step as from the plane to three-dimensional space4 ,i do not say that the step is difficult1 but ,i want to point out that it must be taken4 ,when we have formed the conception of four-dimensional motion1 we can ask a rational question of ,nature4 ,before we have elaborated our conceptions we are asking if an unknown is like an unknown--a futile inquiry4 ,as a matter of fact1 four-dimensional movements are in every way simple and more easy to calculate than three-dimensional movements1 for four-dimensional movements are simply two sets of plane movements put together4 ,without the formation of an experience of four-dimensional bodies1 their shapes and motions1 the subject can be but formal--logically conclusive1 not intuitively evident4 ,it is to this logical apprehension that ,i must appeal4 ,it is perfectly simple to form an experiential familiarity with the facts of four-dimensional movement4 ,the method is analogous to that which a plane being would have to adopt to form an experiential familiarity with three-dimensional movements1 and may be briefly summed up as the formation of a compound sense by means of which duration is regarded as equivalent to extension4 ,consider a being confined to a plane4 ,a square enclosed by four lines will be to him a solid1 the interior of which can only be examined by breaking through the lines4 ,if such a square were to pass transverse to his plane1 it would immediately disappear4 ,it would vanish1 going in no direction to which he could point4 ,if1 now1 a cube be placed in contact with his plane1 its surface of contact would appear like the square which we have just mentioned4 ,but if it were to pass transverse to his plane1 breaking through it1 it would appear as a lasting square4 ,the three-dimensional matter will give a lasting appearance in circumstances under which two-dimensional matter will at once disappear4 ,similarly1 a four-dimensional cube1 or1 as we may call it1 a tesseract1 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 cube1 if it moved transverse to our space1 would appear as a lasting cube4 ,a cube of three-dimensional matter1 since it extends to no distance at all in the fourth dimension1 would instantly disappear1 if subjected to a motion transverse to our space4 ,it would disappear and be gone1 without it being possible to point to any direction in which it had moved4 ,all attempts to visualise a fourth dimension are futile4 ,it must be connected with a time experience in three space4 ,the most difficult notion for a plane being to acquire would be that of rotation about a line4 ,consider a plane being facing a square4 ,if he were told that rotation about a line were possible1 he would move his square this way and that4 ,a square in a plane can rotate about a point1 but to rotate about a line would seem to the plane being perfectly impossible4 ,how could those parts of his square which were on one side of an edge come to the other side without the edge moving8 ,he could understand their reflection in the edge4 ,he could form an idea of the looking-glass image of his square lying on the opposite side of the line of an edge1 but by no motion that he knows of can he make the actual square assume that position4 ,the result of the rotation would be like reflection in the edge1 but it would be a physical impossibility to produce it in the plane4 ,the demonstration of rotation about a line must be to him purely formal4 ,if he conceived the notion of a cube stretching out in an unknown direction away from his plane1 then he can see the base of it1 his square in the plane1 rotating round a point4 ,he can likewise apprehend that every parallel section taken at successive intervals in the unknown direction rotates in like manner round a point4 ,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 rotate4 ,thus1 given three axes1 x1 y1 z1 if x rotates to take the place of y1 and y turns so as to point to negative x1 then the third axis remaining unaffected by this turning is the axis about which the rotation takes place4 ,this1 then1 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 place4 ,there is another way in which a plane being can think about three-dimensional movements2 and1 as it affords the type by which we can most conveniently think about four-dimensional movements1 it will be no loss of time to consider it in detail4 ,illustration3 ,fig4 #a 7#abi74 ,we can represent the plane being and his object by figures cut out of paper1 which slip on a smooth surface4 ,the thickness of these bodies must be taken as so minute that their extension in the third dimension escapes the observation of the plane being1 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 surface4 ,let ,ax1 ,ay be two axes and ,,abcd a square4 ,as far as movements in the plane are concerned1 the square can rotate about a point ,a1 for example4 ,it cannot rotate about a side1 such as ,,ac4 ,but if the plane being is aware of the existence of a third dimension he can study the movements possible in the ample space1 taking his figure portion by portion4 ,his plane can only hold two axes4 ,but1 since it can hold two1 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 plane4 ,he can make a drawing in his plane of what stands up perpendicularly from his plane4 ,let ,az be the axis1 which stands perpendicular to his plane at ,a4 ,he can draw in his plane two lines to represent the two axes1 ,ax and ,az4 ,let ,fig4 #b be this drawing4 ,here the z axis has taken the place of the y axis1 and the plane of ,ax ,az is represented in his plane4 ,in this figure all that exists of the square ,,abcd will be the line ,,ab4 ,illustration3 ,fig4 #b 7#acj74 ,the square extends from this line in the y direction1 but more of that direction is represented in ,fig4 #b4 ,the plane being can study the turning of the line ,,ab in this diagram4 ,it is simply a case of plane turning around the point ,a4 ,the line ,,ab occupies intermediate portions like ,,ab#a and after half a revolution will lie on ,ax produced through ,a4 ,now1 in the same way1 the plane being can take another point1 ,a1 and another line1 ,a,b1 in his square4 ,he can make the drawing of the two directions at ,a1 one along ,a,b1 the other perpendicular to his plane4 ,he will obtain a figure precisely similar to ,fig4 #b1 and will see that1 as ,,ab can turn around ,a1 so ,a,c around ,a4 ,in this turning ,,ab and ,a,b would not interfere with each other1 as they would if they moved in the plane around the separate points ,a and ,a4 ,hence the plane being would conclude that a rotation round a line was possible4 ,he could see his square as it began to make this turning4 ,he could see it half way round when it came to lie on the opposite side of the line ,,ac4 ,but in intermediate portions he could not see it1 for it runs out of the plane4 ,coming now to the question of a four-dimensional body1 let us conceive of it as a series of cubic sections1 the first in our space1 the rest at intervals1 stretching away from our space in the unknown direction4 ,we must not think of a four-dimensional body as formed by moving a three-dimensional body in any direction which we can see4 ,refer for a moment to ,fig4 #c4 ,the point ,a1 moving to the right1 traces out the line ,,ac4 ,the line ,,ac1 moving away in a new direction1 traces out the square ,,aceg at the base of the cube4 ,the square ,,aegc1 moving in a new direction1 will trace out the cube ,,acegbdhf4 ,the vertical direction of this last motion is not identical with any motion possible in the plane of the base of the cube4 ,it is an entirely new direction1 at right angles to every line that can be drawn in the base4 ,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 cube4 ,the cubic sections of the tesseract are related to the cube we see1 as the square sections of the cube are related to the square of its base which a plane being sees4 ,let us imagine the cube in our space1 which is the base of a tesseract1 to turn about one of its edges4 ,the rotation will carry the whole body with it1 and each of the cubic sections will rotate4 ,the axis we see in our space will remain unchanged1 and likewise the series of axes parallel to it about which each of the parallel cubic sections rotates4 ,the assemblage of all of these is a plane4 ,hence in four dimensions a body rotates about a plane4 ,there is no such thing as rotation round an axis4 ,we may regard the rotation from a different point of view4 ,consider four independent axes each at right angles to all the others1 drawn in a four-dimensional body4 ,of these four axes we can see any three4 ,the fourth extends normal to our space4 ,rotation is the turning of one axis into a second1 and the second turning to take the place of the negative of the first4 ,it involves two axes4 ,thus1 in this rotation of a four-dimensional body1 two axes change and two remain at rest4 ,four-dimensional rotation is therefore a turning about a plane4 ,as in the case of a plane being1 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 line1 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 plane4 ,the plane would be the axis of the rotation1 and the path of the body between its two appearances would be unimaginable in three-dimensional space4 ,illustration3 ,fig4 #c 7#aca74 ,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 plane4 ,fig4 #c represents a cube in our space1 the three axes x1 y1 z denoting its three dimensions4 ,let w represent the fourth dimension4 ,now1 since in our space we can represent any three dimensions1 we can1 if we choose1 make a representation of what is in the space determined by the three axes x1 z1 w4 ,this is a three-dimensional space determined by two of the axes we have drawn1 x and z1 and in place of y the fourth axis1 w4 ,we cannot1 keeping x and z1 have both y and w in our space2 so we will let y go and draw w in its place4 ,what will be our view of the cube8 ,evidently we shall have simply the square that is in the plane of xz1 the square ,,acdb4 ,the rest of the cube stretches in the y direction1 and1 as we have none of the space so determined1 we have only the face of the cube4 ,this is represented in fig4 #d4 ,illustration3 ,fig4 #d 7#acb74 ,now1 suppose the whole cube to be turned from the x to the w direction4 ,conformably with our method1 we will not take the whole of the cube into consideration at once1 but will begin with the face ,,abcd4 ,let this face begin to turn4 ,fig4 #e represents one of the positions it will occupy2 the line ,,ab remains on the z axis4 ,the rest of the face extends between the x and the w direction4 ,illustration3 ,fig4 #e 7#acc74 ,now1 since we can take any three axes1 let us look at what lies in the space of zyw1 and examine the turning there4 ,we must now let the z axis disappear and let the w axis run in the direction in which the z ran4 ,making this representation1 what do we see of the cube8 ,obviously we see only the lower face4 ,the rest of the cube lies in the space of xyz4 ,in the space of xyz we have merely the base of the cube lying in the plane of xy1 as shown in fig4 #f4 ,illustration3 ,fig4 #f 7#acd74 ,now let the x to w turning take place4 ,the square ,,aceg will turn about the line ,,ae4 ,this edge will remain along the y axis and will be stationary1 however far the square turns4 ,thus1 if the cube be turned by an x to w turning1 both the edge ,,ab and the edge ,,ac remain stationary2 hence the whole face ,,abef in the yz plane remains fixed4 ,the turning has taken place about the face ,,abef4 ,illustration3 ,fig4 #g 7#ace74 ,suppose this turning to continue till ,,ac runs to the left from ,a4 ,the cube will occupy the position shown in fig4 #h4 ,this is the looking-glass image of the cube in fig4 #c4 ,by no rotation in three-dimensional space can the cube be brought from the position in fig4 #c to that shown in fig4 #h4 ,illustration3 ,fig4 #h 7#acf74 ,we can think of this turning as a turning of the face ,,abcd about ,,ab1 and a turning of each section parallel to ,,abcd round the vertical line in which it intersects the face ,,abef1 the space in which the turning takes place being a different one from that in which the cube lies4 ,one of the conditions1 then1 of our inquiry in the direction of the infinitely small is that we form the conception of a rotation about a plane4 ,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 rotation4 ,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 polarisation2 but this is not a point to which any very great importance can be attached4 ,still1 in this connection1 let me quote a remark from ,prof4 ,john ,g4 ,mc,kendrick's address on ,physiology before the ,british ,association at ,glasgow4 ,discussing the possibility of the hereditary production of characteristics through the material structure of the ovum1 he estimates that in it there exist #ab1jjj1jjj1jjj biophors1 or ultimate particles of living matter1 a sufficient number to account for hereditary transmission1 and observes3 8,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 matter4 ,it may be different in kind from some of the motions known to physicists1 and it is conceivable that life may be the transmission to dead matter1 the molecules of which have already a special kind of motion1 of a form of motion sui generis40 ,now1 in the realm of organic beings symmetrical structures--those with a right and left symmetry--are everywhere in evidence4 ,granted that four dimensions exist1 the simplest turning produces the image form1 and by a folding-over structures could be produced1 duplicated right and left1 just as is the case of symmetry in a plane4 ,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 life4 ,but whether four-dimensional motions correspond in other respects to the physiologist's demand for a special kind of motion1 or not1 ,i do not know4 ,our business is with the evidence for their existence in physics4 ,for this purpose it is necessary to examine into the significance of rotation round a plane in the case of extensible and of fluid matter4 ,let us dwell a moment longer on the rotation of a rigid body4 ,looking at the cube in fig4 #c1 which turns about the face of ,,abfe1 we see that any line in the face can take the place of the vertical and horizontal lines we have examined4 ,take the diagonal line ,,af and the section through it to ,,gh4 ,the portions of matter which were on one side of ,,af in this section in fig4 #c are on the opposite side of it in fig4 #h4 ,they have gone round the line ,,af4 ,thus the rotation round a face can be considered as a number of rotations of sections round parallel lines in it4 ,the turning about two different lines is impossible in three-dimensional space4 ,to take another illustration1 suppose ,a and ,b are two parallel lines in the xy plane1 and let ,,cd and ,,ef be two rods crossing them4 ,now1 in the space of xyz if the rods turn round the lines ,a and ,b in the same direction they will make two independent circles4 ,when the end ,f is going down the end ,c will be coming up4 ,they will meet and conflict4 ,illustration3 ,fig4 #i 7#acg74 ,but if we rotate the rods about the plane of ,,ab by the z to w rotation these movements will not conflict4 ,suppose all the figure removed with the exception of the plane xz1 and from this plane draw the axis of w1 so that we are looking at the space of xzw4 ,here1 fig4 #aj1 we cannot see the lines ,a and ,b4 ,we see the points ,g and ,h1 in which ,a and ,b intercept the x axis1 but we cannot see the lines themselves1 for they run in the y direction1 and that is not in our drawing4 ,now1 if the rods move with the z to w rotation they will turn in parallel planes1 keeping their relative positions4 ,the point ,d1 for instance1 will describe a circle4 ,at one time it will be above the line ,a1 at another time below it4 ,hence it rotates round ,a4 ,illustration3 ,fig4 #aj 7#ach74 ,not only two rods but any number of rods crossing the plane will move round it harmoniously4 ,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 it1 the relative positions of all the rods being preserved4 ,now1 if the rods are thick together1 they may represent a disk of matter1 and we see that a disk of matter can rotate round a central plane4 ,rotation round a plane is exactly analogous to rotation round an axis in three dimensions4 ,if we want a rod to turn round1 the ends must be free2 so if we want a disk of matter to turn round its central plane by a four-dimensional turning1 all the contour must be free4 ,the whole contour corresponds to the ends of the rod4 ,each point of the contour can be looked on as the extremity of an axis in the body1 round each point of which there is a rotation of the matter in the disk4 ,if the one end of a rod be clamped1 we can twist the rod1 but not turn it round2 so if any part of the contour of a disk is clamped we can impart a twist to the disk1 but not turn it round its central plane4 ,in the case of extensible materials a long1 thin rod will twist round its axis1 even when the axis is curved1 as1 for instance1 in the case of a ring of ,india rubber4 ,in an analogous manner1 in four dimensions we can have rotation round a curved plane1 if ,i may use the expression4 ,a sphere can be turned inside out in four dimensions4 ,illustration3 ,fig4 #aa 7#aci74 ,let fig4 #aa represent a spherical surface1 on each side of which a layer of matter exists4 ,the thickness of the matter is represented by the rods ,,cd and ,,ef1 extending equally without and within4 ,illustration3 ,fig4 #ab 7#adj74 ,now1 take the section of the sphere by the yz plane we have a circle--fig4 #ab4 ,now1 let the w axis be drawn in place of the x axis so that we have the space of yzw represented4 ,in this space all that there will be seen of the sphere is the circle drawn4 ,here we see that there is no obstacle to prevent the rods turning round4 ,if the matter is so elastic that it will give enough for the particles at ,e and ,c to be separated as they are at ,f and ,d1 they can rotate round to the position ,d and ,f1 and a similar motion is possible for all other particles4 ,there is no matter or obstacle to prevent them from moving out in the w direction1 and then on round the circumference as an axis4 ,now1 what will hold for one section will hold for all1 as the fourth dimension is at right angles to all the sections which can be made of the sphere4 ,we have supposed the matter of which the sphere is composed to be three-dimensional4 ,if the matter had a small thickness in the fourth dimension1 there would be a slight thickness in fig4 #ab above the plane of the paper--a thickness equal to the thickness of the matter in the fourth dimension4 ,the rods would have to be replaced by thin slabs4 ,but this would make no difference as to the possibility of the rotation4 ,this motion is discussed by ,newcomb in the first volume of the ,american ,journal of ,mathematics4 ,let us now consider1 not a merely extensible body1 but a liquid one4 ,a mass of rotating liquid1 a whirl1 eddy1 or vortex1 has many remarkable properties4 ,on first consideration we should expect the rotating mass of liquid immediately to spread off and lose itself in the surrounding liquid4 ,the water flies off a wheel whirled round1 and we should expect the rotating liquid to be dispersed4 ,but see the eddies in a river strangely persistent4 ,the rings that occur in puffs of smoke and last so long are whirls or vortices curved round so that their opposite ends join together4 ,a cyclone will travel over great distances4 ,helmholtz was the first to investigate the properties of vortices4 ,he studied them as they would occur in a perfect fluid--that is1 one without friction of one moving portion or another4 ,in such a medium vortices would be indestructible4 ,they would go on for ever1 altering their shape1 but consisting always of the same portion of the fluid4 ,but a straight vortex could not exist surrounded entirely by the fluid4 ,the ends of a vortex must reach to some boundary inside or outside the fluid4 ,a vortex which is bent round so that its opposite ends join is capable of existing1 but no vortex has a free end in the fluid4 ,the fluid round the vortex is always in motion1 and one produces a definite movement in another4 ,lord ,kelvin has proposed the hypothesis that portions of a fluid segregated in vortices account for the origin of matter4 ,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 rigidity4 ,it is difficult to conceive1 however1 of any arrangement of the vortex rings and endless vortex filaments in the ether4 ,now1 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 thinkable4 ,to understand the nature of this vortex1 we must go on and take a step by which we accept the full significance of the four-dimensional hypothesis4 ,granted four-dimensional axes1 we have seen that a rotation of one into another leaves two unaltered1 and these two form the axial plane about which the rotation takes place4 ,but what about these two8 ,do they necessarily remain motionless8 ,there is nothing to prevent a rotation of these two1 one into the other1 taking place concurrently with the first rotation4 ,this possibility of a double rotation deserves the most careful attention1 for it is the kind of movement which is distinctly typical of four dimensions4 ,rotation round a plane is analogous to rotation round an axis4 ,but in three-dimensional space there is no motion analogous to the double rotation1 in which1 while axis #a changes into axis #b1 axis #c changes into axis #d4 ,consider a four-dimensional body1 with four independent axes1 x1 y1 z1 w4 ,a point in it can move in only one direction at a given moment4 ,if the body has a velocity of rotation by which the x axis changes into the y axis and all parallel sections move in a similar manner1 then the point will describe a circle4 ,if1 now1 in addition to the rotation by which the x axis changes into the y axis the body has a rotation by which the z axis turns into the w axis1 the point in question will have a double motion in consequence of the two turnings4 ,the motions will compound1 and the point will describe a circle1 but not the same circle which it would describe in virtue of either rotation separately4 ,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 axis4 ,it is in no different condition from that in which it is subjected to one movement of rotation4 ,the direction of the axis changes2 that is all4 ,the same is not true about a four-dimensional body4 ,the two rotations1 x to y and z to w1 are independent4 ,a body subject to the two is in a totally different condition to that which it is in when subject to one only4 ,when subject to a rotation such as that of x to y1 a whole plane in the body1 as we have seen1 is stationary4 ,when subject to the double rotation no part of the body is stationary except the point common to the two planes of rotation4 ,if the two rotations are equal in velocity1 every point in the body describes a circle4 ,all points equally distant from the stationary point describe circles of equal size4 ,we can represent a four-dimensional sphere by means of two diagrams1 in one of which we take the three axes1 x1 y1 z2 in the other the axes x1 w1 and z4 ,in fig4 #ac we have the view of a four-dimensional sphere in the space of xyz4 ,fig4 #ac shows all that we can see of the four sphere in the space of xyz1 for it represents all the points in that space1 which are at an equal distance from the centre4 ,let us now take the xz section1 and let the axis of w take the place of the y axis4 ,here1 in fig4 #ad1 we have the space of xzw4 ,in this space we have to take all the points which are at the same distance from the centre1 consequently we have another sphere4 ,if we had a three-dimensional sphere1 as has been shown before1 we should have merely a circle in the xzw space1 the xz circle seen in the space of xzw4 ,but now1 taking the view in the space of xzw1 we have a sphere in that space also4 ,in a similar manner1 whichever set of three axes we take1 we obtain a sphere4 ,illustration3 ,showing axes xyz ,fig4 #ac 7#ada74 ,illustration3 ,showing axes xwz ,fig4 #ad 7#adb74 ,in fig4 #ac1 let us imagine the rotation in the direction xy to be taking place4 ,the point x will turn to y1 and p to p4 ,the axis zz remains stationary1 and this axis is all of the plane zw which we can see in the space section exhibited in the figure4 ,in fig4 #ad1 imagine the rotation from z to w to be taking place4 ,the w axis now occupies the position previously occupied by the y axis4 ,this does not mean that the w axis can coincide with the y axis4 ,it indicates that we are looking at the four-dimensional sphere from a different point of view4 ,any three-space view will show us three axes1 and in fig4 #ad we are looking at xzw4 ,the only part that is identical in the two diagrams is the circle of the x and z axes1 which axes are contained in both diagrams4 ,thus the plane zxz is the same in both1 and the point p represents the same point in both diagrams4 ,now1 in fig4 #ad let the zw rotation take place1 the z axis will turn toward the point w of the w axis1 and the point p will move in a circle about the point x4 ,thus in fig4 #ac the point p moves in a circle parallel to the xy plane2 in fig4 #ad it moves in a circle parallel to the zw plane1 indicated by the arrow4 ,now1 suppose both of these independent rotations compounded1 the point p will move in a circle1 but this circle will coincide with neither of the circles in which either one of the rotations will take it4 ,the circle the point p will move in will depend on its position on the surface of the four sphere4 ,in this double rotation1 possible in four-dimensional space1 there is a kind of movement totally unlike any with which we are familiar in three-dimensional space4 ,it is a requisite preliminary to the discussion of the behaviour of the small particles of matter1 with a view to determining whether they show the characteristics of four-dimensional movements1 to become familiar with the main characteristics of this double rotation4 ,and here ,i must rely on a formal and logical assent rather than on the intuitive apprehension1 which can only be obtained by a more detailed study4 ,in the first place this double rotation consists in two varieties or kinds1 which we will call the ,a and ,b kinds4 ,consider four axes1 x1 y1 z1 w4 ,the rotation of x to y can be accompanied with the rotation of z to w4 ,call this the ,a kind4 ,but also the rotation of x to y can be accompanied by the rotation1 of not z to w1 but w to z4 ,call this the ,b kind4 ,they differ in only one of the component rotations4 ,one is not the negative of the other4 ,it is the semi-negative4 ,the opposite of an x to y1 z to w rotation would be y to x1 w to z4 ,the semi-negative is x to y and w to z4 ,if four dimensions exist and we cannot perceive them1 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-dimensional1 we should not observe these double rotations1 but only the effects of them in three-dimensional movements of the type with which we are familiar4 ,if matter in its small particles is four-dimensional1 we should expect this double rotation to be a universal characteristic of the atoms and molecules1 for no portion of matter is at rest4 ,the consequences of this corpuscular motion can be perceived1 but only under the form of ordinary rotation or displacement4 ,thus1 if the theory of four dimensions is true1 we have in the corpuscles of matter a whole world of movement1 which we can never study directly1 but only by means of inference4 ,the rotation ,a1 as ,i have defined it1 consists of two equal rotations--one about the plane of zw1 the other about the plane of xy4 ,it is evident that these rotations are not necessarily equal4 ,a body may be moving with a double rotation1 in which these two independent components are not equal2 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 and1 in addition1 a rotation about a plane4 ,if we combine an ,a and a ,b movement1 we obtain a rotation about a plane2 for1 the first being x to y and z to w1 and the second being x to y and w to z1 when they are put together the z to w and w to z rotations neutralise each other1 and we obtain an x to y rotation only1 which is a rotation about the plane of zw4 ,similarly1 if we take a ,b rotation1 y to x and z to w1 we get1 on combining this with the ,a rotation1 a rotation of z to w about the xy plane4 ,in this case the plane of rotation is in the three-dimensional space of xyz1 and we have--what has been described before--a twisting about a plane in our space4 ,consider now a portion of a perfect liquid having an ,a motion4 ,it can be proved that it possesses the properties of a vortex4 ,it forms a permanent individuality--a separated-out portion of the liquid--accompanied by a motion of the surrounding liquid4 ,it has properties analogous to those of a vortex filament4 ,but it is not necessary for its existence that its ends should reach the boundary of the liquid4 ,it is self-contained and1 unless disturbed1 is circular in every section4 ,illustration3 ,fig4 #ae 7#adc74 ,if we suppose the ether to have its properties of transmitting vibration given it by such vortices1 we must inquire how they lie together in four-dimensional space4 ,placing a circular disk on a plane and surrounding it by six others1 we find that if the central one is given a motion of rotation1 it imparts to the others a rotation which is antagonistic in every two adjacent ones4 ,if ,a goes round1 as shown by the arrow1 ,b and ,c will be moving in opposite ways1 and each tends to destroy the motion of the other4 ,now1 if we suppose spheres to be arranged in a corresponding manner in three-dimensional space1 they will be grouped in figures which are for three-dimensional space what hexagons are for plane space4 ,if a number of spheres of soft clay be pressed together1 so as to fill up the interstices1 each will assume the form of a fourteen-sided figure called a tetrakaidecagon4 ,now1 assuming space to be filled with such tetrakaidecagons1 and placing a sphere in each1 it will be found that one sphere is touched by eight others4 ,the remaining six spheres of the fourteen which surround the central one will not touch it1 but will touch three of those in contact with it4 ,hence1 if the central sphere rotates1 it will not necessarily drive those around it so that their motions will be antagonistic to each other1 but the velocities will not arrange themselves in a systematic manner4 ,in four-dimensional space the figure which forms the next term of the series hexagon1 tetrakaidecagon1 is a thirty-sided figure4 ,it has for its faces ten solid tetrakaidecagons and twenty hexagonal prisms4 ,such figures will exactly fill four-dimensional space1 five of them meeting at every point4 ,if1 now1 in each of these figures we suppose a solid four-dimensional sphere to be placed1 any one sphere is surrounded by thirty others4 ,of these it touches ten1 and1 if it rotates1 it drives the rest by means of these4 ,now1 if we imagine the central sphere to be given an ,a or a ,b rotation1 it will turn the whole mass of sphere round in a systematic manner4 ,suppose four-dimensional space to be filled with such spheres1 each rotating with a double rotation1 the whole mass would form one consistent system of motion1 in which each one drove every other one1 with no friction or lagging behind4 ,every sphere would have the same kind of rotation4 ,in three-dimensional space1 if one body drives another round the second body rotates with the opposite kind of rotation2 but in four-dimensional space these four-dimensional spheres would each have the double negative of the rotation of the one next it1 and we have seen that the double negative of an ,a or ,b rotation is still an ,a or ,b rotation4 ,thus four-dimensional space could be filled with a system of self-preservative living energy4 ,if we imagine the four-dimensional spheres to be of liquid and not of solid matter1 then1 even if the liquid were not quite perfect and there were a slight retarding effect of one vortex on another1 the system would still maintain itself4 ,in this hypothesis we must look on the ether as possessing energy1 and its transmission of vibrations1 not as the conveying of a motion imparted from without1 but as a modification of its own motion4 ,we are now in possession of some of the conceptions of four-dimensional mechanics1 and will turn aside from the line of their development to inquire if there is any evidence of their applicability to the processes of nature4 ,is there any mode of motion in the region of the minute which1 giving three-dimensional movements for its effect1 still in itself escapes the grasp of our mechanical theories8 ,i would point to electricity4 ,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 medium2 but there is still a gap to be bridged over in their explanation--the laws of elasticity1 which ,maxwell assumes1 are not those of ordinary matter4 ,and1 to take another instance3 a magnetic pole in the neighbourhood of a current tends to move4 ,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 current3 but we cannot point out the definite mechanical explanation of these pressures4 ,there must be some mode of motion of a body or of the medium in virtue of which a body is said to be electrified4 ,take the ions which convey charges of electricity #ejj times greater in proportion to their mass than are carried by the molecules of hydrogen in electrolysis4 ,in respect of what motion can these ions be said to be electrified8 ,it can be shown that the energy they possess is not energy of rotation4 ,think of a short rod rotating4 ,if it is turned over it is found to be rotating in the opposite direction4 ,now1 if rotation in one direction corresponds to positive electricity1 rotation in the opposite direction corresponds to negative electricity1 and the smallest electrified particles would have their charges reversed by being turned over--an absurd supposition4 ,if we fix on a mode of motion as a definition of electricity1 we must have two varieties of it1 one for positive and one for negative2 and a body possessing the one kind must not become possessed of the other by any change in its position4 ,all three-dimensional motions are compounded of rotations and translations1 and none of them satisfy this first condition for serving as a definition of electricity4 ,but consider the double rotation of the ,a and ,b kinds4 ,a body rotating with the ,a motion cannot have its motion transformed into the ,b kind by being turned over in any way4 ,suppose a body has the rotation x to y and z to w4 ,turning it about the xy plane1 we reverse the direction of the motion x to y4 ,but we also reverse the z to w motion1 for the point at the extremity of the positive z axis is now at the extremity of the negative z axis1 and since we have not interfered with its motion it goes in the direction of position w4 ,hence we have y to x and w to z1 which is the same as x to y and z to w4 ,thus both components are reversed1 and there is the ,a motion over again4 ,the ,b kind is the semi-negative1 with only one component reversed4 ,hence a system of molecules with the ,a motion would not destroy it in one another1 and would impart it to a body in contact with them4 ,thus ,a and ,b motions possess the first requisite which must be demanded in any mode of motion representative of electricity4 ,let us trace out the consequences of defining positive electricity as an ,a motion and negative electricity as a ,b motion4 ,the combination of positive and negative electricity produces a current4 ,imagine a vortex in the ether of the ,a kind and unite with this one of the ,b kind4 ,an ,a motion and ,b motion produce rotation round a plane1 which is in the ether a vortex round an axial surface4 ,it is a vortex of the kind we represent as a part of a sphere turning inside out4 ,now such a vortex must have its rim on a boundary of the ether--on a body in the ether4 ,let us suppose that a conductor is a body which has the property of serving as the terminal abutment of such a vortex4 ,then the conception we must form of a closed current is of a vortex sheet having its edge along the circuit of the conducting wire4 ,the whole wire will then be like the centres on which a spindle turns in three-dimensional space1 and any interruption of the continuity of the wire will produce a tension in place of a continuous revolution4 ,as the direction of the rotation of the vortex is from a three-space direction into the fourth dimension and back again1 there will be no direction of flow to the current2 but it will have two sides1 according to whether z goes to w or z goes to negative w4 ,we can draw any line from one part of the circuit to another2 then the ether along that line is rotating round its points4 ,this geometric image corresponds to the definition of an electric circuit4 ,it is known that the action does not lie in the wire1 but in the medium1 and it is known that there is no direction of flow in the wire4 ,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 boundary1 as is the case in an electric current4 ,but this phenomenon corresponds exactly to the definition of a four-dimensional vortex4 ,if we take a very long magnet1 so long that one of its poles is practically isolated1 and put this pole in the vicinity of an electric circuit1 we find that it moves4 ,now1 assuming for the sake of simplicity that the wire which determines the current is in the form of a circle1 if we take a number of small magnets and place them all pointing in the same direction normal to the plane of the circle1 so that they fill it and the wire binds them round1 we find that this sheet of magnets has the same effect on the magnetic pole that the current has4 ,the sheet of magnets may be curved1 but the edge of it must coincide with the wire4 ,the collection of magnets is then equivalent to the vortex sheet1 and an elementary magnet to a part of it4 ,thus1 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 poles4 ,if a current is started in a circuit1 we must imagine vortices like bowls turning themselves inside out1 starting from the contour4 ,in reaching a parallel circuit1 if the vortex sheet were interrupted and joined momentarily to the second circuit by a free rim1 the axis plane would lie between the two circuits1 and a point on the second circuit opposite a point on the first would correspond to a point opposite to it on the first2 hence we should expect a current in the opposite direction in the second circuit4 ,thus the phenomena of induction are not inconsistent with the hypothesis of a vortex about an axial plane4 ,in four-dimensional space1 in which all four dimensions were commensurable1 the intensity of the action transmitted by the medium would vary inversely as the cube of the distance4 ,now1 the action of a current on a magnetic pole varies inversely as the square of the distance2 hence1 over measurable distances the extension of the ether in the fourth dimension cannot be assumed as other than small in comparison with those distances4 ,if we suppose the ether to be filled with vortices in the shape of four-dimensional spheres rotating with the ,a motion1 the ,b motion would correspond to electricity in the one-fluid theory4 ,there would thus be a possibility of electricity existing in two forms1 statically1 by itself1 and1 combined with the universal motion1 in the form of a current4 ,to arrive at a definite conclusion it will be necessary to investigate the resultant pressures which accompany the collocation of solid vortices with surface ones4 ,to recapitulate3 ,the movements and mechanics of four-dimensional space are definite and intelligible4 ,a vortex with a surface as its axis affords a geometric image of a closed circuit1 and there are rotations which by their polarity afford a possible definition of statical electricity4#g #g ,these double rotations of the ,a and ,b kinds ,i should like to call ,hamiltons and co-,hamiltons1 for it is a singular fact that in his 8,quaternions0 ,sir ,wm4 ,rowan ,hamilton has given the theory of either the ,a or the ,b kind4 ,they follow the laws of his symbols1 ,i1 ,j1 ,k4 ,hamiltons and co-,hamiltons seem to be natural units of geometrical expression4 ,in the paper in the 8,proceedings of the ,royal ,irish ,academy10 ,nov4 #aijc1 already alluded to1 ,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 notation1 which enables his system to be applied to both the ,a and ,b kinds of rotations4 ,the objection which has been often made to ,hamilton's system1 namely1 that it is only under special conditions of application that his processes give geometrically interpretable results1 can be removed1 if we assume that he was really dealing with a four-dimensional motion1 and alter his notation to bring this circumstance into explicit recognition4 ,,appendix ,i ,,the ,,models ,in ,chapter ,,xi4 a description has been given which will enable any one to make a set of models illustrative of the tesseract and its properties4 ,the set here supposed to be employed consists of3-- #a4 ,three sets of twenty-seven cubes each4 #b4 ,twenty-seven slabs4 #c4 ,twelve cubes with points1 lines1 faces1 distinguished by colours1 which will be called the catalogue cubes4 ,the preparation of the twelve catalogue cubes involves the expenditure of a considerable amount of time4 ,it is advantageous to use them1 but they can be replaced by the drawing of the views of the tesseract or by a reference to figs4 #ajc1 #ajd1 #aje1 #ajf of the text4 ,the slabs are coloured like the twenty-seven cubes of the first cubic block in fig4 #aja1 the one with red1 white1 yellow axes4 ,the colours of the three sets of twenty-seven cubes are those of the cubes shown in fig4 #aja4 ,the slabs are used to form the representation of a cube in a plane1 and can well be dispensed with by any one who is accustomed to deal with solid figures4 ,but the whole theory depends on a careful observation of how the cube would be represented by these slabs4 ,in the first step1 that of forming a clear idea how a plane being would represent three-dimensional space1 only one of the catalogue cubes and one of the three blocks is needed4 ,,application ,,to ,,the ,,step ,,from ,,plane ,,to ,,solid4 ,look at fig4 #a of the views of the tesseract1 or1 what comes to the same thing1 take catalogue cube ,no4 #a and place it before you with the red line running up1 the white line running to the right1 the yellow line running away4 ,the three dimensions of space are then marked out by these lines or axes4 ,now take a piece of cardboard1 or a book1 and place it so that it forms a wall extending up and down not opposite to you1 but running away parallel to the wall of the room on your left hand4 ,placing the catalogue cube against this wall we see that it comes into contact with it by the red and yellow lines1 and by the included orange face4 ,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 points4 ,now1 keeping the red line fixed1 turn the cube about it so that the yellow line goes out to the right1 and the white line comes into contact with the plane4 ,in this case a different aspect is presented to the plane being1 a square1 namely1 surrounded by red and white lines and grey points4 ,you should particularly notice that when the yellow line goes out1 at right angles to the plane1 and the white comes in1 the latter does not run in the same sense that the yellow did4 ,from the fixed grey point at the base of the red line the yellow line ran away from you4 ,the white line now runs towards you4 ,this turning at right angles makes the line which was out of the plane before1 come into it in an opposite sense to that in which the line ran which has just left the plane4 ,if the cube does not break through the plane this is always the rule4 ,again turn the cube back to the normal position with red running up1 white to the right1 and yellow away1 and try another turning4 ,you can keep the yellow line fixed1 and turn the cube about it4 ,in this case the red line going out to the right the white line will come in pointing downwards4 ,you will be obliged to elevate the cube from the table in order to carry out this turning4 ,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 below4 ,having looked at the three ways of turning the cube so as to present different faces to the plane1 examine what would be the appearance if a square hole were cut in the piece of cardboard1 and the cube were to pass through it4 ,a hole can be actually cut1 and it will be seen that in the normal position1 with red axis running up1 yellow away1 and white to the right1 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 face4 ,the line above is light yellow1 below is light yellow and on the opposite side away from you is pink4 ,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 into4 ,in each case the plane being will perceive a different set of contour lines4 ,having observed these facts about the catalogue cube1 turn now to the first block of twenty-seven cubes4 ,you notice that the colour scheme on the catalogue cube and that of this set of blocks is the same4 ,place them before you1 a grey or null cube on the table1 above it a red cube1 and on the top a null cube again4 ,then away from you place a yellow cube1 and beyond it a null cube4 ,then to the right place a white cube and beyond it another null4 ,then complete the block1 according to the scheme of the catalogue cube1 putting in the centre of all an ochre cube4 ,you have now a cube like that which is described in the text4 ,for the sake of simplicity1 in some cases1 this cubic block can be reduced to one of eight cubes1 by leaving out the terminations in each direction4 ,thus1 instead of null1 red1 null1 three cubes1 you can take null1 red1 two cubes1 and so on4 ,it is useful1 however1 to practise the representation in a plane of a block of twenty-seven cubes4 ,for this purpose take the slabs1 and build them up against the piece of cardboard1 or the book in such a way as to represent the different aspects of the cube4 ,proceed as follows3-- ,first1 cube in normal position4 ,place nine slabs against the cardboard to represent the nine cubes in the wall of the red and yellow axes1 facing the cardboard2 these represent the aspect of the cube as it touches the plane4 ,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 being1 if it were to pass half way through the plane4 ,there would be a white slab1 above it a pink one1 above that another white one1 and six others1 representing what would be the nature of a section across the middle of the block of cubes4 ,the section can be thought of as a thin slice cut out by two parallel cuts across the cube4 ,having arranged these nine slabs1 push them along the plane1 and make another set of nine to represent what would be the appearance of the cube when it had almost completely gone through4 ,this set of nine will be the same as the first set of nine4 ,now we have in the plane three sets of nine slabs each1 which represent three sections of the twenty-seven block4 ,they are put alongside one another4 ,we see that it does not matter in what order the sets of nine are put4 ,as the cube passes through the plane they represent appearances which follow the one after the other4 ,if they were what they represented1 they could not exist in the same plane together4 ,this is a rather important point1 namely1 to notice that they should not co-exist on the plane1 and that the order in which they are placed is indifferent4 ,when we represent a four-dimensional body our solid cubes are to us in the same position that the slabs are to the plane being4 ,you should also notice that each of these slabs represents only the very thinnest slice of a cube4 ,the set of nine slabs first set up represents the side surface of the block4 ,it is1 as it were1 a kind of tray--a beginning from which the solid cube goes off4 ,the slabs as we use them have thickness1 but this thickness is a necessity of construction4 ,they are to be thought of as merely of the thickness of a line4 ,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 by3-- #a4 ,the first set of nine slabs lasting for one minute4 #b4 ,the second set of nine slabs lasting for one minute4 #c4 ,the third set of nine slabs lasting for one minute4 ,now the appearances which the cube would present to the plane being in other positions can be shown by means of these slabs4 ,the use of such slabs would be the means by which a plane being could acquire a familiarity with our cube4 ,turn the catalogue cube 7or imagine the coloured figure turned7 so that the red line runs up1 the yellow line out to the right1 and the white line towards you4 ,then turn the block of cubes to occupy a similar position4 ,the block has now a different wall in contact with the plane4 ,its appearance to a plane being will not be the same as before4 ,he has1 however1 enough slabs to represent this new set of appearances4 ,but he must remodel his former arrangement of them4 ,he must take a null1 a red1 and a null slab from the first of his sets of slabs1 then a white1 a pink1 and a white from the second1 and then a null1 a red1 and a null from the third set of slabs4 ,he takes the first column from the first set1 the first column from the second set1 and the first column from the third set4 ,to represent the half-way-through appearance1 which is as if a very thin slice were cut out half way through the block1 he must take the second column of each of his sets of slabs1 and to represent the final appearance1 the third column of each set4 ,now turn the catalogue cube back to the normal position1 and also the block of cubes4 ,there is another turning--a turning about the yellow line1 in which the white axis comes below the support4 ,you cannot break through the surface of the table1 so you must imagine the old support to be raised4 ,then the top of the block of cubes in its new position is at the level at which the base of it was before4 ,now representing the appearance on the plane1 we must draw a horizontal line to represent the old base4 ,the line should be drawn three inches high on the cardboard4 ,below this the representative slabs can be arranged4 ,it is easy to see what they are4 ,the old arrangements have to be broken up1 and the layers taken in order1 the first layer of each for the representation of the aspect of the block as it touches the plane4 ,then the second layers will represent the appearance half way through1 and the third layers will represent the final appearance4 ,it is evident that the slabs individually do not represent the same portion of the cube in these different presentations4 ,in the first case each slab represents a section or a face perpendicular to the white axis1 in the second case a face or a section which runs perpendicularly to the yellow axis1 and in the third case a section or a face perpendicular to the red axis4 ,but by means of these nine slabs the plane being can represent the whole of the cubic block4 ,he can touch and handle each portion of the cubic block1 there is no part of it which he cannot observe4 ,taking it bit by bit1 two axes at a time1 he can examine the whole of it4 ,,our ,,representation ,,of ,a ,,block ,,of ,,tesseracts4 ,look at the views of the tesseract #a1 #b1 #c1 or take the catalogue cubes #a1 #b1 #c1 and place them in front of you1 in any order1 say running from left to right1 placing #a in the normal position1 the red axis running up1 the white to the right1 and yellow away4 ,now notice that in catalogue cube #b the colours of each region are derived from those of the corresponding region of cube #a by the addition of blue4 ,thus null blue blue1 and the corners of number #b are blue4 ,again1 red blue purple1 and the vertical lines of #b are purple4 ,blue yellow green1 and the line which runs away is coloured green4 ,by means of these observations you may be sure that catalogue cube #b is rightly placed4 ,catalogue cube #c is just like number #a4 ,having these cubes in what we may call their normal position1 proceed to build up the three sets of blocks4 ,this is easily done in accordance with the colour scheme on the catalogue cubes4 ,the first block we already know4 ,build up the second block1 beginning with a blue corner cube1 placing a purple on it1 and so on4 ,having these three blocks we have the means of representing the appearances of a group of eighty-one tesseracts4 ,let us consider a moment what the analogy in the case of the plane being is4 ,he has his three sets of nine slabs each4 ,we have our three sets of twenty-seven cubes each4 ,our cubes are like his slabs4 ,as his slabs are not the things which they represent to him1 so our cubes are not the things they represent to us4 ,the plane being's slabs are to him the faces of cubes4 ,our cubes then are the faces of tesseracts1 the cubes by which they are in contact with our space4 ,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 out1 so our three blocks of cubes may be considered as three-space trays1 each of which is the beginning of an inch of the solid contents of the four-dimensional solids starting from them4 ,we want now to use the names null1 red1 white1 etc41 for tesseracts4 ,the cubes we use are only tesseract faces4 ,let us denote that fact by calling the cube of null colour1 null face2 or1 shortly1 null f41 meaning that it is the face of a tesseract4 ,to determine which face it is let us look at the catalogue cube #a or the first of the views of the tesseract1 which can be used instead of the models4 ,it has three axes1 red1 white1 yellow1 in our space4 ,hence the cube determined by these axes is the face of the tesseract which we now have before us4 ,it is the ochre face4 ,it is enough1 however1 simply to say null f41 red f4 for the cubes which we use4 ,to impress this in your mind1 imagine that tesseracts do actually run from each cube4 ,then1 when you move the cubes about1 you move the tesseracts about with them4 ,you move the face but the tesseract follows with it1 as the cube follows when its face is shifted in a plane4 ,the cube null in the normal position is the cube which has in it the red1 yellow1 white axes4 ,it is the face having these1 but wanting the blue4 ,in this way you can define which face it is you are handling4 ,i will write an 8f40 after the name of each tesseract just as the plane being might call each of his slabs null slab1 yellow slab1 etc41 to denote that they were representations4 ,we have then in the first block of twenty-seven cubes1 the following--null f41 red f41 null f41 going up2 white f41 null f41 lying to the right1 and so on4 ,starting from the null point and travelling up one inch we are in the null region1 the same for the away and the right-hand directions4 ,and if we were to travel in the fourth dimension for an inch we should still be in a null region4 ,the tesseract stretches equally all four ways4 ,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 distance4 ,for anything less than an inch of their transverse motion we should still have the same appearance4 ,you must notice1 however1 that we should not have null face after the motion had begun4 ,when the tesseract1 null for instance1 had moved ever so little we should not have a face of null but a section of null in our space4 ,hence1 when we think of the motion across our space we must call our cubes tesseract sections4 ,thus on null passing across we should see first null f41 then null s41 and then1 finally1 null f4 again4 ,imagine now the whole first block of twenty-seven tesseracts to have moved tranverse to our space a distance of one inch4 ,then the second set of tesseracts1 which originally were an inch distant from our space1 would be ready to come in4 ,their colours are shown in the second block of twenty-seven cubes which you have before you4 ,these represent the tesseract faces of the set of tesseracts that lay before an inch away from our space4 ,they are ready now to come in1 and we can observe their colours4 ,in the place which null f4 occupied before we have blue f41 in place of red f4 we have purple f41 and so on4 ,each tesseract is coloured like the one whose place it takes in this motion with the addition of blue4 ,now if the tesseract block goes on moving at the rate of an inch a minute1 this next set of tesseracts will occupy a minute in passing across4 ,we shall see1 to take the null one for instance1 first of all null face1 then null section1 then null face again4 ,at the end of the second minute the second set of tesseracts has gone through1 and the third set comes in4 ,this1 as you see1 is coloured just like the first4 ,altogether1 these three sets extend three inches in the fourth dimension1 making the tesseract block of equal magnitude in all dimensions4 ,we have now before us a complete catalogue of all the tesseracts in our group4 ,we have seen them all1 and we shall refer to this arrangement of the blocks as the 8normal position40 ,we have seen as much of each tesseract at a time as could be done in a three-dimensional space4 ,each part of each tesseract has been in our space1 and we could have touched it4 ,the fourth dimension appeared to us as the duration of the block4 ,if a bit of our matter were to be subjected to the same motion it would be instantly removed out of our space4 ,being thin in the fourth dimension it is at once taken out of our space by a motion in the fourth dimension4 ,but the tesseract block we represent having length in the fourth dimension remains steadily before our eyes for three minutes1 when it is subjected to this transverse motion4 ,we have now to form representations of the other views of the same tesseract group which are possible in our space4 ,let us then turn the block of tesseracts so that another face of it comes into contact with our space1 and then by observing what we have1 and what changes come when the block traverses our space1 we shall have another view of it4 ,the dimension which appeared as duration before will become extension in one of our known dimensions1 and a dimension which coincided with one of our space dimensions will appear as duration4 ,leaving catalogue cube #a in the normal position1 remove the other two1 or suppose them removed4 ,we have in space the red1 the yellow1 and the white axes4 ,let the white axis go out into the unknown1 and occupy the position the blue axis holds4 ,then the blue axis1 which runs in that direction now will come into space4 ,but it will not come in pointing in the same way that the white axis does now4 ,it will point in the opposite sense4 ,it will come in running to the left instead of running to the right as the white axis does now4 ,when this turning takes place every part of the cube #a will disappear except the left-hand face--the orange face4 ,and the new cube that appears in our space will run to the left from this orange face1 having axes1 red1 yellow1 blue4 ,take models #d1 #e1 #f4 ,place #d1 or suppose ,no4 #d of the tesseract views placed1 with its orange face coincident with the orange face of #a1 red line to red line1 and yellow line to yellow line1 with the blue line pointing to the left4 ,then remove cube #a and we have the tesseract face which comes in when the white axis runs in the positive unknown1 and the blue axis comes into our space4 ,now place catalogue cube #e in some position1 it does not matter which1 say to the left2 and place it so that there is a correspondence of colour corresponding to the colour of the line that runs out of space4 ,the line that runs out of space is white1 hence1 every part of this cube #e should differ from the corresponding part of #d by an alteration in the direction of white4 ,thus we have white points in #e corresponding to the null points in #d4 ,we have a pink line corresponding to a red line1 a light yellow line corresponding to a yellow line1 an ochre face corresponding to an orange face4 ,this cube section is completely named in ,chapter ,,xi4 ,finally cube #f is a replica of #a4 ,these catalogue cubes will enable us to set up our models of the block of tesseracts4 ,first of all for the set of tesseracts1 which beginning in our space reach out one inch in the unknown1 we have the pattern of catalogue cube #d4 ,we see that we can build up a block of twenty-seven tesseract faces after the colour scheme of cube #d1 by taking the left-hand wall of block #a1 then the left-hand wall of block #b1 and finally that of block #c4 ,we take1 that is1 the three first walls of our previous arrangement to form the first cubic block of this new one4 ,this will represent the cubic faces by which the group of tesseracts in its new position touches our space4 ,we have running up1 null f41 red f41 null f4 ,in the next vertical line1 on the side remote from us1 we have yellow f41 orange f41 yellow f41 and then the first colours over again4 ,then the three following columns are1 blue f41 purple f41 blue f42 green f41 brown f41 green f42 blue f41 purple f41 blue f4 ,the last three columns are like the first4 ,these tesseracts touch our space1 and none of them are by any part of them distant more than an inch from it4 ,what lies beyond them in the unknown8 ,this can be told by looking at catalogue cube #e4 ,according to its scheme of colour we see that the second wall of each of our old arrangements must be taken4 ,putting them together we have1 as the corner1 white f4 above it1 pink f4 above it1 white f4 ,the column next to this remote from us is as follows3--light yellow f41 ochre f41 light yellow f41 and beyond this a column like the first4 ,then for the middle of the block1 light blue f41 above it light purple1 then light blue4 ,the centre column has1 at the bottom1 light green f41 light brown f4 in the centre and at the top light green f4 ,the last wall is like the first4 ,the third block is made by taking the third walls of our previous arrangement1 which we called the normal one4 ,you may ask what faces and what sections our cubes represent4 ,to answer this question look at what axes you have in our space4 ,you have red1 yellow1 blue4 ,now these determine brown4 ,the colours red1 yellow1 blue are supposed by us when mixed to produce a brown colour4 ,and that cube which is determined by the red1 yellow1 blue axes we call the brown cube4 ,when the tesseract block in its new position begins to move across our space each tesseract in it gives a section in our space4 ,this section is transverse to the white axis1 which now runs in the unknown4 ,as the tesseract in its present position passes across our space1 we should see first of all the first of the blocks of cubic faces we have put up--these would last for a minute1 then would come the second block and then the third4 ,at first we should have a cube of tesseract faces1 each of which would be brown4 ,directly the movement began1 we should have tesseract sections transverse to the white line4 ,there are two more analogous positions in which the block of tesseracts can be placed4 ,to find the third position1 restore the blocks to the normal arrangement4 ,let us make the yellow axis go out into the positive unknown1 and let the blue axis1 consequently1 come in running towards us4 ,the yellow ran away1 so the blue will come in running towards us4 ,put catalogue cube #a in its normal position4 ,take catalogue cube #g and place it so that its pink face coincides with the pink face of cube #a1 making also its red axis coincide with the red axis of #a and its white with the white4 ,moreover1 make cube #g come towards us from cube #a4 ,looking at it we see in our space1 red1 white1 and blue axes4 ,the yellow runs out4 ,place catalogue cube #h in the neighbourhood of #g--observe that every region in #h has a change in the direction of yellow from the corresponding region in #g4 ,this is because it represents what you come to now in going in the unknown1 when the yellow axis runs out of our space4 ,finally catalogue cube #i1 which is like number #g1 shows the colours of the third set of tesseracts4 ,now evidently1 starting from the normal position1 to make up our three blocks of tesseract faces we have to take the near wall from the first block1 the near wall from the second1 and then the near wall from the third block4 ,this gives us the cubic block formed by the faces of the twenty-seven tesseracts which are now immediately touching our space4 ,following the colour scheme of catalogue cube #h1 we make the next set of twenty-seven tesseract faces1 representing the tesseracts1 each of which begins one inch off from our space1 by putting the second walls of our previous arrangement together1 and the representation of the third set of tesseracts is the cubic block formed of the remaining three walls4 ,since we have red1 white1 blue axes in our space to begin with1 the cubes we see at first are light purple tesseract faces1 and after the transverse motion begins we have cubic sections transverse to the yellow line4 ,restore the blocks to the normal position1 there remains the case in which the red axis turns out of space4 ,in this case the blue axis will come in downwards1 opposite to the sense in which the red axis ran4 ,in this case take catalogue cubes #aj1 #aa1 #ab4 ,lift up catalogue cube #a and put #aj underneath it1 imagining that it goes down from the previous position of #a4 ,we have to keep in space the white and the yellow axes1 and let the red go out1 the blue come in4 ,now1 you will find on cube #aj a light yellow face2 this should coincide with the base of #a1 and the white and yellow lines on the two cubes should coincide4 ,then the blue axis running down you have the catalogue cube correctly placed1 and it forms a guide for putting up the first representative block4 ,catalogue cube #aa will represent what lies in the fourth dimension--now the red line runs in the fourth dimension4 ,thus the change from #aj to #aa should be towards red1 corresponding to a null point is a red point1 to a white line is a pink line1 to a yellow line an orange line1 and so on4 ,catalogue cube #ab is like #aj4 ,hence we see that to build up our blocks of tesseract faces we must take the bottom layer of the first block1 hold that up in the air1 underneath it place the bottom layer of the second block1 and finally underneath this last the bottom layer of the last of our normal blocks4 ,similarly we make the second representative group by taking the middle courses of our three blocks4 ,the last is made by taking the three topmost layers4 ,the three axes in our space before the transverse motion begins are blue1 white1 yellow1 so we have light green tesseract faces1 and after the motion begins sections transverse to the red light4 ,these three blocks represent the appearances as the tesseract group in its new position passes across our space4 ,the cubes of contact in this case are those determinal by the three axes in our space1 namely1 the white1 the yellow1 the blue4 ,hence they are light green4 ,it follows from this that light green is the interior cube of the first block of representative cubic faces4 ,practice in the manipulations described1 with a realization in each case of the face or section which is in our space1 is one of the best means of a thorough comprehension of the subject4 ,we have to learn how to get any part of these four-dimensional figures into space1 so that we can look at them4 ,we must first learn to swing a tesseract1 and a group of tesseracts about in any way4 ,when these operations have been repeated and the method of arrangement of the set of blocks has become familiar1 it is a good plan to rotate the axes of the normal cube #a about a diagonal1 and then repeat the whole series of turnings4 ,thus1 in the normal position1 red goes up1 white to the right1 yellow away4 ,make white go up1 yellow to the right1 and red away4 ,learn the cube in this position by putting up the set of blocks of the normal cube1 over and over again till it becomes as familiar to you as in the normal position4 ,then when this is learned1 and the corresponding changes in the arrangements of the tesseract groups are made1 another change should be made3 let1 in the normal cube1 yellow go up1 red to the right1 and white away4 ,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 goes4 ,then carry out all the tesseract arrangements and turnings4 ,if you want to understand the subject1 but do not see your way clearly1 if it does not seem natural and easy to you1 practise these turnings4 ,practise1 first of all1 the turning of a block of cubes round1 so that you know it in every position as well as in the normal one4 ,practise by gradually putting up the set of cubes in their new arrangements4 ,then put up the tesseract blocks in their arrangements4 ,this will give you a working conception of higher space1 you will gain the feeling of it1 whether you take up the mathematical treatment of it or not4 ,,appendix ,,ii ,a ,,language ,,of ,,space ,the mere naming the parts of the figures we consider involves a certain amount of time and attention4 ,this time and attention leads to no result1 for with each new figure the nomenclature applied is completely changed1 every letter or symbol is used in a different significance4 ,surely it must be possible in some way to utilise the labour thus at present wasted6 ,why should we not make a language for space itself1 so that every position we want to refer to would have its own name8 ,then every time we named a figure in order to demonstrate its properties we should be exercising ourselves in the vocabulary of place4 ,if we use a definite system of names1 and always refer to the same space position by the same name1 we create as it were a multitude of little hands1 each prepared to grasp a special point1 position1 or element1 and hold it for us in its proper relations4 ,we make1 to use another analogy1 a kind of mental paper1 which has somewhat of the properties of a sensitive plate1 in that it will register1 without effort1 complex1 visual1 or tactual impressions4 ,but of far more importance than the applications of a space language to the plane and to solid space is the facilitation it brings with it to the study of four-dimensional shapes4 ,i have delayed introducing a space language because all the systems ,i made turned out1 after giving them a fair trial1 to be intolerable4 ,i have now come upon one which seems to present features of permanence1 and ,i will here give an outline of it1 so that it can be applied to the subject of the text1 and in order that it may be subjected to criticism4 ,the principle on which the language is constructed is to sacrifice every other consideration for brevity4 ,it is indeed curious that we are able to talk and converse on every subject of thought except the fundamental one of space4 ,the only way of speaking about the spatial configurations that underlie every subject of discursive thought is a co-ordinate system of numbers4 ,this is so awkward and incommodious that it is never used4 ,in thinking also1 in realising shapes1 we do not use it2 we confine ourselves to a direct visualisation4 ,now1 the use of words corresponds to the storing up of our experience in a definite brain structure4 ,a child1 in the endless tactual1 visual1 mental manipulations it makes for itself1 is best left to itself1 but in the course of instruction the introduction of space names would make the teachers work more cumulative1 and the child's knowledge more social4 ,their full use can only be appreciated1 if they are introduced early in the course of education2 but in a minor degree any one can convince himself of their utility1 especially in our immediate subject of handling four-dimensional shapes4 ,the sum total of the results obtained in the preceding pages can be compendiously and accurately expressed in nine words of the ,space ,language4 ,in one of ,plato's dialogues ,socrates makes an experiment on a slave boy standing by4 ,he makes certain perceptions of space awake in the mind of ,meno's slave by directing his close attention on some simple facts of geometry4 ,by means of a few words and some simple forms we can repeat ,plato's experiment on new ground4 ,do we by directing our close attention on the facts of four dimensions awaken a latent faculty in ourselves8 ,the old experiment of ,plato's1 it seems to me1 has come down to us as novel as on the day he incepted it1 and its significance not better understood through all the discussion of which it has been the subject4 ,imagine a voiceless people living in a region where everything had a velvety surface1 and who were thus deprived of all opportunity of experiencing what sound is4 ,they could observe the slow pulsations of the air caused by their movements1 and arguing from analogy1 they would no doubt infer that more rapid vibrations were possible4 ,from the theoretical side they could determine all about these more rapid vibrations4 ,they merely differ1 they would say1 from slower ones1 by the number that occur in a given time2 there is a merely formal difference4 ,but suppose they were to take the trouble1 go to the pains of producing these more rapid vibrations1 then a totally new sensation would fall on their rudimentary ears4 ,probably at first they would only be dimly conscious of ,sound1 but even from the first they would become aware that a merely formal difference1 a mere difference in point of number in this particular respect1 made a great difference practically1 as related to them4 ,and to us the difference between three and four dimensions is merely formal1 numerical4 ,we can tell formally all about four dimensions1 calculate the relations that would exist4 ,but that the difference is merely formal does not prove that it is a futile and empty task1 to present to ourselves as closely as we can the phenomena of four dimensions4 ,in our formal knowledge of it1 the whole question of its actual relation to us1 as we are1 is left in abeyance4 ,possibly a new apprehension of nature may come to us through the practical1 as distinguished from the mathematical and formal1 study of four dimensions4 ,as a child handles and examines the objects with which he comes in contact1 so we can mentally handle and examine four-dimensional objects4 ,the point to be determined is this4 ,do we find something cognate and natural to our faculties1 or are we merely building up an artificial presentation of a scheme only formally possible1 conceivable1 but which has no real connection with any existing or possible experience8 ,this1 it seems to me1 is a question which can only be settled by actually trying4 ,this practical attempt is the logical and direct continuation of the experiment ,plato devised in the 8,meno40 ,why do we think true8 ,why1 by our processes of thought1 can we predict what will happen1 and correctly conjecture the constitution of the things around us8 ,this is a problem which every modern philosopher has considered1 and of which ,descartes1 ,leibnitz1 ,kant1 to name a few1 have given memorable solutions4 ,plato was the first to suggest it4 ,and as he had the unique position of being the first devisor of the problem1 so his solution is the most unique4 ,later philosophers have talked about consciousness and its laws1 sensations1 categories4 ,but ,plato never used such words4 ,consciousness apart from a conscious being meant nothing to him4 ,his was always an objective search4 ,he made man's intuitions the basis of a new kind of natural history4 ,in a few simple words ,plato puts us in an attitude with regard to psychic phenomena--the mind--the ego--8what we are10 which is analogous to the attitude scientific men of the present day have with regard to the phenomena of outward nature4 ,behind this first apprehension of ours of nature1 there is an infinite depth to be learned and known4 ,plato said that behind the phenomena of mind that ,meno's slave boy exhibited1 there was a vast1 an infinite perspective4 ,and his singularity1 his originality1 comes out most strongly marked in this1 that the perspective1 the complex phenomena beyond were1 according to him1 phenomena of personal experience4 ,a footprint in the sand means a man to a being that has the conception of a man4 ,but to a creature that has no such conception1 it means a curious mark1 somehow resulting from the concatenation of ordinary occurrences4 ,such a being would attempt merely to explain how causes known to him could so coincide as to produce such a result2 he would not recognise its significance4 ,plato introduced the conception which made a new kind of natural history possible4 ,he said that ,meno's slave boy thought true about things he had never learned1 because his 8soul0 had experience4 ,i know this will sound absurd to some people1 and it flies straight in the face of the maxim1 that explanation consists in showing how an effect depends on simple causes4 ,but what a mistaken maxim that is6 ,can any single instance be shown of a simple cause8 ,take the behaviour of spheres for instance2 say those ivory spheres1 billiard balls1 for example4 ,we can explain their behaviour by supposing they are homogeneous elastic solids4 ,we can give formul which will account for their movements in every variety4 ,but are they homogeneous elastic solids8 ,no1 certainly not4 ,they are complex in physical and molecular structure1 and atoms and ions beyond open an endless vista4 ,our simple explanation is false1 false as it can be4 ,the balls act as if they were homogeneous elastic spheres4 ,there is a statistical simplicity in the resultant of very complex conditions1 which makes that artificial conception useful4 ,but its usefulness must not blind us to the fact that it is artificial4 ,if we really look deep into nature1 we find a much greater complexity than we at first suspect4 ,and so behind this simple 8,i10 this myself1 is there not a parallel complexity8 ,plato's 8soul0 would be quite acceptable to a large class of thinkers1 if by 8soul0 and the complexity he attributes to it1 he meant the product of a long course of evolutionary changes1 whereby simple forms of living matter endowed with rudimentary sensation had gradually developed into fully conscious beings4 ,but ,plato does not mean by 8soul0 a being of such a kind4 ,his soul is a being whose faculties are clogged by its bodily environment1 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 organs4 ,at the same time ,plato's soul is not incorporeal4 ,it is a real being with a real experience4 ,the question of whether ,plato had the conception of non-spatial existence has been much discussed4 ,the verdict is1 ,i believe1 that even his 8ideas0 were conceived by him as beings in space1 or1 as we should say1 real4 ,plato's attitude is that of ,science1 inasmuch as he thinks of a world in ,space4 ,but1 granting this1 it cannot be denied that there is a fundamental divergence between ,plato's conception and the evolutionary theory1 and also an absolute divergence between his conception and the genetic account of the origin of the human faculties4 ,the functions and capacities of ,plato's 8soul0 are not derived by the interaction of the body and its environment4 ,plato was engaged on a variety of problems1 and his religious and ethical thoughts were so keen and fertile that the experimental investigation of his soul appears involved with many other motives4 ,in one passage ,plato will combine matter of thought of all kinds and from all sources1 overlapping1 interrunning4 ,and in no case is he more involved and rich than in this question of the soul4 ,in fact1 ,i wish there were two words1 one denoting that being1 corporeal and real1 but with higher faculties than we manifest in our bodily actions1 which is to be taken as the subject of experimental investigation2 and the other word denoting 8soul0 in the sense in which it is made the recipient and the promise of so much that men desire4 ,it is the soul in the former sense that ,i wish to investigate1 and in a limited sphere only4 ,i wish to find out1 in continuation of the experiment in the ,meno1 what the 8soul0 in us thinks about extension1 experimenting on the grounds laid down by ,plato4 ,he made1 to state the matter briefly1 the hypothesis with regard to the thinking power of a being in us1 a 8soul40 ,this soul is not accessible to observation by sight or touch1 but it can be observed by its functions2 it is the object of a new kind of natural history1 the materials for constructing which lie in what it is natural to us to think4 ,with ,plato 8thought0 was a very wide-reaching term1 but still ,i would claim in his general plan of procedure a place for the particular question of extension4 ,the problem comes to be1 8,what is it natural to us to think about matter qua extended80 ,first of all1 ,i find that the ordinary intuition of any simple object is extremely imperfect4 ,take a block of differently marked cubes1 for instance1 and become acquainted with them in their positions4 ,you may think you know them quite well1 but when you turn them round--rotate the block round a diagonal1 for instance--you will find that you have lost track of the individuals in their new positions4 ,you can mentally construct the block in its new position1 by a rule1 by taking the remembered sequences1 but you don't know it intuitively4 ,by observation of a block of cubes in various positions1 and very expeditiously by a use of ,space names applied to the cubes in their different presentations1 it is possible to get an intuitive knowledge of the block of cubes1 which is not disturbed by any displacement4 ,now1 with regard to this intuition1 we moderns would say that ,i had formed it by my tactual visual experiences 7aided by hereditary pre-disposition74 ,plato would say that the soul had been stimulated to recognise an instance of shape which it knew4 ,plato would consider the operation of learning merely as a stimulus2 we as completely accounting for the result4 ,the latter is the more common-sense view4 ,but1 on the other hand1 it presupposes the generation of experience from physical changes4 ,the world of sentient experience1 according to the modern view1 is closed and limited2 only the physical world is ample and large and of ever-to-be-discovered complexity4 ,plato's world of soul1 on the other hand1 is at least as large and ample as the world of things4 ,let us now try a crucial experiment4 ,can ,i form an intuition of a four-dimensional object8 ,such an object is not given in the physical range of my sense contacts4 ,all ,i can do is to present to myself the sequences of solids1 which would mean the presentation to me under my conditions of a four-dimensional object4 ,all ,i can do is to visualise and tactualise different series of solids which are alternative sets of sectional views of a four-dimensional shape4 ,if now1 on presenting these sequences1 ,i find a power in me of intuitively passing from one of these sets of sequences to another1 of1 being given one1 intuitively constructing another1 not using a rule1 but directly apprehending it1 then ,i have found a new fact about my soul1 that it has a four-dimensional experience2 ,i have observed it by a function it has4 ,i do not like to speak positively1 for ,i might occasion a loss of time on the part of others1 if1 as may very well be1 ,i am mistaken4 ,but for my own part1 ,i think there are indications of such an intuition2 from the results of my experiments1 ,i adopt the hypothesis that that which thinks in us has an ample experience1 of which the intuitions we use in dealing with the world of real objects are a part2 of which experience1 the intuition of four-dimensional forms and motions is also a part4 ,the process we are engaged in intellectually is the reading the obscure signals of our nerves into a world of reality1 by means of intuitions derived from the inner experience4 ,the image ,i form is as follows4 ,imagine the captain of a modern battle-ship directing its course4 ,he has his charts before him2 he is in communication with his associates and subordinates2 can convey his messages and commands to every part of the ship1 and receive information from the conning-tower and the engine-room4 ,now suppose the captain immersed in the problem of the navigation of his ship over the ocean1 to have so absorbed himself in the problem of the direction of his craft over the plane surface of the sea that he forgets himself4 ,all that occupies his attention is the kind of movement that his ship makes4 ,the operations by which that movement is produced have sunk below the threshold of his consciousness1 his own actions1 by which he pushes the buttons1 gives the orders1 are so familiar as to be automatic1 his mind is on the motion of the ship as a whole4 ,in such a case we can imagine that he identifies himself with his ship2 all that enters his conscious thought is the direction of its movement over the plane surface of the ocean4 ,such is the relation1 as ,i imagine it1 of the soul to the body4 ,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 craft4 ,as the captain is capable of a kind of movement1 an amplitude of motion1 which does not enter into his thoughts with regard to the directing the ship over the plane surface of the ocean1 so the soul is capable of a kind of movement1 has an amplitude of motion1 which is not used in its task of directing the body in the three-dimensional region in which the body's activity lies4 ,if for any reason it became necessary for the captain to consider three-dimensional motions with regard to his ship1 it would not be difficult for him to gain the materials for thinking about such motions2 all he has to do is to call his own intimate experience into play4 ,as far as the navigation of the ship1 however1 is concerned1 he is not obliged to call on such experience4 ,the ship as a whole simply moves on a surface4 ,the problem of three-dimensional movement does not ordinarily concern its steering4 ,and thus with regard to ourselves all those movements and activities which characterise our bodily organs are three-dimensional2 we never need to consider the ampler movements4 ,but we do more than use the movements of our body to effect our aims by direct means2 we have now come to the pass when we act indirectly on nature1 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 whole4 ,when we come to the problem of what goes on in the minute1 and apply ourselves to the mechanism of the minute1 we find our habitual conceptions inadequate4 ,the captain in us must wake up to his own intimate nature1 realise those functions of movement which are his own1 and in virtue of his knowledge of them apprehend how to deal with the problems he has come to4 ,think of the history of man4 ,when has there been a time1 in which his thoughts of form and movement were not exclusively of such varieties as were adapted for his bodily performance8 ,we have never had a demand to conceive what our own most intimate powers are4 ,but1 just as little as by immersing himself in the steering of his ship over the plane surface of the ocean1 a captain can lose the faculty of thinking about what he actually does1 so little can the soul lose its own nature4 ,it can be roused to an intuition that is not derived from the experience which the senses give4 ,all that is necessary is to present some few of those appearances which1 while inconsistent with three-dimensional matter1 are yet consistent with our formal knowledge of four-dimensional matter1 in order for the soul to wake up and not begin to learn1 but of its own intimate feeling fill up the gaps in the presentiment1 grasp the full orb of possibilities from the isolated points presented to it4 ,in relation to this question of our perceptions1 let me suggest another illustration1 not taking it too seriously1 only propounding it to exhibit the possibilities in a broad and general way4 ,in the heavens1 amongst the multitude of stars1 there are some which1 when the telescope is directed on them1 seem not to be single stars1 but to be split up into two4 ,regarding these twin stars through a spectroscope1 an astronomer sees in each a spectrum of bands of colour and black lines4 ,comparing these spectrums with one another1 he finds that there is a slight relative shifting of the dark lines1 and from that shifting he knows that the stars are rotating round one another1 and can tell their relative velocity with regard to the earth4 ,by means of his terrestrial physics he reads this signal of the skies4 ,this shifting of lines1 the mere slight variation of a black line in a spectrum1 is very unlike that which the astronomer knows it means4 ,but it is probably much more like what it means than the signals which the nerves deliver are like the phenomena of the outer world4 ,no picture of an object is conveyed through the nerves4 ,no picture of motion1 in the sense in which we postulate its existence1 is conveyed through the nerves4 ,the actual deliverances of which our consciousness takes account are probably identical for eye and ear1 sight and touch4 ,if for a moment ,i take the whole earth together and regard it as a sentient being1 ,i find that the problem of its apprehension is a very complex one1 and involves a long series of personal and physical events4 ,similarly the problem of our apprehension is a very complex one4 ,i only use this illustration to exhibit my meaning4 ,it has this especial merit1 that1 as the process of conscious apprehension takes place in our case in the minute1 so1 with regard to this earth being1 the corresponding process takes place in what is relatively to it very minute4 ,now1 ,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 event1 both physically and personally4 ,he does not seek to explain what an intuition is2 he makes it a basis from whence he sets out on a voyage of discovery4 ,knowledge means knowledge2 he puts conscious being to account for conscious being4 ,he makes an hypothesis of the kind that is so fertile in physical science--an hypothesis making no claim to finality1 which marks out a vista of possible determination behind determination1 like the hypothesis of space itself1 the type of serviceable hypotheses4 ,and1 above all1 ,plato's hypothesis is conducive to experiment4 ,he gives the perspective in which real objects can be determined2 and1 in our present enquiry1 we are making the simplest of all possible experiments--we are enquiring what it is natural to the soul to think of matter as extended4 ,aristotle says we always use a 8phantasm0 in thinking1 a phantasm of our corporeal senses a visualisation or a tactualisation4 ,but we can so modify that visualisation or tactualisation that it represents something not known by the senses4 ,do we by that representation wake up an intuition of the soul8 ,can we by the presentation of these hypothetical forms1 that are the subject of our present discussion1 wake ourselves up to higher intuitions8 ,and can we explain the world around by a motion that we only know by our souls8 ,apart from all speculation1 however1 it seems to me that the interest of these four-dimensional shapes and motions is sufficient reason for studying them1 and that they are the way by which we can grow into a fuller apprehension of the world as a concrete whole4 ,,space ,,names4 ,if the words written in the squares drawn in fig4 #a are used as the names of the squares in the positions in which they are placed1 it is evident that a combination of these names will denote a figure composed of the designated squares4 ,it is found to be most convenient to take as the initial square that marked with an asterisk1 so that the directions of progression are towards the observer and to his right4 ,the directions of progression1 however1 are arbitrary1 and can be chosen at will4 ,illustration3 ,fig4 #a4 ,thus et1 at1 it1 an1 al will denote a figure in the form of a cross composed of five squares4 ,here1 by means of the double sequence1 e1 a1 i and n1 t1 l1 it is possible to name a limited collection of space elements4 ,the system can obviously be extended by using letter sequences of more members4 ,but1 without introducing such a complexity1 the principles of a space language can be exhibited1 and a nomenclature obtained adequate to all the considerations of the preceding pages4 #a4 ,extension4 ,call the large squares in fig4 #b by the name written in them4 ,it is evident that each can be divided as shown in fig4 #a4 ,then the small square marked #a will be 8en0 in 8,en10 or 8,enen40 ,the square marked #b will be 8et0 in 8,en0 or 8,enet10 while the square marked #d will be 8en0 in 8,et0 or 8,eten40 ,thus the square #e will be called 8,ilil40 ,illustration3 ,fig4 #b4 ,this principle of extension can be applied in any number of dimensions4 #b4 ,application to ,three-,dimensional ,space4 ,to name a three-dimensional collocation of cubes take the upward direction first1 secondly the direction towards the observer1 thirdly the direction to his right hand4 ,illustration ,these form a word in which the first letter gives the place of the cube upwards1 the second letter its place towards the observer1 the third letter its place to the right4 ,we have thus the following scheme1 which represents the set of cubes of column #a1 fig4 #aja1 page #afe4 ,we begin with the remote lowest cube at the left hand1 where the asterisk is placed 7this proves to be by far the most convenient origin to take for the normal system74 ,thus 8nen0 is a 8null0 cube1 8ten0 a red cube on it1 and 8len0 a 8null0 cube above 8ten40 ,by using a more extended sequence of consonants and vowels a larger set of cubes can be named4 ,to name a four-dimensional block of tesseracts it is simply necessary to prefix an 8e10 an 8a10 or an 8i0 to the cube names4 ,thus the tesseract blocks schematically represented on page #afe1 fig4 #aja are named as follows3-- ,illustration3 #a #b #c #b4 ,,derivation ,,of ,,point1 ,,line1 ,,face1 ,,etc41 ,,names4 ,illustration ,the principle of derivation can be shown as follows3 ,taking the square of squares the number of squares in it can be enlarged and the whole kept the same size4 ,illustration ,compare fig4 #gi1 p4 #ach1 for instance1 or the bottom layer of fig4 #hd4 ,now use an initial 8s0 to denote the result of carrying this process on to a great extent1 and we obtain the limit names1 that is the point1 line1 area names for a square4 8,sat0 is the whole interior4 ,the corners are 8sen10 8sel10 8sin10 8sil10 while the lines are 8san10 8sal10 8set10 8sit40 ,illustration ,i find that by the use of the initial 8s0 these names come to be practically entirely disconnected with the systematic names for the square from which they are derived4 ,they are easy to learn1 and when learned can be used readily with the axes running in any direction4 ,to derive the limit names for a four-dimensional rectangular figure1 like the tesseract1 is a simple extension of this process4 ,these point1 line1 etc41 names include those which apply to a cube1 as will be evident on inspection of the first cube of the diagrams which follow4 ,all that is necessary is to place an 8s0 before each of the names given for a tesseract block4 ,we then obtain apellatives which1 like the colour names on page #agd1 fig4 #ajc1 apply to all the points1 lines1 faces1 solids1 and to the hyper-solid of the tesseract4 ,these names have the advantage over the colour marks that each point1 line1 etc41 has its own individual name4 ,in the diagrams ,i give the names corresponding to the positions shown in the coloured plate or described on p4 #agd4 ,by comparing cubes #a1 #b1 #c with the first row of cubes in the coloured plate1 the systematic names of each of the points1 lines1 faces1 etc41 can be determined4 ,the asterisk shows the origin from which the names run4 ,these point1 line1 face1 etc41 names should be used in connection with the corresponding colours4 ,the names should call up coloured images of the parts named in their right connection4 ,illustration ,it is found that a certain abbreviation adds vividness of distinction to these names4 ,if the final 8en0 be dropped wherever it occurs the system is improved4 ,thus instead of 8senen10 8seten10 8selen10 it is preferable to abbreviate to 8sen10 8set10 8sel10 and also use 8san10 8sin0 for 8sanen10 8sinen40 ,illustration ,illustration ,we can now name any section4 ,take e4g4 the line in the first cube from senin to senel1 we should call the line running from senin to senel1 senin senat senel1 a line light yellow in colour with null points4 ,illustration ,here senat is the name for all of the line except its ends4 ,using 8senat0 in this way does not mean that the line is the whole of senat1 but what there is of it is senat4 ,it is a part of the senat region4 ,thus also the triangle1 which has its three vertices in senin1 senel1 selen1 is named thus3 ,area3 setat4 ,sides3 setan1 senat1 setet4 ,vertices3 senin1 senel1 sel4 ,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 fig4 #aad1 p4 #aia4 ,in b#j the section is the one written above4 ,in b#a the section is made by a plane which cuts the three edges from sanen intermediate of their lengths and thus will be3 ,area3 satat4 ,sides3 satan1 sanat1 satet4 ,vertices3 sanan1 sanet1 sat4 ,the sections in b#b1 b#c will be like the section in b#a but smaller4 ,finally in b#d the section plane simply passes through the corner named sin4 ,hence1 putting these sections together in their right relation1 from the face setat1 surrounded by the lines and points mentioned above1 there run3 #c faces3 satan1 sanat1 satet #c lines3 sanan1 sanet1 sat and these faces and lines run to the point sin4 ,thus the tetrahedron is completely named4 ,the octahedron section of the tesseract1 which can be traced from fig4 #gb1 p4 #abi by extending the lines there drawn1 is named3 ,front triangle selin1 selat1 selel1 setal1 senil1 setit1 selin with area setat4 ,the sections between the front and rear triangle1 of which one is shown in #a;b1 another in #b;b1 are thus named1 points and lines1 salan1 salat1 salet1 satet1 satel1 satal1 sanal1 sanat1 sanit1 satit1 satin1 satan1 salan4 ,the rear triangle found in #c;b by producing lines is sil1 sitet1 sinel1 sinat1 sinin1 sitan1 sil4 ,the assemblage of sections constitute the solid body of the octahedron satat with triangular faces4 ,the one from the line selat to the point sil1 for instance1 is named selin1 selat1 selel1 salet1 salat1 salan1 sil4 ,the whole interior is salat4 ,shapes can easily be cut out of cardboard which1 when folded together1 form not only the tetrahedron and the octahedron1 but also samples of all the sections of the tesseract taken as it passes cornerwise through our space4 ,to name and visualise with appropriate colours a series of these sections is an admirable exercise for obtaining familiarity with the subject4 ,,extension ,,and ,,connection ,,with ,,numbers4 ,by extending the letter sequence it is of course possible to name a larger field4 ,by using the limit names the corners of each square can be named4 ,thus 8en sen10 8an sen10 etc41 will be the names of the points nearest the origin in 8en0 and in 8an40 ,a field of points of which each one is indefinitely small is given by the names written below4 ,illustration ,the squares are shown in dotted lines1 the names denote the points4 ,these points are not mathematical points1 but really minute areas4 ,instead of starting with a set of squares and naming them1 we can start with a set of points4 ,by an easily remembered convention we can give names to such a region of points4 ,let the space names with a final 8e0 added denote the mathematical points at the corner of each square nearest the origin4 ,we have then for the set of mathematical points indicated4 ,this system is really completely independent of the area system and is connected with it merely for the purpose of facilitating the memory processes4 ,the word 8ene0 is pronounced like 8eny10 with just sufficient attention to the final vowel to distinguish it from the word 8en40 ,illustration ,now1 connecting the numbers #j1 #a1 #b with the sequence e1 a1 i1 and also with the sequence n1 t1 l1 we have a set of points named as with numbers in a co-ordinate system4 ,thus 8ene0 is 7#j1 #j7 8ate0 is 7#a1 #a7 8ite0 is 7#b1 #a74 ,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 e4 ,by using a notation analogous to the decimal system a larger field of points can be named4 ,it remains to assign a letter sequence to the numbers from positive #j to positive #i1 and from negative #j to negative #i1 to obtain a system which can be used to denote both the usual co-ordinate system of mapping and a system of named squares4 ,the names denoting the points all end with e4 ,those that denote squares end with a consonant4 ,there are many considerations which must be attended to in extending the sequences to be used1 such as uniqueness in the meaning of the words formed1 ease of pronunciation1 avoidance of awkward combinations4 ,i drop 8s0 altogether from the consonant series and short 8u0 from the vowel series4 ,it is convenient to have unsignificant letters at disposal4 ,a double consonant like 8st0 for instance can be referred to without giving it a local significance by calling it 8ust40 ,i increase the number of vowels by considering a sound like 8ra0 to be a vowel1 using1 that is1 the letter 8r0 as forming a compound vowel4 ,the series is as follows3-- ,,consonants4 #j #a #b #c #d #e #f #g #h #i positive n t l p f sh k ch nt st negative z d th b v m g j nd sp ,,vowels4 #j #a #b #c #d #e #f #g #h #i positive e a i ee ae ai ar ra ri ree negative er o oo io oe iu or ro roo rio ,pronunciation4--e as in men2 a as in man2 i as in in2 ee as in between2 ae as ay in may2 ai as i in mine2 ar as in art2 er as ear in earth2 o as in on2 oo as oo in soon2 io as in clarion2 oe as oa in oat2 iu pronounced like yew4 ,to name a point such as 7#bc1 #da7 it is considered as 7#c1 #a7 on from 7#bj1 #dj7 and is called 8ifeete40 ,it is the initial point of the square ifeet of the area system4 ,the preceding amplification of a space language has been introduced merely for the sake of completeness4 ,as has already been said nine words and their combinations1 applied to a few simple models suffice for the purposes of our present enquiry4 ,printed by ,hazell1 ,watson ,viney1 ,ld41 ,london and ,aylesbury4