,transcriber's ,notes ,text printed in italics is represented between underscores1 bold face text between tildes4 ,small capitals have been transcribed as ,,all ,,capitals4 ,a letter in square brackets preceded by an equal sign1 as ,x1 represents a letter with bar above4 ,more ,transcriber's ,notes may be found at the end of this text4 ,a ,,new ,,era ,,of ,,thought4 ,,scientific ,,romances4 ,by ,c4 ,,howard ,,hinton1 ,m4,a4 ,crown #hvo1 cloth gilt1 #fs42 or separately1 #as4 each4 #a4 ,what is the ,fourth ,dimension8 #as4 ,,ghosts ,,explained4 8,a short treatise of admirable clearness4 4 4 4 ,mr4 ,hinton brings us1 panting but delighted1 to at least a momentary faith in the ,fourth ,dimension1 and upon the eye of this faith there opens a vista of interesting problems4 4 4 4 ,his pamphlet exhibits a boldness of speculation1 and a power of conceiving and expressing even the inconceivable1 which rouses one's faculties like a tonic40--,pall ,mall4 #b4 ,the ,persian ,king2 or1 ,the ,law of the ,valley1 #as4 ,,the ,,mystery ,,of ,,pleasure ,,and ,,pain4 8,a very suggestive and well-written speculation1 by the inheritor of an honoured name40--,mind4 8,will arrest the attention of the reader at once40--,knowledge4 #c4 ,a ,plane ,world1 #as4 #d4 ,a ,picture of our ,universe1 #as4 #e4 ,casting out the ,self1 #as4 ,,second ,,series4 #a4 ,on the ,education of the ,imagination4 #b4 ,many ,dimensions1 #as4 ,,london3 ,,swan ,,sonnenschein ,,co4 ,a ,new ,era of ,thought4 ,,by ,,charles ,,howard ,,hinton1 ,m4,a41 ,,oxon4 ,author of 8,what is the ,fourth ,dimension10 and other 8,scientific ,romances40 ,illustration ,london3 ,,swan ,,sonnenschein ,,co41 ,,paternoster ,,square4 #ahhh4 ,,butler ,,tanner1 ,,the ,,selwood ,,printing ,,works1 ,,frome1 ,,and ,,london4 ,,preface4 ,the ,,mss4 which formed the basis of this book were committed to us by the author1 on his leaving ,england for a distant foreign appointment4 ,it was his wish that we should construct upon them a much more complete treatise than we have effected1 and with that intention he asked us to make any changes or additions we thought desirable4 ,but long alliance with him in this work has convinced us that his thought 7especially that of a general philosophical character7 loses much of its force if subjected to any extraneous touch4 ,this feeling has induced us to print ,part ,i4 almost exactly as it came from his hands1 although it would probably have received much rearrangement if he could have watched it through the press himself4 ,part ,,ii4 has been written from a hurried sketch1 which he considered very inadequate1 and which we have consequently corrected and supplemented4 ,chapter ,,xi4 of this part has been entirely re-written by us1 and has thus not had the advantage of his supervision4 ,this remark also applies to ,appendix ,e1 which is an elaboration of a theorem he suggested4 ,appendix ,h1 and all the exercises have1 in accordance with his wish1 been written solely by us4 ,it will be apparent to the reader that ,appendix ,h is little more than a brief introduction to a very large subject1 which1 being concerned with tessaracts and solids1 is really beyond treatment in writing and diagrams4 ,this difficulty recalls us to the one great fact1 upon which we feel bound to insist1 that the matter of this book must receive objective treatment from the reader1 who will find it quite useless even to attempt to apprehend it without actually building in squares and cubes all the facts of space which we ask him to impress on his consciousness4 ,indeed1 we consider that printing1 as a method of spreading space-knowledge1 is but a 8pis aller10 and we would go back to that ancient and more fruitful method of the ,greek geometers1 and1 while describing figures on the sand1 or piling up pebbles in series1 would communicate to others that spirit of learning and generalization begotten in our consciousness by continuous contact with facts1 and only by continuous contact with facts vitally maintained4 ,,alicia ,,boole1 ,h4 ,,john ,,falk4 ,n4,b4 ,models4--,it is unquestionably a most important part of the process of learning space to construct these1 and the reader should do so1 however roughly and hastily4 ,but1 if ,models are required as patterns1 they may be ordered from ,messrs4 ,swan ,sonnenschein ,co41 ,paternoster ,square1 ,london1 and will be supplied as soon as possible1 the uncertainty as to demand for same not allowing us to have a large number made in advance4 ,much of the work can be done with plain cubes by using names without colours1 but further on the reader will find colours necessary to enable him to grasp and retain the complex series of observations4 ,coloured models can easily be made by covering ,kindergarten cubes with white paper and painting them with water-colour1 and1 if permanence be desired1 dipping them in size and copal varnish4 ,,table ,,of ,,contents4 ,,part ,i4 ,,page ,,introduction #a-g ,,chapter ,i4 ,scepticism and ,science4 ,beginning of ,knowledge #h-ac ,,chapter ,,ii4 ,apprehension of ,nature4 ,intelligence4 ,study of ,arrangement or ,shape #ad-bj ,,chapter ,,iii4 ,the ,elements of ,knowledge #ba-bc ,,chapter ,,iv4 ,theory and ,practice #bd-bh ,,chapter ,v4 ,knowledge3 ,self-,elements #bi-cd ,,chapter ,,vi4 ,function of ,mind4 ,space against ,metaphysics4 ,self-,limitation and its ,test4 ,a ,plane ,world #ce-df ,,chapter ,,vii4 ,self ,elements in our ,consciousness #dg-ej ,,chapter ,,viii4 ,relation of ,lower to ,higher ,space4 ,theory of the ther #ea-fj ,,chapter ,,ix4 ,another ,view of the ther4 ,material and therial ,bodies #fa-ff ,,chapter ,x4 ,higher ,space and ,higher ,being4 ,perception and ,inspiration #fg-hd ,,chapter ,,xi4 ,space the ,scientific ,basis of ,altruism and ,religion #he-ii ,,part ,,ii4 ,,chapter ,i4 ,three-space4 ,genesis of a ,cube4 ,appearances of a ,cube to a ,plane-being #aja-aab ,,chapter ,,ii4 ,further ,appearances of a ,cube to a ,plane-being #aac-aag ,,chapter ,,iii4 ,four-space4 ,genesis of a ,tessaract2 its ,representation in ,three-space #aah-abi ,,chapter ,,iv4 ,tessaract moving through ,three-space4 ,models of the ,sections #acj-acd ,,chapter ,v4 ,representation of ,three-space by ,names and in a ,plane #ace-adh ,,chapter ,,vi4 ,the ,means by which a ,plane-being would ,acquire a ,conception of our ,figures #adi-aee ,,chapter ,,vii4 ,four-space3 its ,representation in ,three-space #aef-aff ,,chapter ,,viii4 ,representation of ,four-space by ,name4 ,study of ,tessaracts #afg-agf ,,chapter ,,ix4 ,further ,study of ,tessaracts #agg-agi ,,chapter ,x4 ,cyclical ,projections #ahj-ahc ,,chapter ,,xi4 ,a ,tessaractic ,figure and its ,projections #ahd-aid ,,appendices4 ,a4 #ajj ,names used for ,plane ,space #aig ,b4 #baf ,names used for ,cubic ,space #aih ,c4 #bef ,names used for ,tessaractic ,space #bjj-bja ,d4 ,list of ,colours1 ,names1 and ,symbols #bjb-bjc ,e4 ,a ,theorem in ,four-space #bjd-bje ,f4 ,exercises on ,shapes of ,three ,dimensions #bje-bjg ,g4 ,exercises on ,shapes of ,four ,dimensions #bjg-bji ,h4 ,sections of the ,tessaract #bji-bag ,k4 ,drawings of the ,cubic ,sides and ,sections of the ,tessaract 7,models #a-ab7 with ,colours and ,names #bai-bda ,,introductory ,,note ,,to ,,part ,i4 ,at the completion of a work1 or at the completion of the first part of a work1 the feelings are necessarily very different from those with which the work was begun2 and the meaning and value of the work itself bear a very different appearance4 ,it will therefore be the simplest and shortest plan1 if ,i tell the reader briefly what the work is to which these pages are a guide1 and what ,i consider to be its value when done4 ,the task was to obtain a sense of the properties of higher space1 or space of four dimensions1 in the same way as that by which we reach a sense of our ordinary three-dimensional space4 ,i now prefer to call the task that of obtaining a familiarity with higher matter1 which shall be as intuitive to the mind as that of ordinary matter has become4 ,the expression 8higher matter0 is preferable to 8higher space10 because it is a somewhat hasty proceeding to split this concrete matter1 which we touch and feel1 into the abstractions of extension and impenetrability4 ,it seems to me that ,i cannot think of space without matter1 and therefore1 as no necessity compels me to such a course1 ,i do not split up the concrete object into subtleties1 but ,i simply ask3 8,what is that which is to a cube or block or shape of any kind as the cube is to a square80 ,in entering upon this inquiry we find the task is twofold4 ,firstly1 there is the theoretical part1 which is easy1 viz4 to set clearly before us the relative conditions which would obtain if there were a matter physically higher than this matter of ours1 and to choose the best means of liberating our minds from the limitations imposed on it by the particular conditions under which we are placed4 ,the second part of the task is somewhat laborious1 and consists of a constant presentation to the senses of those appearances which portions of higher matter would present1 and of a continual dwelling on them1 until the higher matter becomes familiar4 ,the reader must undertake this task1 if he accepts it at all1 as an experiment4 ,those of us who have done it1 are satisfied that there is that in the results of the experiment which make it well worthy of a trial4 ,and in a few words ,i may state the general bearings of this work1 for every branch of work has its general bearings4 ,it is an attempt1 in the most elementary and simple domain1 to pass from the lower to the higher4 ,in pursuing it the mind passes from one kind of intuition to a higher one1 and with that transition the horizon of thought is altered4 ,it becomes clear that there is a physical existence transcending the ordinary physical existence2 and one becomes inclined to think that the right direction to look is1 not away from matter to spiritual existences1 but towards the discovery of conceptions of higher matter1 and thereby of those material existences whose definite relations to us are apprehended as spiritual intuitions4 ,thus1 8material0 would simply mean 8grasped by the intellect1 become known and familiar40 ,our apprehension of anything which is not expressed in terms of matter1 is vague and indefinite4 ,to realize and live with that which we vaguely discern1 we need to apply the intuition of higher matter to the world around us4 ,and this seems to me the great inducement to this study4 ,let us form our intuition of higher space1 and then look out upon the world4 ,secondly1 in this progress from ordinary to higher matter1 as a general type of progress from lower to higher1 we make the following observations4 ,firstly1 we become aware that there are certain limitations affecting our regard4 ,secondly1 we discover by our reason what those limitations are1 and then force ourselves to go through the experience which would be ours if the limitations did not affect us4 ,thirdly1 we become aware of a capacity within us for transcending those limitations1 and for living in the higher mode as we had lived in the previous one4 ,we may remark that this progress from the ordinary to the higher kind of matter demands an absolute attention to details4 ,it is only in the retention of details that such progress becomes possible4 ,and as1 in this question of matter1 an absolute and unconventional examination gives us the indication of a higher1 so1 doubtless1 in other questions1 if we but come to facts without presupposition1 we begin to know that there is a higher and to discover indications of the way whereby we can approach4 ,that way lies in the fulness of detail rather than in the generalization4 ,biology has shown us that there is a universal order of forms or organisms1 passing from lower to higher4 ,therein we find an indication that we ourselves take part in this progress4 ,and in using the little cubes we can go through the process ourselves1 and learn what it is in a little instance4 ,but of all the ways in which the confidence gained from this lesson can be applied1 the nearest to us lies in the suggestion it gives1--and more than the suggestion1 if inclination to think be counted for anything1--in the suggestion of that which is higher than ourselves4 ,we1 as individuals1 are not the limit and end-all1 but there is a higher being than ours4 ,what our relation to it is1 we cannot tell1 for that is unlike our relation to anything we know4 ,but1 perhaps all that happens to us is1 could we but grasp it1 our relation to it4 ,at any rate1 the discovery of it is the great object beside which all else is as secondary as the routine of mere existence is to companionship4 ,and the method of discovery is full knowledge of each other4 ,thereby is the higher being to be known4 ,in as much as the least of us knows and is known by another1 in so much does he know the higher4 ,thus1 scientific prayer is when two or three meet together1 and1 in the belief of one higher than themselves1 mutually comprehend that vision of the higher1 which each one is1 and1 by absolute fulness of knowledge of the facts of each other's personality1 strive to attain a knowledge of that which is to each of their personalities as a higher figure is to its solid sides4 ,c4 ,h4 ,h4 ,a ,,new ,,era ,,of ,,thought4 ,,part ,i4 ,,introduction4 ,there are no new truths in this book1 but it consists of an effort to impress upon and bring home to the mind some of the more modern developments of thought4 ,a few sentences of ,kant1 a few leading ideas of ,gauss and ,lobatschewski form the material out of which it is built up4 ,it may be thought to be unduly long2 but it must be remembered that in these times there is a twofold process going on--one of discovery about external nature1 one of education1 by which our minds are brought into harmony with that which we know4 ,in certain respects we find ourselves brought on by the general current of ideas--we feel that matter is permanent and cannot be annihilated1 and it is almost an axiom in our minds that energy is persistent1 and all its transformations remains the same in amount4 ,but there are other directions in which there is need of definite training if we are to enter into the thoughts of the time4 ,and it seems to me that a return to ,kant1 the creator of modern philosophy1 is the first condition4 ,now of ,kant's enormous work only a small part is treated here1 but with the difference that should be found between the work of a master and that of a follower4 ,kant's statements are taken as leading ideas1 suggesting a field of work1 and it is in detail and manipulation merely that there is an opportunity for workmanship4 ,of ,kant's work it is only his doctrine of space which is here experimented upon4 ,with ,kant the perception of things as being in space is not treated as it seems so obvious to do4 ,we should naturally say that there is space1 and there are things in it4 ,from a comparison of those properties which are common to all things we obtain the properties of space4 ,but ,kant says that this property of being in space is not so much a quality of any definable objects1 as the means by which we obtain an apprehension of definable objects--it is the condition of our mental work4 ,now as ,kant's doctrine is usually commented on1 the negative side is brought into prominence1 the positive side is neglected4 ,it is generally said that the mind cannot perceive things in themselves1 but can only apprehend them subject to space conditions4 ,and in this way the space conditions are as it were considered somewhat in the light of hindrances1 whereby we are prevented from seeing what the objects in themselves truly are4 ,but if we take the statement simply as it is--that we apprehend by means of space--then it is equally allowable to consider our space sense as a positive means by which the mind grasps its experience4 ,there is in so many books in which the subject is treated a certain air of despondency--as if this space apprehension were a kind of veil which shut us off from nature4 ,but there is no need to adopt this feeling4 ,the first postulate of this book is a full recognition of the fact1 that it is by means of space that we apprehend what is4 ,space is the instrument of the mind4 ,and here for the purposes of our work we can avoid all metaphysical discussion4 ,very often a statement which seems to be very deep and abstruse and hard to grasp1 is simply the form into which deep thinkers have thrown a very simple and practical observation4 ,and for the present let us look on ,kant's great doctrine of space from a practical point of view1 and it comes to this--it is important to develop the space sense1 for it is the means by which we think about real things4 ,there is a doctrine which found much favour with the first followers of ,kant1 that also affords us a simple and practical rule of work4 ,it was considered by ,fichte that the whole external world was simply a projection from the ego1 and the manifold of nature was a recognition by the spirit of itself4 ,what this comes to as a practical rule is1 that we can only understand nature in virtue of our own activity2 that there is no such thing as mere passive observation1 but every act of sight and thought is an activity of our own4 ,now according to ,kant the space sense1 or the intuition of space1 is the most fundamental power of the mind4 ,but ,i do not find anywhere a systematic and thoroughgoing education of the space sense4 ,in every practical pursuit it is needed--in some it is developed4 ,in geometry it is used2 but the great reason of failure in education is that1 instead of a systematic training of the space sense1 it is left to be organized by accident and is called upon to act without having been formed4 ,according to ,kant and according to common experience it will be found that a trained thinker is one in whom the space sense has been well developed4 ,with regard to the education of the space sense1 ,i must ask the indulgence of the reader4 ,it will seem obvious to him that any real pursuit or real observation trains the space sense1 and that it is going out of the way to undertake any special discipline4 ,to this ,i would answer that1 according to my own experience1 ,i was perfectly ignorant of space relations myself before ,i actually worked at the subject1 and that directly ,i got a true view of space facts a whole series of conceptions1 which before ,i had known merely by repute and grasped by an effort1 became perfectly simple and clear to me4 ,moreover1 to take one instance3 in studying the relations of space we always have to do with coloured objects1 we always have the sense of weight2 for if the things themselves have no weight1 there is always a direction of up and down--which implies the sense of weight1 and to get rid of these elements requires careful sifting4 ,but perhaps the best point of view to take is this--if the reader has the space sense well developed he will have no difficulty in going through the part of the book which relates to it1 and the phraseology will serve him for the considerations which come next4 ,amongst the followers of ,kant1 those who pursued one of the lines of thought in his works have attracted the most attention and have been considered as his successors4 ,fichte1 ,schelling1 ,hegel have developed certain tendencies and have written remarkable books4 ,but the true successors of ,kant are ,gauss and ,lobatchewski4 ,for if our intuition of space is the means by which we apprehend1 then it follows that there may be different kinds of intuitions of space4 ,who can tell what the absolute space intuition is8 ,this intuition of space must be coloured1 so to speak1 by the conditions of the being which uses it4 ,now1 after ,kant had laid down his doctrine of space1 it was important to investigate how much in our space intuition is due to experience--is a matter of the physical circumstances of the thinking being--and how much is the pure act of the mind4 ,the only way to investigate this is the practical way1 and by a remarkable analysis the great geometers above mentioned have shown that space is not limited as ordinary experience would seem to inform us1 but that we are quite capable of conceiving different kinds of space4 ,our space as we ordinarily think of it is conceived as limited--not in extent1 but in a certain way which can only be realized when we think of our ways of measuring space objects4 ,it is found that there are only three independent directions in which a body can be measured--it must have height1 length and breadth1 but it has no more than these dimensions4 ,if any other measurement be taken in it1 this new measurement will be found to be compounded of the old measurements4 ,it is impossible to find a point in the body which could not be arrived at by travelling in combinations of the three directions already taken4 ,but why should space be limited to three independent directions8 ,geometers have found that there is no reason why bodies should be thus limited4 ,as a matter of fact all the bodies which we can measure are thus limited4 ,so we come to this conclusion1 that the space which we use for conceiving ordinary objects in the world is limited to three dimensions4 ,but it might be possible for there to be beings living in a world such that they would conceive a space of four dimensions4 ,all that we can say about such a supposition is1 that it is not demanded by our experience4 ,it may be that in the very large or the very minute a fourth dimension of space will have to be postulated to account for parts--but with regard to objects of ordinary magnitudes we are certainly not in a four dimensional world4 ,and this was the point at which about ten years ago ,i took up the inquiry4 ,it is possible to say a great deal about space of higher dimensions than our own1 and to work out analytically many problems which suggest themselves4 ,but can we conceive four-dimensional space in the same way in which we can conceive our own space8 ,can we think of a body in four dimensions as a unit having properties in the same way as we think of a body having a definite shape in the space with which we are familiar8 ,now this question1 as every other with which ,i am acquainted1 can only be answered by experiment4 ,and ,i commenced a series of experiments to arrive at a conclusion one way or the other4 ,it is obvious that this is not a scientific inquiry--but one for the practical teacher4 ,and just as in experimental researches the skilful manipulator will demonstrate a law of nature1 the less skilled manipulator will fail2 so here1 everything depended on the manipulation4 ,i was not sure that this power lay hidden in the mind1 but to put the question fairly would surely demand every resource of the practical art of education4 ,and so it proved to be2 for after many years of work1 during which the conception of four-dimensional bodies lay absolutely dark1 at length1 by a certain change of plan1 the whole subject of four-dimensional existence became perfectly clear and easy to impart4 ,there is really no more difficulty in conceiving four-dimensional shapes1 when we go about it the right way1 than in conceiving the idea of solid shapes1 nor is there any mystery at all about it4 ,when the faculty is acquired--or rather when it is brought into consciousness1 for it exists in every one in imperfect form--a new horizon opens4 ,the mind acquires a development of power1 and in this use of ampler space as a mode of thought1 a path is opened by using that very truth which1 when first stated by ,kant1 seemed to close the mind within such fast limits4 ,our perception is subject to the condition of being in space4 ,but space is not limited as we at first think4 ,the next step after having formed this power of conception in ampler space1 is to investigate nature and see what phenomena are to be explained by four-dimensional relations4 ,but this part of the subject is hardly one for the same worker as the one who investigates how to think in four-dimensional space4 ,the work of building up the power is the work of the practical educator1 the work of applying it to nature is the work of the scientific man4 ,and it is not possible to accomplish both tasks at the same time4 ,consequently the crown is still to be won4 ,here the method is given of training the mind2 it will be an exhilarating moment when an investigator comes upon phenomena which show that external nature cannot be explained except by the assumption of a four-dimension space4 ,the thought of the past ages has used the conception of a three-dimensional space1 and by that means has classified many phenomena and has obtained rules for dealing with matters of great practical utility4 ,the path which opens immediately before us in the future is that of applying the conception of four-dimensional space to the phenomena of nature1 and of investigating what can be found out by this new means of apprehension4 ,in fact1 what has been passed through may be called the three-dimensional era2 ,gauss and ,lobatchewski have inaugurated the four-dimensional era4 ,,chapter ,i4 ,,scepticism ,,and ,,science4 ,,beginning ,,of ,,knowledge4 ,the following pages have for their object to induce the reader to apply himself to the study1 in the first place of ,space1 and then of ,higher ,space2 and1 therefore1 ,i have tried by indirect means to show forth those thoughts and conceptions to which the practical work leads4 ,and ,i feel that ,i have a great advantage in this project1 inasmuch as many of the thoughts which spring up in the mind of one who studies higher space1 and many of the conceptions to which he is driven1 turn out to be nothing more nor less than old truths--the property of every mind that thinks and feels--truths which are not generally associated with the scientific apprehension of the world1 but which are not for that reason any the less valuable4 ,and for my own part ,i cannot do more than put them forward in a very feeble and halting manner4 ,for ,i have come upon them1 not in the way of feeling or direct apprehension1 but as the result of a series of works undertaken purely with the desire to know--a desire which did not lift itself to the height of expecting or looking for the beautiful or the good1 but which simply asked for something to know4 ,for ,i found myself--and many others ,i find do so also--,i found myself in respect to knowledge like a man who is in the midst of plenty and yet who cannot find anything to eat4 ,all around me were the evidences of knowledge--the arts1 the sciences1 interesting talk1 useful inventions--and yet ,i myself was profited nothing at all2 for somehow1 amidst all this activity1 ,i was left alone1 ,i could get nothing which ,i could know4 ,the dialect was foreign to me--its inner meaning was hidden4 ,if ,i would1 imitating the utterance of my fellows1 say a few words1 the effort was forced1 the whole result was an artificiality1 and1 if successful1 would be but a plausible imposture4 ,the word 8sceptical0 has a certain unpleasant association attached to it1 for it has been used by so many people who are absolutely certain in a particular line1 and attack other people's convictions4 ,but to be sceptical in the real sense is a far more unpleasant state of mind to the sceptic than to any one of his companions4 ,for to a mind that inquires into what it really does know1 it is hardly possible to enunciate complete sentences1 much less to put before it those complex ideas which have so large a part in true human life4 ,every word we use has so wide and fugitive a meaning1 and every expression touches or rather grazes fact by so very minute a point1 that1 if we wish to start with something which we do know1 and thence proceed in a certain manner1 we are forced away from the study of reality and driven to an artificial system1 such as logic or mathematics1 which1 starting from postulates and axioms1 develops a body of ideal truth which rather comes into contact with nature than is nature4 ,scientific achievement is reserved for those who are content to absorb into their consciousness1 by any means and by whatever way they come1 the varied appearances of nature1 whence and in which by reflection they find floating as it were on the sea of the unknown1 certain similarities1 certain resemblances and analogies1 by means of which they collect together a body of possible predictions and inferences2 and in nature they find correspondences which are actually verified4 ,hence science exists1 although the conceptions in the mind cannot be said to have any real correspondence in nature4 ,we form a set of conceptions in the mind1 and the relations between these conceptions give us relations which we find actually vibrating in the world around us4 ,but the conceptions themselves are essentially artificial4 ,we have a conception of atoms2 but no one supposes that atoms actually exist4 ,we suppose a force varying inversely as the square of the distance2 but no one supposes such a mysterious thing to really be in nature4 ,and when we come to the region of descriptive science1 when we come to simple observation1 we do not find ourselves any better provided with a real knowledge of nature4 ,if1 for instance1 we think of a plant1 we picture to ourselves a certain green shape1 of a more or less definite character4 ,this green shape enables us to recognise the plant we think of1 and to describe it to a certain extent4 ,but if we inquire into our imagination of it1 we find that our mental image very soon diverges from the fact4 ,if1 for instance1 we cut the plant in half1 we find cells and tissues of various kinds4 ,if we examine our idea of the plant1 it has merely an external and superficial resemblance to the plant itself4 ,it is a mental drawing meeting the real plant in external appearance2 but the two things1 the plant and our thought of it1 come as it were from different sides--they just touch each other as far as the colour and shape are concerned1 but as structures and as living organisms they are as wide apart as possible4 ,of course by observation and study the image of a plant which we bear in our minds may be made to resemble a plant as found in the fields more and more4 ,but the agreement with nature lies in the multitude of points superadded on to the notion of greenness which we have at first--there is no natural starting-point where the mind meets nature1 and whence they can travel hand in hand4 ,it almost seems as if1 by sympathy and feeling1 a human being was easier to know than the simplest object4 ,to know any object1 however simple1 by the reason and observation requires an endless process of thought and looking1 building up the first vague impression into something like in more and more respects4 ,while1 on the other hand1 in dealing with human beings there is an inward sympathy and capacity for knowing which is independent of1 though called into play by1 the observation of the actions and outward appearance of the human being4 ,but for the purpose of knowing we must leave out these human relationships4 ,they are an affair of instinct and inherited unconscious experience4 ,the mind may some day rise to the level of these inherited apprehensions1 and be able to explain them2 but at present it is far more than overtasked to give an account of the simplest portions of matter1 and is quite inadequate to give an account of the nature of a human being4 ,asking1 then1 what there was which ,i could know1 ,i found no point of beginning4 ,there were plenty of ways of accumulating observations1 but none in which one could go hand in hand with nature4 ,a child is provided in the early part of its life with a provision of food adapted for it4 ,but it seemed that our minds are left without a natural subsistence1 for on the one hand there are arid mathematics1 and on the other there is observation1 and in observation there is1 out of the great mass of constructed mental images1 but little which the mind can assimilate4 ,to the worker at science of course this crude and omnivorous observation is everything2 but if we ask for something which we can know1 it is like a vast mass of indigestible material with every here and there a fibre or thread which we can assimilate4 ,in this perplexity ,i was reduced to the last condition of mental despair2 and in default of finding anything which ,i could understand in nature1 ,i was sufficiently humbled to learn anything which seemed to afford a capacity of being known4 ,and the objects which came before me for this endeavour were the simple ones which will be plentifully used in the practical part of this book4 ,for ,i found that the only assertion ,i could make about external objects1 without bringing in unknown and unintelligible relations1 was this3 ,i could say how things were arranged4 ,if a stone lay between two others1 that was a definite and intelligible fact1 and seemed primary4 ,as a stone itself1 it was an unknown somewhat which one could get more and more information about the more one studied the various sciences4 ,but granting that there were some things there which we call stones1 the way they were arranged was a simple and obvious fact which could be easily expressed and easily remembered4 ,and so in despair of being able to obtain any other kind of mental possession in the way of knowledge1 ,i commenced to learn arrangements1 and ,i took as the objects to be arranged certain artificial objects of a simple shape4 ,i built up a block of cubes1 and giving each a name ,i learnt a mass of them4 ,now ,i do not recommend this as a thing to be done4 ,all ,i can say is that genuinely then and now it seemed and seems to be the only kind of mental possession which one can call knowledge4 ,it is perfectly definite and certain4 ,i could tell where each cube came and how it was related to each of the others4 ,as to the cube itself1 ,i was profoundly ignorant of that2 but assuming that as a necessary starting-point1 taking that as granted1 ,i had a definite mass of knowledge4 ,but ,i do not wish to say that this is better than any kind of knowledge which other people may find come home to them4 ,all ,i want to do is to take this humble beginning of knowledge and show how inevitably1 by devotion to it1 it leads to marvellous and far-distant truths1 and how1 by a strange path1 it leads directly into the presence of some of the highest conceptions which great minds have given us4 ,i do not think it ought to be any objection to an inquiry1 that it begins with obvious and common details4 ,in fact ,i do not think that it is possible to get anything simpler1 with less of hypothesis about it1 and more obviously a simple taking in of facts than the study of the arrangement of a block of cubes4 ,many philosophers have assumed a starting point for their thought4 ,i want the reader to accept a very humble one and see what comes of it4 ,if this leads us to anything1 no doubt greater results will come from more ambitious beginnings4 ,and now ,i feel that ,i have candidly exposed myself to the criticism of the reader4 ,if he will have the patience to go on1 we will begin and build up on our foundations4 ,,chapter ,,ii4 ,,apprehension ,,of ,,nature4 ,,intelligence4 ,,study ,,of ,,arrangement ,,or ,,shape4 ,nature is that which is around us4 ,but it is by no means easy to get to nature4 ,the savage living we may say in the bosom of nature1 is certainly unapprehensive of it1 in fact it has needed the greatness of a ,wordsworth and of generations of poets and painters to open our eyes even in a slight measure to the wonder of nature4 ,thus it is clear that it is not by mere passivity that we can comprehend nature2 it is the goal of an activity1 not a free gift4 ,and there are many ways of apprehending nature4 ,there are the sounds and sights of nature which delight the senses1 and the involved harmonies and the secret affinities which poetry makes us feel2 then1 moreover1 there is the definite knowledge of natural facts in which the memory and reason are employed4 ,thus we may divide our means of coming into contact with nature into three main channels3 the senses1 the imagination1 and the mind4 ,the imagination is perhaps the highest faculty1 but we leave it out of consideration here1 and ask3 ,how can we bring our minds into contact with nature8 ,now when we see two people of diverse characters we sometimes say that they cannot understand one another--there is nothing in the one by which he can understand the other--he is shut out by a limitation of his own faculties4 ,and thus our power of understanding nature depends on our own possession2 it is in virtue of some mental activity of our own that we can apprehend that outside activity which we call nature4 ,and thus the training to enable us to approach nature with our minds will be some active process on our own part4 ,in the course of my experience as a teacher ,i have often been struck by the want of the power of reason displayed by pupils2 they are not able to put two and two together1 as the saying goes1 and ,i have been at some pains to investigate wherein this curious deficiency lies1 and how it can be supplied4 ,and ,i have found that there is in the curriculum no direct cure for it--the discipline which supplies it is not one which comes into school methods1 it is a something which most children obtain in the natural and unsupervised education of their first contact with the world1 and lies before any recognised mode of distinction4 ,they can only understand in virtue of an activity of their own1 and they have not had sufficient exercise in this activity4 ,in the present state of education it is impossible to diverge from the ordinary routine4 ,but it is always possible to experiment on children who are out of the common line of education4 ,and ,i believe ,i am amply justified by the result of my experiments4 ,i have seen that the same activity which ,i have found makes that habit of mind which we call intelligence in a child1 is the source of our common and everyday rational intellectual work1 and that just as the faculties of a child can be called forth by it1 so also the powers of a man are best prepared by the same means1 but on an ampler scale4 ,a more detailed development of the practical work of ,part ,,ii41 would be the best training for the mind of a child4 ,an extension of the work of that ,part would be the training which1 hand in hand with observation and recapitulation1 would best develop a man's thought power4 ,in order to tell what the activity is by the prosecution of which we can obtain mental contact with nature we should observe what it is which we say we 8understand0 in any phenomenon of nature which has become clear to us4 ,when we look at a bright object it seems very different from a dull one4 ,a piece of bright steel hardly looks like the same substance as a piece of dull steel4 ,but the difference of appearance in the two is easily accounted for by the different nature of the surface in the two cases2 in the one all the irregularities are done away with1 and the rays of light which fall on it are sent off again without being dispersed and broken up4 ,in the case of the dull iron the rays of light are broken up and divided1 so that they are not transmitted with intensity in any one direction1 but flung off in all sorts of directions4 ,here the difference between the bright object and the dull object lies in the arrangement of the particles on its surface and their influence on the rays of light4 ,again1 with light itself the differences of colour are explained as being the effect on us of rays of different rates of vibration4 ,now a vibration is essentially this1 a series of arrangements of matter which follow each other in a closed order1 so that when the set has been run through1 the first arrangement follows again4 ,the whole theory of light is an account of arrangements of the particles in the transmitting medium1 only the arrangements alter--are not permanent in any one characteristic1 but go through a complete cycle of varieties4 ,again1 when the movements of the heavenly bodies are deduced from the theory of universal gravitation1 what we primarily do is to take account of arrangement2 for the law of gravity connects the movements which the attracted bodies tend to make with their distances1 that is1 it shows how their movements depend on their arrangement4 ,and if gravity as a force is to be explained itself1 the suppositions which have been put forward resolve it into the effect of the movements of small bodies2 that is to say1 gravity1 if explained at all1 is explained as the result of the arrangement and altering arrangements of small particles4 ,again1 to take the idea which proceeding from ,goethe casts such an influence on botanical observation4 ,goethe 7and also ,wolf7 laid down that the parts of a flower were modified leaves--and traced the stages and intermediate states between the ordinary green leaf and the most gorgeous petal or stamen or carpel1 so unlike a leaf in form and function4 ,now the essential value in this conception is1 that it enables us to look1 upon these different organs of a plant as modifications of one and the same organ--it enables us to think about the different varieties of the flower head as modifications of one single plant form4 ,we can trace correspondences between them1 and are led to possible explanations of their growth4 ,and all this is done by getting rid of pistil and stamen as separate entities1 and looking on them as leaves1 and their parts due to different arrangement of the leaf structure4 ,we have reduced these diverse objects to a common element1 we have found the unit by whose arrangements the whole is produced4 ,and in this department of thought1 as also to take another instance1 in chemistry1 to understand is practically this3 we find units 7leaves or atoms7 combinations of which account for the results which we see4 ,thus we see that that which the mind essentially apprehends is arrangement4 ,and this holds over the whole range of mental work1 from the simplest observation to the most complex theory4 ,when the eye takes in the form of an external object there is something more than a sense impression1 something more than a sensation of greenness and light and dark4 ,the mind works as well as the sense1 and these sense impressions are definitely grouped in what we call the shape of the object4 ,the essential act of perceiving lies in the apprehension of a shape1 and a shape is an arrangement of parts4 ,it does not matter what these parts are2 if we take meaningless dots of colour and arrange them we obtain a shape which represents the appearance of a stone or a leaf to a certain degree4 ,if we want to make our representation still more like1 we must treat each of the dots as in themselves arrangements1 we must compose each of them by many strokes and dots of the brush4 ,but even in this case we have not got anything else besides arrangement4 ,the ultimate element1 the small items of light and shade or of colour1 are in themselves meaningless2 it is in their arrangement that the likeness of the representation consists4 ,thus1 from a drawing to our notion of the planetary system1 all our contact with nature lies in this1 in an appreciation of arrangement4 ,hence to prepare ourselves for the understanding of nature1 we must 8arrange40 ,in virtue of our activity in making arrangements we prepare ourselves to do what is called understand nature4 ,or we may say1 that which we call understanding nature is to discern something similar in nature to that which we do when we arrange elements into compounded groups4 ,now if we study arrangement in the active way1 we must have something to arrange2 and the things we work with may be either all alike1 or each of them varying from every other4 ,if the elements are not alike then we are not studying pure arrangement2 but our knowledge is affected by the compound nature of that with which we deal4 ,if the elements are all alike1 we have what we call units4 ,hence the discipline preparatory for the understanding of nature is the active arrangement of like units4 ,and this is very much the case with all educational processes2 only the things chosen to arrange are in general words1 which are so complicated and carry such a train of association that1 unless the mind has already acquired a knowledge of arrangement1 it is puzzled and hampered1 and never gets a clear apprehension of what its work is4 ,now what shall we choose for our units8 ,any unit would do2 but it ought to be a real thing--it ought to be something which can be touched and seen1 not something which no one has ever touched or seen1 and which is even incapable of definition1 like a 8number40 ,i would divide studies into two classes3 those which create the faculty of arrangement1 and those which use it and exercise it4 ,mathematics exercises it1 but ,i do not think it creates it2 and unfortunately1 in mathematics as it is now often taught1 the pupil is launched 1 into a vast system of symbols--the whole use and meaning of symbols 7namely1 as means to acquire a clear grasp of facts7 is lost to him4 ,of the possible units which will serve1 ,i take the cube2 and ,i have found that whenever ,i took any other unit ,i got wrong1 puzzled and lost my way4 ,with the cube one does not get along very fast1 but everything is perfectly obvious and simple1 and builds up into a whole of which every part is evident4 ,and ,i must ask the reader to absolutely erase from his mind all desire or wish to be able to predict or assert anything about nature1 and he must please look with horror on any mental process by which he gets at a truth in an ingenious but obscure and inexplicable way4 ,let him take nothing which is not perfectly clear1 patent and evident1 demonstrable to his senses1 a simple repetition of obvious fact4 ,our work will then be this3 a study1 by means of cubes1 of the facts of arrangement4 ,and the process of learning will be an active one of actually putting up the cubes4 ,in this way we do for the mind what ,wordsworth does for the imagination--we bring it into contact with nature4 ,,chapter ,,iii4 ,,the ,,elements ,,of ,,knowledge4 ,there are two elements which enter into our knowledge with respect to any phenomenon4 ,if1 for instance1 we take the sun1 and ask ourselves what we observe1 we notice that it is a bright1 moving body2 and of these two qualities1 the brightness and the movement1 each seems equally predicable of the sun4 ,it does move1 and it is bright4 ,now further study discloses to us that there is a difference between these two affirmations4 ,the motion of the sun in its diurnal course round the earth is only apparent2 but it is really a bright1 hot body4 ,now of these two assertions which the mind naturally makes about the sun1 one--that it is moving--depends on the relation of the beholder to the sun1 the other is true about the sun itself4 ,the observed motion depends on a fact affecting oneself and having nothing to do with the sun1 while the brightness is really a quality of the sun itself4 ,now we will call those qualities or appearances which we notice in a body which are due to the particular conditions under which oneself is placed in observing it1 the self elements2 those facts about it which are independent of the observer's particular relationship we will call the residual element4 ,thus the sun's motion is a self element in our thought of the sun1 its brightness is a residual element4 ,it is not1 of course1 possible to draw a line distinctly between the self elements and the residual elements4 ,for instance1 some people have denied that brightness is a quality of things1 but that it depends on the capacity of the being for receiving sensations2 and for brightness they would substitute the assertion that the sun is giving forth a great deal of energy in the form of heat and light4 ,but there is no object in pursuing the discussion further4 ,the main distinction is sufficiently obvious4 ,and it is important to separate the self elements involved in our knowledge as far as possible1 so that the residual elements may be kept for our closer attention4 ,by getting rid of the self elements we put ourselves in a position in which we can propound sensible questions4 ,by getting rid of the notion of its circular motion round the earth we prepare our way to study the sun as it really is4 ,we get the subject clear of complications and extraneous considerations4 ,it would hardly be worth while to dwell on this consideration were it not of importance in our study of arrangement4 ,but the fact is that directly a subject has been cleared of the self elements1 it seems so absurd to have had them introduced at all that the great difficulty there was in getting rid of them is forgotten4 ,with regard to the knowledge we have at the present day about scientific matters1 there do not seem to be any self elements present4 ,but the worst about a self element is1 that its presence is never dreamed of till it is got rid of2 to know that it is there is to have done away with it4 ,and thus our body of knowledge is like a fluid which keeps clear1 not because there are no substances in solution1 but because directly they become evident they fall down as precipitates4 ,now one of our serious pieces of work will be to get rid of the self elements in the knowledge of arrangement4 ,and the kind of knowledge which we shall try to obtain will be somewhat different from the kind of knowledge which we have about events or natural phenomena4 ,in the large subjects which generally occupy the mind the things thought of are so complicated that every detail cannot possibly be considered4 ,the principles of the whole are realized1 and then at any required time the principles can be worked out4 ,thus1 with regard to a knowledge of the planetary system1 it is said to be known if the law of movement of each of the planets is recognized1 and their positions at any one time committed to memory4 ,it is not our habit to remember their relative positions with regard to one another at many intervals1 so as to have an exhaustive catalogue of them in our minds4 ,but with regard to the elements of knowledge with which we shall work1 the subject is so simple that we may justly demand of ourselves that we will know every detail4 ,and the knowledge we shall acquire will be much more one of the sense and feeling than of the reason4 ,we do not want to have a rule in our minds by which we can recall the positions of the different cubes1 but we want to have an immediate apprehension of them4 ,it was ,kant who first pointed out how much of thought there was embodied in the sense impressions2 and it is this embodied thought which we wish to form4 ,,chapter ,,iv4 ,,theory ,,and ,,practice4 ,both in science and in morals there is an important distinction to be drawn between theory and practice4 ,a knowledge of chemistry does not consist in the intellectual appreciation of different theories and principles1 but in being able to act in accordance with the facts of chemical combination1 so that by means of the appliances of chemistry practical results are produced4 ,and so in morals--the theoretic acquaintance with the principles of human action may consist with a marked degree of ignorance of how to act amongst other human beings4 ,now the use of the word 8learn0 has been much restricted to merely theoretic studies4 ,it requires to be enlarged to the scientific meaning4 ,and to know1 should include practice in actual manipulation4 ,let us take an instance4 ,we all know what justice is1 and any child can be taught to tell the difference between acting justly and acting unjustly4 ,but it is a different thing to teach them to act with justice4 ,something is done which affects them unpleasantly4 ,they feel an impulse to retaliate4 ,in order to see what justice demands they have to put their personal feeling on one side4 ,they have to get rid of those conditions under which they apprehended the effects of the action at first1 and they have to look on it from another point of view4 ,then they have to act in accordance with this view4 ,now there are two steps--one an intellectual one of understanding1 one a practical one of carrying out the view4 ,neither is a moral step4 ,one demands intelligence1 the other the formation of a habit1 and this habit can be inculcated by precept1 by reward and punishment1 by various means4 ,but as human nature is constituted1 if the habit of justice is inculcated it touches a part of the being4 ,there is an emotional response4 ,we know but little of a human being1 but we can safely say that there are depths in it1 beyond the feelings of momentary resentment and the stimulus of pleasurable or painful sensation1 to which justice is natural4 ,how little adequate is our physical knowledge of a human being as a bodily frame to explain the fact of human life4 ,now and again we see one of these isolated beings bound up in another1 as if there was an undiscovered physical bond between them4 ,and in all there is this sense of justice--a kind of indwelling verdict of the universal mind1 if we may use such an expression1 in virtue of which a man feels not as a single individual but as all men4 ,with respect to justice1 it is not only necessary to take the view of one other person than oneself1 but that of many4 ,there may be justice which is very good justice from the point of view of a party1 but very bad justice from the point of view of a nation4 ,and if we suppose an agency outside the human race1 gifted with intelligence1 and affecting the race1 in the way for instance of causing storms or disturbances of the ground1 in order to judge it with justice we should have to take a standpoint outside the race of men altogether4 ,we could not say that this agency was bad4 ,we should have to judge it with reference to its effect on other sentient beings4 ,there are some words which are often used in contrast with each other--egoism and altruism2 and each seems to me unmeaning except as terms in a contrast4 ,let us take an instance4 ,a boy has a bag of cakes1 and is going to enjoy them by himself4 ,his parent stops him1 and makes him set up some stumps and begin to learn to play cricket with another boy4 ,the enjoyment of the cakes is lost--he has given that up2 but after a little while he has a pleasure which is greater than that of cakes in solitude4 ,he enters into the life of the game4 ,he has given up1 or been forced to give up1 the pleasure he knew1 and he has found a greater one4 ,what he thought about himself before was that he liked cakes1 now what he thinks about himself is that he likes cricket4 ,which of these is the true thought about himself8 ,neither1 probably1 but at any rate it is more near the truth to say that he likes the cricket4 ,if now we use the word self to mean that which a person knows of himself1 and it is difficult to see what other meaning it can have1 his self as he knew it at first was thwarted1 was given up1 and through that he discovered his true self4 ,and again with the cricket2 he will make the sacrifice of giving that up1 voluntarily or involuntarily1 and will find a truer self still4 ,in general there is not much difficulty in making a boy find out that he likes cricket2 and it is quite possible for him to eat his cakes first and learn to play cricket afterwards--the cricket will not come to him as a thwarting in any sense of what he likes better4 ,but this ease in entering in to the pursuit only shows that the boy's nature is already developed to the level of enjoying the game4 ,the distinct moral advance would come in such a case when something which at first was hard to him to do was presented to him--and the hardness1 the unpleasantness is of a double kind1 the giving up of a pursuit or indulgence to which he is accustomed1 and the exertion of forming the habits demanded by the new pursuit4 ,now it is unimportant whether the renunciation is forced or willingly taken4 ,but as a general rule it may be laid down1 that by giving up his own desires as he feels them at the moment1 to the needs and advantage of those around him1 or to the objects which he finds before him demanding accomplishment1 a human being passes to the discovery of his true self on and on4 ,the process is limited by the responsibilities which a man finds come upon him4 ,the method of moral advance is to acquire a practical knowledge2 he must first see what the advantage of some one other than himself would be1 and then he must act in accordance with that view of things4 ,then having acted and formed a habit1 he discovers a response in himself4 ,he finds that he really cares1 and that his former limited life was not really himself4 ,his body and the needs of his body1 so far as he can observe them1 externally are the same as before2 but he has obtained an inner and unintellectual1 but none the less real1 apprehension of what he is4 ,thus altruism1 or the sacrifice of egoism to others1 is followed by a truer egoism1 or assertion of self1 and this process flashed across by the transcendent lights of religion1 wherein1 as in the sense of justice and duty1 untold depths in the nature of man are revealed entirely unexpressed by the intellectual apprehension which we have of him as an animal frame of a very high degree of development1 is the normal one by which from childhood a human being develops into the full responsibilities of a man4 ,now both in science and in conduct there are self elements4 ,in science1 getting rid of the self elements means a truer apprehension of the facts about one2 in conduct1 getting rid of the self elements means obtaining a truer knowledge of what we are--in the way of feeling more strongly and deeply and being bound and linked in a larger scale4 ,thus without pretending to any scientific accuracy in the use of terms1 we can assign a certain amount of meaning to the expression--getting rid of self elements4 ,and all that we can do is to take the rough idea of this process1 and then taking our special subject matter1 apply it4 ,in affairs of life experiments lead to disaster4 ,but happily science is provided wherein the desire to put theories into practice can be safely satisfied--and good results sometimes follow4 ,were it not for this the human race might before now have been utopiad from off the face of the earth4 ,in experiment1 manipulation is everything2 we must be certain of all our conditions1 otherwise we fail assuredly and have not even the satisfaction of knowing that our failure is due to the wrongness of our conjectures4 ,and for our purposes we use a subject matter so simple that the manipulation is easy4 ,,chapter ,v4 ,,knowledge3 ,,self-,,elements4 ,i must now go with somewhat of detail into the special subject in which these general truths will be exhibited4 ,everything ,i have to say would be conceived much more clearly by a very little practical manipulation4 ,but here ,i want to put the subject in as general a light as possible1 so that there may be no hindrance to the judgment of the reader4 ,and when ,i use the word 8know10 ,i assume something else than the possession of a rule1 by which it can be said how facts are4 ,by knowing ,i mean that the facts of a subject all lie in the mind ready to come out vividly into consciousness when the attention is directed on them4 ,michael ,angelo knew the human frame1 he could tell every little fact about it2 if he chose to call up the image1 he would see mentally how each muscle and fold of the skin lay with regard to the surrounding parts4 ,we want to obtain a knowledge as good as ,michael ,angelo's4 ,there is a great difference between ,michael ,angelo and us2 but let that difference be expressed1 not in our way of knowing1 but in the difference between the things he knew and the things we know4 ,we take a very simple structure and know it as absolutely as he knew the complicated structure of the human body4 ,and let us take a block of cubes2 any number will do1 but for convenience sake let us take a set of twenty-seven cubes put together so as to form a large cube of twenty-seven parts4 ,and let each of these cubes be marked so as to be recognized1 and let each have a name so that it can be referred to4 ,and let us suppose that we have learnt this block of cubes so that each one is known--that is to say1 its position in the block is known and its relation to the other blocks4 ,now having obtained this knowledge of the block as it stands in front of us1 let us ask ourselves if there is any self element present in our knowledge of it4 ,and there is obviously this self element present4 ,we have learnt the cubes as they stand in accordance with our own convenience in putting them up4 ,we put the lowest ones first1 and the others on the top of them1 and we distinctly conceive the lower ones as supporting the upper ones4 ,now this fact of support has nothing to do with the block of cubes itself1 it depends on the conditions under which we come to apprehend the block of cubes1 it depends on our position on the surface of the earth1 whereby gravity is an all important factor in our experience4 ,in fact our sight has got so accustomed to take gravity into consideration in its view of things1 that when we look at a landscape or object with our head upside down we do not see it inverted1 but we superinduce on the direct sense impressions our knowledge of the action of gravity1 and obtain a view differing very little from what we see when in an upright position4 ,it will be found that every fact about the cubes has involved in it a reference to up and down4 ,it is by being above or below that we chiefly remember where the cubes are4 ,but above and below is a relation which depends simply on gravity4 ,if it were not for gravity above and below would be interchangeable terms1 instead of expressing a difference of marked importance to us under our conditions of existence4 ,now we put 8being above0 or 8being below0 into the cubes themselves and feel it a quality in them--it defines their position4 ,but this above or below really comes from the conditions in which we are4 ,it is a self element1 and as such1 to obtain a true knowledge of the cubes we must get rid of it4 ,and now1 for the sake of a process which will be explained afterwards1 let us suppose that we cannot move the block of cubes which we have put up4 ,let us keep it fixed4 ,in order to learn how it is independent of gravity the best way would be to go to a place where gravity has virtually ceased to act2 at the centre of the earth1 for instance1 or in a freely falling shell4 ,but this is impossible1 so we must choose another way4 ,let us1 then1 since we cannot get rid of gravity1 see what we have done already4 ,we have learnt the cubes1 and however they are learnt1 it will be found that there is a certain set of them round which the others are mentally grouped1 as being on the right or left1 above or below4 ,now to get our knowledge as perfect as we can before getting rid of the self element up and down1 we have to take as central cubes in our mind different sets again and again1 until there are none which are primary to us4 ,then there remains only the distinction of some being above others4 ,now this can only be made to sink out of the primary place in our thoughts by reversing the relation4 ,if we turned the block upside down1 and learnt it in this new position1 then we should learn the position of the cubes with regard to each other with that element in them1 which comes from the action of gravity1 reversed4 ,and the true nature of the arrangement to which we added something in virtue of our sensation of up and down1 would become purer and more isolated in our minds4 ,we have1 however1 supposed that the cubes are fixed4 ,then1 in order to learn them1 we must put up another block showing what they would be like in the supposed new position4 ,we then take a set of cubes1 models of the original cubes1 and by consideration we can put them in such positions as to be an exact model of what the block of cubes would be if turned upside down4 ,and here is the whole point on which the process depends4 ,we can tell where each cube would come1 but we do not know the block in this new position4 ,i draw a distinction between the two acts1 8to tell where it would be10 and to 8know40 ,telling where it would be is the preparation for knowing4 ,the power of assigning the positions may be called the theory of the block4 ,the actual knowledge is got by carrying out the theory practically1 by putting up the blocks and becoming able to realize without effort where each one is4 ,it is not enough to put up the model blocks in the reverse position4 ,it is found that this up and down is a very obstinate element indeed1 and a good deal of work is requisite to get rid of it completely4 ,but when it is got rid of in one set of cubes1 the faculty is formed of appreciating shape independently of the particular parts which are above or below on first examination4 ,we discover in our own minds the faculty of appreciating the facts of position independent of gravity and its influence on us4 ,i have found a very great difference in different minds in this respect4 ,to some it is easy1 to some it is hard4 ,and to use our old instance1 the discovery of this capacity is like the discovery of a love of justice in the being who has forced himself to act justly4 ,it is a capacity for being able to take a view independent of the conditions under which he is placed1 and to feel in accordance with that view4 ,there is1 so far as ,i know1 no means of arriving immediately at this impartial appreciation of shape4 ,it can only be done by1 as it were1 extending our own body so as to include certain cubes1 and appreciating then the relation of the other cubes to those4 ,and after this1 by identifying ourselves with other cubes1 and in turn appreciating the relation of the other cubes to these4 ,and the practical putting up of the cubes is the way in which this power is gained4 ,it springs up with a repetition of the mechanical acts4 ,thus there are three processes4 #ast1 ,an apprehension of what the position of the cubes would be4 #bnd1 ,an actual putting of them up in accordance with that apprehension1 #crd1 ,the springing up in the mind of a direct feeling of what the block is1 independent of any particular presentation4 ,thus the self element of up and down can be got rid of out of a block of cubes4 ,and when even a little block is known like this1 the mind has gained a great deal4 ,yet in the apprehension and knowledge of the block of cubes with the up and down relation in them1 there is more than in the absolute apprehension of them4 ,for there is the apprehension of their position and also of the effect of gravity on them in their position4 ,imagine ourselves to be translated suddenly to another part of the universe1 and to find there intelligent beings1 and to hold conversation with them4 ,if we told them that we came from a world1 and were to describe the sun to them1 saying that it was a bright1 hot body which moved round us1 they would reply3 ,you have told us something about the sun1 but you have also told us something about yourselves4 ,thus in the apprehension of the sun as a body moving round us there is more than in the apprehension of it as not moving round1 for we really in this case apprehend two things--the sun and our own conditions4 ,but for the purpose of further knowledge it is most important that the more abstract knowledge should be acquired4 ,the self element introduced by the motion of the earth must be got rid of before the true relations of the solar system can be made out4 ,and in our block of cubes1 it will be found that feelings about arrangement1 and knowledge of space1 which are quite unattainable with our ordinary view of position1 become simple and clear when this discipline has been gone through4 ,and there can be no possible mental harm in going through this bit of training1 for all that it comes to is looking at a real thing as it actually is--turning it round and over and learning it from every point of view4 ,,chapter ,,vi4 ,,function ,,of ,,mind4 ,,space ,,against ,,metaphysics4 ,,self-,,limitation ,,and ,,its ,,test4 ,a ,,plane ,,world4 ,we now pass on to the question3 ,are there any other self elements present in our knowledge of the block of cubes8 ,when we have learnt to free it from up and down1 is there anything else to be got rid of8 ,it seems as if1 when the cubes were thus learnt1 we had got as abstract and impersonal a bit of knowledge as possible4 ,but1 in reality1 in the relations of the cubes as we thus apprehend them there is present a self element to which the up and down is a mere trifle4 ,if we think we have got absolute knowledge we are indeed walking on a thin crust in unconsciousness of the depths below4 ,we are so certain of that which we are habituated to1 we are so sure that the world is made up of the mechanical forces and principles which we familiarly deal with1 that it is more of a shock than a welcome surprise to us to find how mistaken we were4 ,and after all1 do we suppose that the facts of distance and size and shape are the ultimate facts of the world--is it in truth made up like a machine out of mechanical parts8 ,if so1 where is there room for that other which we know--more certainly1 because inwardly--that reverence and love which make life worth having8 ,no2 these mechanical relations are our means of knowing about the world2 they are not reality itself1 and their primary place in our imaginations is due to the familiarity which we have with them1 and to the peculiar limitations under which we are4 ,but ,i do not for a moment wish to go in thought beyond physical nature--,i do not suppose that in thought we can4 ,to the mind it is only the body that appears1 and all that ,i hope to do is to show material relations1 mechanism1 arrangements4 ,but much depends on what kind of material relations we perceive outside us4 ,a human being1 an animal and a machine are to the mind all merely portions of matter arranged in certain ways4 ,but the mind can give an exhaustive account of the machine1 account fairly well for the animal1 while the human being it only defines externally1 leaving the real knowledge to be supplied by other faculties4 ,but we must not under-estimate the work of the mind1 for it is only by the observation of and thought about the bodies with which we come into contact that we know human beings4 ,it is the faculty of thought that puts us in a position to recognize a soul4 ,and so1 too1 about the universe--it is only by correct thought about it that we can perceive its true moral nature4 ,and it will be found that the deadness which we ascribe to the external world is not really there1 but is put in by us because of our own limitations4 ,it is really the self elements in our knowledge which make us talk of mechanical necessity1 dead matter4 ,when our limitations fall1 we behold the spirit of the world like we behold the spirit of a friend--something which is discerned in and through the material presentation of a body to us4 ,our thought means are sufficient at present to show us human souls2 but all except human beings is1 as far as science is concerned1 inanimate4 ,one self element must be got rid of from our perception1 and this will be changed4 ,the one thing necessary is1 that in matters of thinking we will not admit anything that is not perfectly clear1 palpable and evident4 ,on the mind the only conceivable demand is to seek for facts4 ,the rock on which so many systems of philosophy have come to grief is the attempt to put moral principles into nature4 ,our only duty is to accept what we find4 ,man is no more the centre of the moral world than he is of the physical world4 ,then relegate the intellect to its right position of dealing with facts of arrangement--it can appreciate structure--and let it simply look on the world and report on it4 ,we have to choose between metaphysics and space thought4 ,in metaphysics we find lofty ideals--principles enthroned high in our souls1 but which reduce the world to a phantom1 and ourselves to the lofty spectators of an arid solitude4 ,on the other hand1 if we follow ,kant's advice1 we use our means and find realities linked together1 and in the physical interplay of forces and connexion of structure we behold the relations between spirits--those dwelling in man and those above him4 ,it is difficult to explain this next self element that has to be removed from the block of cubes2 it requires a little careful preparation1 in fact our language hardly affords us the means4 ,but it is possible to approach indirectly1 and to detect the self-element by means of an analogy4 ,if we suspect there be some condition affecting ourselves which make us perceive things not as they are1 but falsely1 then it is possible to test the matter by making the supposition of other beings subject to certain conditions1 and then examining what the effect on their experience would be of these conditions4 ,thus if we make up the appearances which would present themselves to a being subject to a limitation or condition1 we shall find that this limitation or condition1 when unrecognized by him1 presents itself as a general law of his outward world1 or as properties and qualities of the objects external to him4 ,he will1 moreover1 find certain operations possible1 others impossible1 and the boundary line between the possible and impossible will depend quite as much on the conditions under which he is as on the nature of the operations4 ,and if we find that in our experience of the outward world there are analogous properties and qualities of matter1 analogous possibilities and impossibilities1 then it will show to us that we in our turn are under analogous limitations1 and that what we perceive as the external world is both the external world and our own conditions4 ,and the task before us will be to separate the two4 ,now the problem we take up here is this--to separate the self elements from the true fact4 ,to separate them not merely as an outward theory and intelligent apprehension1 but to separate them in the consciousness itself1 so that our power of perception is raised to a higher level4 ,we find out that we are under limitations4 ,our next step is to so familiarize ourselves with the real aspect of things1 that we perceive like beings not under our limitations4 ,or more truly1 we find that inward soul which itself not subject to these limitations1 is awakened to its own natural action1 when the verdicts conveyed to it through the senses are purged of the self elements introduced by the senses4 ,everything depends on this--,is there a native and spontaneous power of apprehension1 which springs into activity when we take the trouble to present to it a view from which the self elements are eliminated8 ,about this every one must judge for himself4 ,but the process whereby this inner vision is called on is a definite one4 ,and just as a human being placed in natural human relationships finds in himself a spontaneous motive towards the fulfilment of them1 discovers in himself a being whose motives transcend the limits of bodily self-regard1 so we should expect to find in our minds a power which is ready to apprehend a more absolute order of fact than that which comes through the senses4 ,i do not mean a theoretical power4 ,a theory is always about it1 and about it only4 ,i mean an inner view1 a vision whereby the seeing mind as it were identifies itself with the thing seen4 ,not the tree of knowledge1 but of the inner and vital sap which builds up the tree of knowledge4 ,and if this point is settled1 it will be of some use in answering the question3 ,what are we8 ,are we then bodies only8 ,this question has been answered in the negative by our instincts4 ,why should we despair of a rational answer8 ,let us adopt our space thought and develop it4 ,the supposition which we must make is the following4 ,let us imagine a smooth surface--like the surface of a table2 but let the solid body at which we are looking be very thin1 so that our surface is more like the surface of a thin sheet of metal than the top of a table4 ,and let us imagine small particles1 like particles of dust1 to lie on this surface1 and to be attracted downwards so that they keep on the surface4 ,but let us suppose them to move freely over the surface4 ,let them never in their movements rise one over the other2 let them all singly and collectively be close to the surface4 ,and let us suppose all sorts of attractions and repulsions between these atoms1 and let them have all kinds of movements like the atoms of our matter have4 ,then there may be conceived a whole world1 and various kinds of beings as formed out of this matter4 ,the peculiarity about this world and these beings would be1 that neither the inanimate nor the animate members of it would move away from the surface4 ,their movements would all lie in one plane1 a plane parallel to and very near the surface on which they are4 ,and if we suppose a vast mass to be formed out of these atoms1 and to lie like a great round disk on the surface1 compact and cohering closely together1 then this great disk would afford a support for the smaller shapes1 which we may suppose to be animate beings4 ,the smaller shapes would be attracted to the great disk1 but would be arrested at its rim4 ,they would tend to the centre of the disk1 but be unable to get nearer to the centre than its rim4 ,thus1 as we are attracted to the centre of the earth1 but walk on its surface1 the beings on this disk would be attracted to its centre1 but walk on its rim4 ,the force of attraction which they would feel would be the attraction of the disk4 ,the other force of attraction1 acting perpendicularly to the plane which keeps them and all the matter of their world to the surface1 they would know nothing about4 ,for they cannot move either towards this force or away from it2 and the surface is quite smooth1 so that they feel no friction in their movement over it4 ,now let us realize clearly one of these beings as he proceeds along the rim of his world4 ,let us imagine him in the form of an outline of a human being1 with no thickness except that of the atoms of his world4 ,as to the mode in which he walks1 we must imagine that he proceeds by springs or hops1 because there would be no room for his limbs to pass each other4 ,imagine a large disk on the table before you1 and a being1 such as the one described1 proceeding round it4 ,let there be small movable particles surrounding him1 which move out of his way as he goes along1 and let these serve him for respiration2 let them constitute an atmosphere4 ,forwards and backwards would be to such a being direction along the rim--the direction in which he was proceeding and its reverse4 ,then up and down would evidently be the direction away from the disk's centre and towards it4 ,thus backwards and forwards1 up and down1 would both lie in the plane in which he was4 ,and he would have no other liberty of movement except these4 ,thus the words right and left would have no meaning to him4 ,all the directions in which he could move1 or could conceive movement possible1 would be exhausted when he had thought of the directions along the rim and at right angles to it1 both in the plane4 ,what he would call solid bodies1 would be groups of the atoms of his world cohering together4 ,such a mass of atoms would1 we know1 have a slight thickness2 namely1 the thickness of a single atom4 ,but of this he would know nothing4 ,he would say1 8,a solid body has two dimensions--height 7by how much it goes away from the rim7 and thickness 7by how much it lies along the rim740 ,thus a solid would be a two-dimensional body1 and a solid would be bounded by lines4 ,lines would be all that he could see of a solid body4 ,thus one of the results of the limitations under which he exists would be1 that he would say1 8,there are only two dimensions in real things40 ,in order for his world to be permanent1 we must suppose the surface on which he is to be very compact1 compared to the particles of his matter2 to be very rigid2 and1 if he is not to observe it by the friction of matter moving on it1 to be very smooth4 ,and if it is very compact with regard to his matter1 the vibrations of the surface must have the effect of disturbing the portions of his matter1 and of separating compound bodies up into simpler ones4 ,illustration3 ,fig4 #a4 ,illustration3 ,fig4 #b4 ,another consequence of the limitation under which this being lies1 would be the following3--,if we cut out from the corners of a piece of paper two triangles1 ,a ,b ,c and ,a ,b ,c1 and suppose them to be reduced to such a thinness that they are capable of being put on to the imaginary surface1 and of being observed by the flat being like other bodies known to him2 he will1 after studying the bounding lines1 which are all that he can see or touch1 come to the conclusion that they are equal and similar in every respect2 and he can conceive the one occupying the same space as the other occupies1 without its being altered in any way4 ,if1 however1 instead of putting down these triangles into the surface on which the supposed being lives1 as shown in ,fig4 #a1 we first of all turn one of them over1 and then put them down1 then the plane-being has presented to him two triangles1 as shown in ,fig4 #b4 ,and if he studies these1 he finds that they are equal in size and similar in every respect4 ,but he cannot make the one occupy the same space as the other one2 this will become evident if the triangles be moved about on the surface of a table4 ,one will not lie on the same portion of the table that the other has marked out by lying on it4 ,hence the plane-being by no means could make the one triangle in this case coincide with the space occupied by the other1 nor would he be able to conceive the one as coincident with the other4 ,the reason of this impossibility is1 not that the one cannot be made to coincide1 but that before having been put down on his plane it has been turned round4 ,it has been turned1 using a direction of motion which the plane-being has never had any experience of1 and which therefore he cannot use in his mental work any more than in his practical endeavours4 ,thus1 owing to his limitations1 there is a certain line of possibility which he cannot overstep4 ,but this line does not correspond to what is actually possible and impossible4 ,it corresponds to a certain condition affecting him1 not affecting the triangle4 ,his saying that it is impossible to make the two triangles coincide1 is an assertion1 not about the triangles1 but about himself4 ,now1 to return to our own world1 no doubt there are many assertions which we make about the external world which are really assertions about ourselves4 ,and we have a set of statements which are precisely similar to those which the plane-being would make about his surroundings4 ,thus1 he would say1 there are only two independent directions2 we say there are only three4 ,he would say that solids are bounded by lines2 we say that solids are bounded by planes4 ,moreover1 there are figures about which we assert exactly the same kind of impossibility as his plane-being did about the triangles in ,fig4 #b4 ,we know certain shapes which are equal the one to the other1 which are exactly similar1 and yet which we cannot make fit into the same portion of space1 either practically or by imagination4 ,if we look at our two hands we see this clearly1 though the two hands are a complicated case of a very common fact of shape4 ,now1 there is one way in which the right hand and the left hand may practically be brought into likeness4 ,if we take the right-hand glove and the left-hand glove1 they will not fit any more than the right hand will coincide with the left hand4 ,but if we turn one glove inside out1 then it will fit4 ,now1 to suppose the same thing done with the solid hand as is done with the glove when it is turned inside out1 we must suppose it1 so to speak1 pulled through itself4 ,if the hand were inside the glove all the time the glove was being turned inside out1 then1 if such an operation were possible1 the right hand would be turned into an exact model of the left hand4 ,such an operation is impossible4 ,but curiously enough there is a precisely similar operation which1 if it were possible1 would1 in a plane1 turn the one triangle in ,fig4 #b into the exact copy of the other4 ,illustration ,look at the triangle in ,fig4 #b1 ,a ,b ,c1 and imagine the point ,a to move into the interior of the triangle and to pass through it1 carrying after it the parts of the lines ,a ,b and ,a ,c to which it is attached1 we should have finally a triangle ,a ,b ,c1 which was quite like the other of the two triangles ,a ,b ,c in ,fig4 #b4 ,thus we know the operation which produces the result of the 8pulling through0 is not an impossible one when the plane-being is concerned4 ,then may it not be that there is a way in which the results of the impossible operation of pulling a hand through could be performed8 ,the question is an open one4 ,our feeling of it being impossible to produce this result in any way1 may be because it really is impossible1 or it may be a useful bit of information about ourselves4 ,now at this point my special work comes in4 ,if there be really a four-dimensional world1 and we are limited to a space or three-dimensional view1 then either we are absolutely three-dimensional with no experience at all or capacity of apprehending four-dimensional facts1 or we may be1 as far as our outward experience goes1 so limited2 but we may really be four-dimensional beings whose consciousness is by certain undetermined conditions limited to a section of the real space4 ,thus we may really be like the plane-beings mentioned above1 or we may be in such a condition that our perceptions1 not ourselves1 are so limited4 ,the question is one which calls for experiment4 ,we know that if we take an animal1 such as a dog or cat1 we can by careful training1 and by using rewards and punishment1 make them act in a certain way1 in certain defined cases1 in accordance with justice2 we can produce the mechanical action4 ,but the feeling of justice will not be aroused2 it will be but a mere outward conformity4 ,but a human being1 if so trained1 and seeing others so acting1 gets a feeling of justice4 ,now1 if we are really four-dimensional1 by going through those acts which correspond to a four-dimensional experience 7so far as we can71 we shall obtain an apprehension of four-dimensional existence--not with the outward eye1 but essentially with the mind4 ,and after a number of years of experiment which were entirely nugatory1 ,i can now lay it down as a verifiable fact1 that by taking the proper steps we can feel four-dimensional existence1 that the human being somehow1 and in some way1 is not simply a three-dimensional being--in what way it is the province of science to discover4 ,all that ,i shall do here is1 to put forward certain suppositions which1 in an arbitrary and forced manner1 give an outline of the relation of our body to four-dimensional existence1 and show how in our minds we have faculties by which we recognise it4 ,,chapter ,,vii4 ,,self ,,elements ,,in ,,our ,,consciousness4 ,it is often taken for granted that our consciousness of ourselves and of our own feelings has a sort of direct and absolute value4 ,it is supposed to afford a testimony which does not require to be sifted like our consciousness of external events4 ,but in reality it needs far more criticism to be applied to it than any other mode of apprehension4 ,to a certain degree we can sift our experience of the external world1 and divide it into two portions4 ,we can determine the self elements and the realities4 ,but with regard to our own nature and emotions1 the discovery which makes a science possible has yet to be made4 ,there are certain indications1 however1 springing from our observation of our own bodies1 which have a certain degree of interest4 ,it is found that the processes of thought and feeling are connected with the brain4 ,if the brain is disturbed1 thoughts1 sights1 and sounds come into the consciousness which have no objective cause in the external world4 ,hence we may conclusively say that the human being1 whatever he is1 is in contact with the brain1 and through the brain with the body1 and through the body with the external world4 ,it is the structures and movements in the brain which the human being perceives4 ,it is by a structure in the brain that he apprehends nature1 not immediately4 ,the most beautiful sights and sounds have no effect on a human being unless there is the faculty in the brain of taking them in and handing them on to the consciousness4 ,hence1 clearly1 it is the movements and structure of the minute portions of matter forming the brain which the consciousness perceives4 ,and it is only by models and representations made in the stuff of the brain that the mind knows external changes4 ,now1 our brains are well furnished with models and representations of the facts and events of the external world4 ,but a most important fact still requires its due weight to be laid upon it4 ,these models and representations are made on a very minute scale--the particles of brain matter which form images and representations are beyond the power of the microscope in their minuteness4 ,hence the consciousness primarily apprehends the movements of matter of a degree of smallness which is beyond the power of observation in any other way4 ,hence we have a means of observing the movements of the minute portions of matter4 ,let us call those portions of the brain matter which are directly instrumental in making representations of the external world--let us call them brain molecules4 ,now1 these brain molecules are very minute portions of matter indeed2 generally they are made to go through movements and form structures in such a way as to represent the movements and structures of the external world of masses around us4 ,but it does not follow that the structures and movements which they perform of their own nature are identical with the movements of the portions of matter which we see around us in the world of matter4 ,it may be that these brain molecules have the power of four-dimensional movement1 and that they can go through four-dimensional movements and form four-dimensional structures4 ,if so1 there is a practical way of learning the movements of the very small particles of matter--by observing1 not what we can see1 but what we can think4 ,for1 suppose these small molecules of the brain were to build up structures and go through movements not in accordance with the rule of representing what goes on in the external world1 but in accordance with their own activity1 then they might go through four-dimensional movements and form four-dimensional structures4 ,and these movements and structures would be apprehended by the consciousness along with the other movements and structures1 and would seem as real as the others--but would have no correspondence in the external world4 ,they would be thoughts and imaginations1 not observations of external facts4 ,now1 this field of investigation is one which requires to be worked at4 ,at present it is only those structures and movements of the brain molecules which correspond to the realities of our three-dimensional space which are in general worked at consistently4 ,but in the practical part of this book it will be found that by proper stimulus the brain molecules will arrange themselves in structures representing a four-dimensional existence4 ,it only requires a certain amount of care to build up mental models of higher space existences4 ,in fact1 it is probably part of the difficulty of forming three-dimensional brain models1 that the brain molecules have to be limited in their own freedom of motion to the requirements of the limited space in which our practical daily life is carried on4 ,note4--,for my own part ,i should say that all those confusions in remembering which come from an image taking the place of the original mental model--as1 for instance1 the difficulty in remembering which way to turn a screw1 and the numerous cases of images in thought transference--may be due to a toppling over in the brain1 four-dimensionalwise1 of the structures formed--which structures would be absolutely safe from being turned into image structures if the brain molecules moved only three-dimensionalwise4 ,it is remarkable how in science 8explaining0 means the reference of the movements and tendencies to movement of the masses about us to the movements and tendencies to movement of the minute portions of matter4 ,thus1 the behaviour of gaseous bodies--the pressure which they exert1 the laws of their cooling and intermixture are explained by tracing the movements of the very minute particles of which they are composed4 ,,chapter ,,viii4 ,,relation ,,of ,,lower ,,to ,,higher ,,space4 ,,theory ,,of ,,the ,,ther4 ,at this point of our inquiries the best plan is to turn to the practical work1 and try if the faculty of thinking in higher space can be awakened in the mind4 ,the general outline of the method is the same as that which has been described for getting rid of the limitation of up and down from a block of cubes4 ,we supposed that the block was fixed2 and to get the sense of what it would be when gravity acted in a different way with regard to it1 we made a model of it as it would be under the new circumstances4 ,we thought out the relations which would exist2 and by practising this new arrangement we gradually formed the direct apprehension4 ,and so with higher-space arrangements4 ,we cannot put them up actually1 but we can say how they would look and be to the touch from various sides4 ,and we can put up the actual appearances of them1 not altogether1 but as models succeeding one another2 and by contemplation and active arrangement of these different views we call upon our inward power to manifest itself4 ,in preparing our general plan of work1 it is necessary to make definite assumptions with regard to our world1 our universe1 or we may call it our space1 in relation to the wider universe of four-dimensional space4 ,what our relation to it may be1 is altogether undetermined4 ,the real relationship will require a great deal of study to apprehend1 and when apprehended will seem as natural to us as the position of the earth among the other planets does to us now4 ,but we have not got to wait for this exploration in order to commence our work of higher-space thought1 for we know definitely that whatever our real physical relationship to this wider universe may be1 we are practically in exactly the same relationship to it as the creature we have supposed living on the surface of a smooth sheet is to the world of threefold space4 ,and this relationship of a surface to a solid or of a solid1 as we conjecture1 to a higher solid1 is one which we often find in nature4 ,a surface is nothing more nor less than the relation between two things4 ,two bodies touch each other4 ,the surface is the relationship of one to the other4 ,again1 we see the surface of water4 ,thus our solid existence may be the contact of two four-dimensional existences with each other2 and just as sensation of touch is limited to the surface of the body1 so sensation on a larger scale may be limited to this solid surface4 ,and it is a fact worthy of notice1 that in the surface of a fluid different laws obtain from those which hold throughout the mass4 ,there are a whole series of facts which are grouped together under the name of surface tensions1 which are of great importance in physics1 and by which the behaviour of the surfaces of liquids is governed4 ,and it may well be that the laws of our universe are the surface tensions of a higher universe4 ,but these expressions1 it is evident1 afford us no practical basis for investigation4 ,we must assume something more definite1 and because more definite 7in the absence of details drawn from experience71 more arbitrary4 ,and we will assume that the conditions under which we human beings are1 exactly resemble those under which the plane-beings are placed1 which have been described4 ,this forms the basis of our work2 and the practical part of it consists in doing1 with regard to higher space1 that which a plane-being would do with regard to our space in order to enable himself to realize what it was4 ,if we imagine one of these limited creatures whose life is cramped and confined studying the facts of space existence1 we find that he can do it in two ways4 ,he can assume another direction in addition to those which he knows2 and he can1 by means of abstract reasoning1 say what would take place in an ampler kind of space than his own4 ,all this would be formal work4 ,the conclusions would be abstract possibilities4 ,the other mode of study is this4 ,he can take some of these facts of his higher space and he can ponder over them in his mind1 and can make up in his plane world those different appearances which one and the same solid body would present to him1 and then he may try to realize inwardly what his higher existence is4 ,now1 it is evident that if the creature is absolutely confined to a two-dimensional existence1 then anything more than such existence will always be a mere abstract and formal consideration to him4 ,but if this higher-space thought becomes real to him1 if he finds in his mind a possibility of rising to it1 then indeed he knows that somehow he is not limited to his apparent world4 ,everything he sees and comes into contact with may be two-dimensional2 but essentially1 somehow1 himself he is not two-dimensional merely4 ,and a precisely similar piece of work is before us4 ,assuming as we must that our outer experience is limited to three-dimensional space1 we shall make up the appearances which the very simplest higher bodies would present to us1 and we shall gradually arrive at a more than merely formal and abstract appreciation of them4 ,we shall discover in ourselves a faculty of apprehension of higher space similar to that which we have of space4 ,and thus we shall discover1 each for himself1 that1 limited as his senses are1 he essentially somehow is not limited4 ,the mode and method in which this consciousness will be made general1 is the same in which the spirit of an army is formed4 ,the individuals enter into the service from various motives1 but each and all have to go through those movements and actions which correspond to the unity of a whole formed out of different members4 ,the inner apprehension which lies in each man of a participation in a life wider than that of his individual body1 is awakened and responds2 and the active spirit of the army is formed4 ,so with regard to higher space1 this faculty of apprehending intuitively four-dimensional relationships will be taken up because of its practical use4 ,individuals will be practically employed to do it by society because of the larger faculty of thought which it gives4 ,in fact1 this higher-space thought means as an affair of mental training simply the power of apprehending the results arising from four independent causes4 ,it means the power of dealing with a greater number of details4 ,and when this faculty of higher-space thought has been formed1 then the faculty of apprehending that higher existence in which men have part1 will come into being4 ,it is necessary to guard here against there being ascribed to this higher-space thought any other than an intellectual value4 ,it has no moral value whatever4 ,its only connexion with moral or ethical considerations is the possibility it will afford of recognizing more of the facts of the universe than we do now4 ,there is a gradual process going on which may be described as the getting rid of self elements4 ,this process is one of knowledge and feeling1 and either may be independent of the other4 ,at present1 in respect of feeling1 we are much further on than in respect to understanding1 and the reason is very much this3 ,when a self element has been got rid of in respect of feeling1 the new apprehension is put into practice1 and we live it into our organization4 ,but when a self element has been got rid of intellectually1 it is allowed to remain a matter of theory1 not vitally entering into the mental structure of individuals4 ,thus up and down was discovered to be a self element more than a thousand years ago2 but1 except as a matter of theory1 we are perfect barbarians in this respect up to the present day4 ,we have supposed a being living in a plane world1 that is1 a being of a very small thickness in a direction perpendicular to the surface on which he is4 ,now1 if we are situated analogously with regard to an ampler space1 there must be some element in our experience corresponding to each element in the plane-being's experience4 ,and it is interesting to ask1 in the case of the plane-being1 what his opinion would be with respect to the surface on which he was4 ,he would not recognize it as a surface with which he was in contact2 he would have no idea of a motion away from it or towards it4 ,but he would discover its existence by the fact that movements were transmitted along it4 ,by its vibrating and quivering1 it would impart movement to the particles of matter lying on it4 ,hence1 he would consider this surface to be a medium lying between bodies1 and penetrating them4 ,it would appear to him to have no weight1 but to be a powerful means of transmitting vibrations4 ,moreover1 it would be unlike any other substance with which he was acquainted1 inasmuch as he could never get rid of it4 ,however perfect a vacuum be made1 there would be in this vacuum just as much of this unknown medium as there was before4 ,moreover1 this surface would not hinder the movement of the particles of matter over it4 ,being smooth1 matter would slide freely over it4 ,and this would seem to him as if matter went freely through the medium4 ,then he would also notice the fact that vibrations of this medium would tear asunder portions of matter4 ,the plane surface1 being very compact1 compared to the masses of matter on it1 would1 by its vibrations1 shake them into their component parts4 ,hence he would have a series of observations which tended to show that this medium was unlike any ordinary matter with which he was acquainted4 ,although matter passed freely through it1 still by its shaking it could tear matter in pieces4 ,these would be very difficult properties to reconcile in one and the same substance4 ,then it is weightless1 and it is everywhere4 ,it might well be that he would regard the supposition of there being a plane surface1 on which he was1 as a preferable one to the hypothesis of this curious medium2 and thus he might obtain a proof of his limitations from his observations4 ,now1 is there anything in our experience which corresponds to this medium which the plane-being gets to observe8 ,do we suppose the existence of any medium through which matter freely moves1 which yet by its vibrations destroys the combinations of matter--some medium which is present in every vacuum1 however perfect1 which penetrates all bodies1 and yet can never be laid hold of8 ,these are precisely observations which have been made4 ,the substance which possesses all these qualities is called the ther4 ,and the properties of the ther are a perpetual object of investigation in science4 ,now1 it is not the place here to go into details1 as all we want is a basis for work2 and however arbitrary it may be1 it will serve if it enables us to investigate the properties of higher space4 ,we will suppose1 then1 that we are not in1 but on the ther1 only not on it in any known direction1 but that the new direction is that which comes in4 ,the ther is a smooth body1 along which we slide1 being distant from it at every point about the thickness of an atom2 or1 if we take our mean distance1 being distant from it by half the thickness of an atom measured in this new direction4 ,then1 just as in space objects1 a cube1 for instance1 can stand on the surface of a table1 or on the surface over which the plane-being moves1 so on the ther can stand a higher solid4 ,all that the plane-being sees or touches of a cube1 is the square on which it rests4 ,so all that we could see or touch of a higher solid would be that part by which it stood on the ther2 and this part would be to us exactly like any ordinary solid body4 ,the base of a cube would be to the plane-being like a square which is to him an ordinary solid4 ,now1 the two ways1 in which a plane-being would apprehend a solid body1 would be by the successive appearances to him of it as it passed through his plane2 and also by the different views of one and the same solid body which he got by turning the body over1 so that different parts of its surface come into contact with his plane4 ,and the practical work of learning to think in four-dimensional space1 is to go through the appearances which one and the same higher solid has4 ,often1 in the course of investigation in nature1 we come across objects which have a certain similarity1 and yet which are in parts entirely different4 ,the work of the mind consists in forming an idea of that whole in which they cohere1 and of which they are simple presentations4 ,the work of forming an idea of a higher solid is the most simple and most definite of all such mental operations4 ,if we imagine a plane world in which there are objects which correspond to our sun1 to the planets1 and1 in fact1 to all our visible universe1 we must suppose a surface of enormous extent on which great disks slide1 these disks being worlds of various orders of magnitude4 ,these disks would some of them be central1 and hot1 like our sun2 round them would circulate other disks1 like our planets4 ,and the systems of sun and planets must be conceived as moving with great velocity over the surface which bears them all4 ,and the movements of the atoms of these worlds will be the course of events in such worlds4 ,as the atoms weave together1 and form bodies altering1 becoming1 and ceasing1 so will bodies be formed and disappear4 ,and the plane which bears them all on its smooth surface will simply be a support to all these movements1 and influence them in no way4 ,is to be conscious of being conscious of being hot1 the same thing as to be conscious of being hot8 ,it is not the same4 ,there is a standing outside1 and objectivation of a state of mind which every one would say in the first state was very different from the simple consciousness4 ,but the consciousness must do as much in the first case as in the second4 ,hence the feeling hot is very different from the consciousness of feeling hot4 ,a feeling which we always have1 we should not be conscious of--a sound always present ceases to be heard4 ,hence consciousness is a concomitant of change1 that is1 of the contact between one state and another4 ,if a being living on such a plane were to investigate the properties1 he would have to suppose the solid to pass through his plane in order to see the whole of its surface4 ,thus we may imagine a cube resting on a table to begin to penetrate through the table4 ,if the cube passes through the surface1 making a clean cut all round it1 so that the plane-being can come up to it and investigate it1 then the different parts of the cube as it passes through the plane will be to him squares1 which he apprehends by the boundary lines4 ,the cut which there is in his plane must be supposed not to be noticed1 he must be able to go right up to the cube without hindrance1 and to touch and see that thin slice of it which is just above the plane4 ,and so1 when we study a higher solid1 we must suppose that it passes through the ther1 and that we only see that thin three-dimensional section of it which is just about to pass from one side to the other of the ther4 ,when we look on a solid as a section of a higher solid1 we have to suppose the ther broken through1 only we must suppose that it runs up to the edge of the body which is penetrating it1 so that we are aware of no breach of continuity4 ,the surface of the ther must then be supposed to have the properties of the surface of a fluid2 only1 of course1 it is a solid three-dimensional surface1 not a two-dimensional surface4 ,,chapter ,,ix4 ,,another ,,view ,,of ,,the ,,ther4 ,,material ,,and ,,therial ,,bodies4 ,we have supposed in the case of a plane world that the surface on which the movements take place is inactive1 except by its vibrations4 ,it is simply a smooth support4 ,for the sake of simplicity let us call this smooth surface 8the ther0 in the case of a plane world4 ,the ther then we have imagined to be simply a smooth1 thin sheet1 not possessed of any definite structure1 but excited by real disturbances of the matter on it into vibrations1 which carry the effect of these disturbances as light and heat to other portions of matter4 ,now1 it is possible to take an entirely different view of the ther in the case of a plane world4 ,let us imagine that1 instead of the ther being a smooth sheet serving simply as a support1 it is definitely marked and grooved4 ,let us imagine these grooves and channels to be very minute1 but to be definite and permanent4 ,then1 let us suppose that1 instead of the matter which slides in the ther having attractions and repulsions of its own1 that it is quite inert1 and has only the properties of inertia4 ,that is to say1 taking a disk or a plane world as a specimen1 the whole disk is sliding on the ther in virtue of a certain momentum which it has1 and certain portions of its matter fit into the grooves in the ther1 and move along those grooves4 ,the size of the portions is determined by the size of the grooves4 ,and let us call those portions of matter which occupy the breadth of a groove1 atoms4 ,then it is evident that the disk sliding along over the ther1 its atoms will move according to the arrangement of the grooves over which the disk slides4 ,if the grooves at any one particular place come close together1 there will be a condensation of matter at that place when the disk passes over it2 and if the grooves separate1 there will be a rarefaction of matter4 ,if we imagine five particles1 each slipping along in its own groove1 if the particles are arranged in the form of a regular pentagon1 and the grooves are parallel1 then these five particles1 moving evenly on1 will maintain their positions with regard to one another1 and a body would exist like a pentagon1 lasting as long as the groves remained parallel4 ,but if1 after some distance had been traversed by the disk1 and these five particles were brought into a region where one of the grooves tended away from the others1 the shape of the pentagon would be destroyed1 it would become some irregular figure4 ,and it is easy to see that if the grooves separated1 and other grooves came in amongst them1 along which other portions of matter were sliding1 that the pentagon would disappear as an isolated body1 that its constituent matter would be separated1 and that its particles would enter into other shapes as constituents of them1 and not of the original pentagon4 ,thus1 in cases of greater complication1 an elaborate structure may be supposed to be formed1 to alter1 and to pass away2 its origin1 growth1 and decay being due1 not to any independent motion of the particles constituting it1 but to the movement of the disk whereby its portions of matter were brought to regions where there was a particular disposition of the grooves4 ,then the nature of the shape would really be determined by the grooves1 not by the portions of matter which passed over them--they would become manifest as giving rise to a material form when a disk passed over them1 but they would subsist independently of the disk2 and if another disk were to pass over the same grooves1 exactly the same material structures would spring up as came into being before4 ,if we make a similar supposition about our ther along which our earth slides1 we may conceive the movements of the particles of matter to be determined1 not by attractions or repulsions exerted on one another1 but to be set in existence by the alterations in the directions of the grooves of the ther along which they are proceeding4 ,if the grooves were all parallel1 the earth would proceed without any other motion than that of its path in the heavens4 ,but with an alteration in the direction of the grooves1 the particles1 instead of proceeding uniformly with the mass of the earth1 would begin to move amongst each other4 ,and by a sufficiently complicated arrangement of grooves it may be supposed that all the movements of the forms we see around us are due to interweaving and variously disposed grooves4 ,thus the movements1 which any body goes through1 would depend on the arrangement of the thereal grooves along which it was passing4 ,as long as the grooves remain grouped together in approximately the same way1 it would maintain its existence as the same body2 but when the grooves separated1 and became involved with the grooves of other objects1 this body would cease to exist separately4 ,thus the separate existences of the earth might conceivably be due to the disposition of those parts of the ther over which the earth passed4 ,and thus any object would have to be separated into two parts1 one the thereal form1 or modification which lasted1 the other the material particles which1 coming on with blind momentum1 were directed into such movements as to produce the actual objects around us4 ,in this way there would be two parts in any organism1 the material part and the thereal part4 ,there would be the material body1 which soon passes and becomes indistinguishable from any other material body1 and the thereal body which remains4 ,now1 if we direct our attention to the material body1 we see the phenomena of growth1 decay1 and death1 the coming and the passing away of a living being1 isolated during his existence1 absolutely merged at his death into the common storehouse of matter4 ,but if we regard the thereal body1 we find something different4 ,we find an organism which is not so absolutely separated from the surrounding organisms--an organism which is part of the ther1 and which is linked to other thereal organisms by its very substance--an organism between which and others there exists a unity incapable of being broken1 and a common life which is rather marked than revealed by the matter which passes over it4 ,the thereal body moreover remains permanently when the material body has passed away4 ,the correspondences between the thereal body and the life of an organism such as we know1 is rather to be found in the emotional region than in the one of outward observation4 ,to the thereal form1 all parts of it are equally one2 but part of this form corresponds to the future of the material being1 part of it to his past4 ,thus1 care for the future and regard for the past would be the way in which the material being would exhibit the unity of the thereal body1 which is both his past1 his present1 and his future4 ,that is to say1 suppose the thereal body capable of receiving an injury1 an injury in one part of it would correspond to an injury in a man's past2 an injury in another part1--that which the material body was traversing1--would correspond to an injury to the man at the present moment2 injury to the thereal body at another part1 would correspond to injury coming to the man at some future time4 ,and the self-preservation of the thereal body1 supposing it to have such a motive1 would in the last case be the motive of regarding his own future to the man4 ,and inasmuch as the man felt the real unity of his thereal body1 and did not confine his attention to his material body1 which is absolutely disunited at every moment from its future and its past--inasmuch as he apprehended his thereal unity1 insomuch would he care for his future welfare1 and consider it as equal in importance to his present comfort4 ,the correspondence between emotion and physical fact would be1 that the emotion of regard corresponded to an undiscerned thereal unity4 ,and then also1 just as the two tips of two fingers put down on a plane1 would seem to a plane-being to be two completely different bodies1 not connected together1 so one and the same thereal body might appear as two distinct material bodies1 and any regard between the two would correspond to an apprehension of their thereal unity4 ,in the supposition of an thereal body1 it is not necessary to keep to the idea of the rigidity and permanence of the grooves defining the motion of the matter which1 passing along1 exhibits the material body4 ,the thereal body may have a life of its own1 relations with other thereal bodies1 and a life as full of vicissitudes as that of the material body1 which in its total orbit expresses in the movements of matter one phase in the life of the thereal body4 ,but there are certain obvious considerations which prevent any serious dwelling on these speculations--they are only introduced here in order to show how the conception of higher space lends itself to the representation of certain indefinite apprehensions1--such as that of the essential unity of the race1--and affords a possible clue to correspondences between the emotional and the physical life4 ,the whole question of our relation to the ther has to be settled4 ,that which we call the ther is far more probably the surface of a liquid1 and the phenomena we observe due to surface tensions4 ,indeed1 the physical questions concern us here nothing at all4 ,it is easy enough to make some supposition which gives us a standing ground to discipline our higher-space perception2 and when that is trained1 we shall turn round and look at the facts4 ,the conception which we shall form of the universe will undoubtedly be as different from our present one1 as the ,copernican view differs from the more pleasant view of a wide immovable earth beneath a vast vault4 ,indeed1 any conception of our place in the universe will be more agreeable than the thought of being on a spinning ball1 kicked into space without any means of communication with any other inhabitants of the universe4 ,,chapter ,x4 ,,higher ,,space ,,and ,,higher ,,being4 ,,perception ,,and ,,inspiration4 ,in the instinctive and sense perception of man and nature there is all hidden1 which reflection afterwards brings into consciousness4 ,we are conscious of somewhat higher than each individual man when we look at men4 ,in some1 this consciousness reaches an extreme pitch1 and becomes a religious apprehension4 ,but in none is it otherwise than instinctive4 ,the apprehension is sufficiently definite to be certain4 ,but it is not expressible to us in terms of the reason4 ,now1 ,i have shown that by using the conception of higher space it is easy enough to make a supposition which shall show all mankind as physical parts of one whole4 ,our apparent isolation as bodies from each other is by no means so necessary to assume as it would appear4 ,but1 of course1 a supposition of that kind is of no value1 except as showing a possibility4 ,if we came to examine into the matter closely1 we should find a natural relationship which accounted for our consciousness being limited as at present it is4 ,the first thing to be done1 is to organize our higher-space perception1 and then look4 ,we cannot tell what external objects will blend together into the unity of a higher being4 ,but just as the riddle of the two hands becomes clear to us from our first inspection of higher space1 so will there grow before our eyes greater unities and greater surprises4 ,we have been subject to a limitation of the most absurd character4 ,let us open our eyes and see the facts4 ,now1 it requires some training to open the eyes4 ,for many years ,i worked at the subject without the slightest success4 ,all was mere formalism4 ,but by adopting the simplest means1 and by a more thorough knowledge of space1 the whole flashed clear4 ,space shapes can only be symbolical of four-dimensional shapes2 and if we do not deal with space shapes directly1 but only treat them by symbols on the plane--as in analytical geometry--we are trying to get a perception of higher space through symbols of symbols1 and the task is hopeless4 ,but a direct study of space leads us to the knowledge of higher space4 ,and with the knowledge of higher space there come into our ken boundless possibilities4 ,all those things may be real1 whereof saints and philosophers have dreamed4 ,looking on the fact of life1 it has become clear to the human mind1 that justice1 truth1 purity1 are to be sought--that they are principles which it is well to serve4 ,and men have invented an abstract devotion to these1 and all comes together in the grand but vague conception of ,duty4 ,but all these thoughts are to those which spring up before us as the shadow on a bank of clouds of a great mountain is to the mountain itself4 ,on the piled-up clouds falls the shadow--vast1 imposing1 but dark1 colourless4 ,if the beholder but turns1 he beholds the mountain itself1 towering grandly with verdant pines1 the snowline1 and the awful peaks4 ,so all these conceptions are the way in which now1 with vision confined1 we apprehend the great existences of the universe4 ,instead of an abstraction1 what we have to serve is a reality1 to which even our real things are but shadows4 ,we are parts of a great being1 in whose service1 and with whose love1 the utmost demands of duty are satisfied4 ,how can it not be a struggle1 when the claims of righteousness mean diminished life1--even death1--to the individual who strives8 ,and yet to a clear and more rational view it will be seen that in his extinction and loss1 that which he loves1--that real being which is to him shadowed forth in the present existence of wife and child1--that being lives more truly1 and in its life those he loves are his for ever4 ,but1 of course1 there are mistakes in what we consider to be our duty1 as in everything else2 and this is an additional reason for pursuing the quest of this reality4 ,for by the rational observance of other material bodies than our own1 we come to the conclusion that there are other beings around us like ourselves1 whom we apprehend in virtue of two processes--the one simply a sense one of observation and reflection--the other a process of direct apprehension4 ,now1 if we did not go through the sense process of observation1 we might1 it is true1 know that there were other human beings around us in some subtle way--in some mesmeric feeling2 but we should not have that organized human life which1 dealing with the things of the world1 grows into such complicated forms4 ,we should for ever be good-humoured babies--a sensuous1 affectionate kind of jelly-fish4 ,and just so now with reference to the high intelligences by whom we are surrounded4 ,we feel them1 but we do not realize them4 ,to realize them1 it will be necessary to develop our power of perception4 ,the power of seeing with our bodily eye is limited to the three-dimensional section4 ,but ,i have shown that the inner eye is not thus limited2 that we can organize our power of seeing in higher space1 and that we can form conceptions of realities in this higher space1 just as we can in our ordinary space4 ,and this affords the groundwork for the perception and study of these other beings than man4 ,just as some mechanical means are necessary for the apprehension of our fellows in space1 so a certain amount of mechanical education is necessary for the perception of higher beings in higher space4 ,let us turn the current of our thought right round2 instead of seeking after abstractions1 and connecting our observations by ideas1 let us train our sense of higher space and build up conceptions of greater realities1 more absolute existences4 ,it is really a waste of time to write or read more generalities4 ,here is the grammar of the knowledge of higher being--let us learn it1 not spend time in speculating as to whither it will lead us4 ,yet one thing more4 ,we are1 with reference to the higher things of life1 like blind and puzzled children4 ,we know that we are members of one body1 limbs of one vine2 but we cannot discern1 except by instinct and feeling1 what that body is1 what the vine is4 ,if to know it would take away our feeling1 then it were well never to know it4 ,but fuller knowledge of other human beings does not take away our love for them2 what reason is there then to suppose that a knowledge of the higher existences would deaden our feelings8 ,and then1 again1 we each of us have a feeling that we ourselves have a right to exist4 ,we demand our own perpetuation4 ,no man1 ,i believe1 is capable of sacrificing his life to any abstract idea2 in all cases it is the consciousness of contact with some being that enables him to make the last human sacrifice4 ,and what we can do by this study of higher space1 is to make this consciousness1 which has been reserved for a few1 the property of all4 ,do we not all feel that there is a limit to our devotion to abstractions1 none to beings whom we love4 ,and to love them1 we must know them4 ,then1 just as our own individual life is empty and meaningless without those we love1 so the life of the human race is empty and meaningless without a knowledge of those that surround it4 ,and although to some an inner knowledge of the oneness of all men is vouchsafed1 it remains to be demonstrated to the many4 ,the perpetual struggle between individual interests and the common good can only be solved by merging both impulses in a love towards one being whose life lies in the fulfilment of each4 ,and this search1 it seems to me1 affords the needful supplement to the inquiries of one with whose thought ,i have been very familiar1 and to which ,i return again1 after having abandoned it for the purely materialistic views which seem forced upon us by the facts of science4 ,all that he said seemed to me unsupported by fact1 unrelated to what we know4 ,but when ,i found that my knowledge was merely an empty pretence1 that it was the vanity of being able to predict and foretell that stood to me in the place of an absolute apprehension of fact--when all my intellectual possessions turned to nothingness1 then ,i was forced into that simple quest for fact1 which1 when persisted in and lived in1 opens out to the thoughts like a flower to the life-giving sun4 ,it is indeed a far safer course1 to believe that which appeals to us as noble1 than simply to ask what is true2 to take that which great minds have given1 than to demand that our puny ones should be satisfied4 ,but ,i suppose there is some good to some one in the scepticism and struggle of those who cannot follow in the safer course4 ,the thoughts of the inquirer to whom ,i allude may roughly be stated thus3-- ,he saw in human life the working out of a great process1 in the toil and strain of our human history he saw the becoming of man4 ,there is a defect whereby we fall short of the true measure of our being1 and that defect is made good in the course of history4 ,it is owing to that defect that we perceive evil2 and in the perception of evil and suffering lies our healing1 for we shall be forced into that path at last1 after trying every other1 which is the true one4 ,and this1 the history of the redemption of man1 is what he saw in all the scenes of life2 each most trivial occurrence was great and significant in relation to this4 ,and1 further1 he put forward a definite statement with regard to this defect1 this lack of true being1 for it lay1 he said1 in the self-centredness of our emotions1 in the limitation of them to our bodily selves4 ,he looked for a time when1 driven from all thoughts of our own pain or pleasure1 good or evil1 we should say1 in view of the miseries of our fellow-creatures1 ,let me be anyhow1 use my body and my mind in any way1 so that ,i serve4 ,and this1 it seems to me1 is the true aspiration2 for1 just as a note of music flings itself into the march of the melody1 and1 losing itself in it1 is used for it and lost as a separate being1 so we should throw these lives of ours as freely into the service of--whom8 ,here comes the difficulty4 ,let it be granted that we should have no self-rights1 limit our service in no way1 still the question comes1 ,what shall we serve8 ,it is far happier to have some concrete object to which we are devoted1 or to be bound up in the ceaseless round of active life1 wherein each day presents so many necessities that we have no room for choice4 ,but besides and apart from all these1 there comes to some the question1 8,what does it all mean80 ,to others1 an unlovable and gloomy aspect is presented1 wherein their life seems to be but used as a material worthless in itself and ungifted with any dignity or honour2 while to others again1 with the love of those they love1 comes a cessation of all personal interest in life1 and a disappointment and feeling of valuelessness4 ,and in all these cases some answer is needed4 ,and here human duty ceases4 ,we cannot make objects to love4 ,we can make machines and works of art1 but nothing which directly excites our love4 ,to give us that which rouses our love1 is the duty of one higher than ourselves4 ,and yet in one respect we have a duty--we must look4 ,what good would it be1 to surround us with objects of loving interest1 if we bury our regards in ourselves and will not see8 ,and does it not seem as if with lowered eyelids1 till only the thinnest slit was open1 we gazed persistently1 not on what is1 but on the thinnest conceivable section of it8 ,let it be granted that our right attitude is1 so to devote ourselves that there is no question as to what we will do or what we will not do1 but we are perfectly obedient servants4 ,the question is1 ,whom are we to serve8 ,it cannot be each individual1 for their claims are conflicting1 and as often as not there is more need of a master than of a servant4 ,moreover1 the aspect of our fellows does not always excite love1 which is the only possible inducer of the right attitude of service4 ,if we do not love1 we can only serve for a self motive1 because it is in some way good for ourselves4 ,thus it seems to me that we are reduced to this3 our only duty is to look for that which it is given us to love4 ,but this looking is not mere gazing4 ,to know1 we must act4 ,let any one try it4 ,he will find that unless he goes through a series of actions corresponding to his knowledge1 he gets merely a theoretic and outside view of any facts4 ,the way to know is this3 ,get somehow a means of telling what your perceptions would be if you knew1 and act in accordance with those perceptions4 ,thus1 with regard to a fellow-creature1 if we knew him we should feel what his feelings are4 ,let us then learn his feelings1 and act as if we had them4 ,it is by the practical work of satisfying his needs that we get to know him4 ,then1 may-be1 we love him2 or perchance it is said we may find that through him we have been brought into contact with one greater than him4 ,this is our duty--to know--to know1 not merely theoretically1 but practically2 and then1 when we know1 we have done our part2 if there is nothing1 we cannot supply it4 ,all we have to do is to look for realities4 ,we must not take this view of education--that we are horribly pressed for time1 and must learn1 somehow1 a knack of saying how things must be1 without looking at them4 ,but rather1 we must say that we have a long time--all our lives1 in which we will press facts closer and closer to our minds2 and we begin by learning the simplest4 ,there is an idea in that home of our inspiration--the fact that there are certain mechanical processes by which men can acquire merit4 ,this is perfectly true4 ,it is by mechanical processes that we become different2 and the science of education consists largely in systematizing these processes4 ,then1 just as space perceptions are necessary for the knowledge of our fellow-men1 and enable us to enter into human relationships with them in all the organized variety of civilized life1 so it is necessary to develop our perceptions of higher space1 so that we can apprehend with our minds the relationship which we have to beings higher than ourselves1 and bring our instinctive knowledge into clearer consciousness4 ,it appears to me self-evident1 that in the particular disposition of any portion of matter1 that is1 in any physical action1 there can be neither right nor wrong2 the thing done is perfectly indifferent4 ,at the same time1 it is only in things done that we come into relationship with the beings about us and higher than us4 ,consequently1 in the things we do lies the whole importance of our lives4 ,now1 many of our impulses are directly signs of a relationship in us to a being of which we are not immediately conscious4 ,the feeling of love1 for instance1 is always directed towards a particular individual2 but by love man tends towards the preservation and improvement of his race2 thus in the commonest and most universal impulses lie his relations to higher beings than the individuals by whom he is surrounded4 ,now1 along with these impulses are many instincts of a modifying tendency2 and1 being altogether in the dark as to the nature of the higher beings to whom we are related1 it is difficult to say in what the service of the higher beings consists1 in what it does not4 ,the only way is1 as in every other pre-rational department of life1 to take the verdict of those with the most insight and inspiration4 ,and any striving against such verdicts1 and discontent with them1 should be turned into energy towards finding out exactly what relation we have towards these higher beings by the study of ,space4 ,human life at present is an art constructed in its regulations and rules on the inspirations of those who love the undiscerned higher beings1 of which we are a part4 ,they love these higher beings1 and know their service4 ,but our perceptions are coarser2 and it is only by labour and toil that we shall be brought also to see1 and then lose the restraints that now are necessary to us in the fulness of love4 ,exactly what relationship there is towards us on the part of these higher beings we cannot say in the least4 ,we cannot even say whether there is more than humanity before the highest2 and any conception which we form now must use the human drama as its only possible mode of presentation4 ,but that there is such a relation seems clear2 and the ludicrous manner1 in which our perceptions have been limited1 is a sufficient explanation of why they have not been scientifically apprehended4 ,the mode1 in which an apprehension of these higher beings or being is at present secured1 is as follows2 and it bears a striking analogy to the mode by which the self is cut out of a block of cubes4 ,when we study a block of cubes1 we first of all learn it1 by starting from a particular cube1 and learning how all the others come with regard to that4 ,all the others are right or left1 up or down1 near or far1 with regard to that particular cube4 ,and the line of cubes starting from this first one1 which we take as the direction in which we look1 is1 as it were1 an axis about which the rest of the cubes are grouped4 ,we learn the block with regard to this axis1 so that we can mentally conceive the disposition of every cube as it comes regarded from one point of view4 ,next we suppose ourselves to be in another cube at the extremity of another axis2 and1 looking from this axis1 we learn the aspects of all the cubes1 and so on4 ,thus we impress on the feeling what the block of cubes is like from every axis4 ,in this way we get a knowledge of the block of cubes4 ,now1 to get a knowledge of humanity1 we must feel with many individuals4 ,each individual is an axis as it were1 and we must regard human beings from many different axes4 ,and as1 in learning the block of cubes1 muscular action1 as used in putting up the block of cubes1 is the means by which we impress on the feeling the different views of the block2 so1 with regard to humanity1 it is by acting with regard to the view of each individual that a knowledge is obtained4 ,that is to say1 that1 besides sympathizing with each individual1 we must act with regard to his view2 and acting so1 we shall feel his view1 and thus get to know humanity from more than one axis4 ,thus there springs up a feeling of humanity1 and of more4 ,those who feel superficially with a great many people1 are like those learners who have a slight acquaintance with a block of cubes from many points of view4 ,those who have some deep attachments1 are like those who know them well from one or two points of view4 ,thus there are two definite paths--one by which the instinctive feeling is called out and developed1 the other by which we gain the faculty of rationally apprehending and learning the higher beings4 ,in the one way it is by the exercise of a sympathetic and active life2 in the other1 by the study of higher space4 ,both should be followed2 but the latter way is more accessible to those who are not good4 ,for we at any rate have the industry to go through mechanical operations1 and know that we need something4 ,and after all1 perhaps1 the difference between the good and the rest of us1 lies rather in the former being aware4 ,there is something outside them which draws them to it1 which they see while we do not4 ,there is no reason1 however1 why this knowledge should not become demonstrable fact4 ,surely1 it is only by becoming demonstrable fact that the errors which have been necessarily introduced into it by human weakness will fall away from it4 ,the rational knowledge will not replace feeling1 but will form the vehicle by which the facts will be presented to our consciousness4 ,just as we learn to know our fellows by watching their deeds1--but it is something beyond the mere power of observing them that makes us regard them1--so the higher existences need to be known2 and1 when known1 then there is a chance that in the depths of our nature they will awaken feelings towards them like the natural response of one human being to another4 ,and when we reflect on what surrounds us1 when we think that the beauty of fruit and flower1 the blue depths of the sky1 the majesty of rock and ocean1--all these are but the chance and arbitrary view which we have of true being1--then we can imagine somewhat of the glories that await our coming4 ,how set out in exquisite loveliness are all the budding trees and hedgerows on a spring day--from here1 where they almost sing to us in their nearness1 to where1 in the distance1 they stand up delicately distant and distinct in the amethyst ocean of the air6 ,and there1 quiet and stately1 revolve the slow moving sun and the stars of the night4 ,all these are the fragmentary views which we have of great beings to whom we are related1 to whom we are linked1 did we but realize it1 by a bond of love and service in close connexions of mutual helpfulness4 ,just as here and there on the face of a woman sits the divine spirit of beauty1 so that all cannot but love who look--so1 presenting itself to us in all this mingled scene of air and ocean1 plain and mountain1 is a being of such loveliness that1 did we but know with one accord in one stream1 all our hearts would be carried in a perfect and willing service4 ,it is not that we need to be made different2 we have but to look and gaze1 and see that centre whereunto with joyful love all created beings move4 ,but not with effortless wonder will our days be filled1 but in toil and strong exertion2 for1 just as now we all labour and strive for an object1 our service is bound up with things which we do--so then we find no rest from labour1 but the sense of solitude and isolation is gone4 ,the bonds of brotherhood with our fellow-men grow strong1 for we know one common purpose4 ,and through the exquisite face of nature shines the spiritual light that gives us a great and never-failing comrade4 ,our task is a simple one--to lift from our mind that veil which somehow has fallen on us1 to take that curious limitation from our perception1 which at present is only transcended by inspiration4 ,and the means to do it is by throwing aside our reason--by giving up the idea that what we think or are has any value4 ,we too often sit as judges of nature1 when all we can be are her humble learners4 ,we have but to drink in of the inexhaustible fulness of being1 pressing it close into our minds1 and letting our pride of being able to foretell vanish into dust4 ,there is a curious passage in the works of ,immanuel ,kant1#a in which he shows that space must be in the mind before we can observe things in space4 8,for10 he says1 8since everything we conceive is conceived as being in space1 there is nothing which comes before our minds from which the idea of space can be derived2 it is equally present in the most rudimentary perception and the most complete40 ,hence he says that space belongs to the perceiving soul itself4 ,without going into this argument to abstract regions1 it has a great amount of practical truth4 ,all our perceptions are of things in space2 we cannot think of any detail1 however limited or isolated1 which is not in space4 #a ,the idea of space can 8nicht aus den ,verhltnissen der usseren ,erscheinung durch ,erfahrung erborgt sein1 sondern diese ussere ,erfahrung ist nur durch gedachte ,vorstellung allererst mglich40 ,hence1 in order to exercise our perceptive powers1 it is well to have prepared beforehand a strong apprehension of space and space relations4 ,and so1 as we pass on1 is it not easily conceivable that1 with our power of higher space perception so rudimentary and so unorganized1 we should find it impossible to perceive higher existences8 ,that mode of perception which it belongs to us to exercise is wanting4 ,what wonder1 then1 that we cannot see the objects which are ready1 were but our own part done8 ,think how much has come into human life through exercising the power of the three-dimensional space perception1 and we can form some measure1 in a faint way1 of what is in store for us4 ,there is a certain reluctance in us in bringing anything1 which before has been a matter of feeling1 within the domain of conscious reason4 ,we do not like to explain why the grass is green1 flowers bright1 and1 above all1 why we have the feelings which we pass through4 ,but this objection and instinctive reluctance is chiefly derived from the fact that explaining has got to mean explaining away4 ,we so often think that a thing is explained1 when it can be shown simply to be another form of something which we know already4 ,and1 in fact1 the wearied mind often does long to have a phenomenon shown to be merely a deduction from certain known laws4 ,but explanation proper is not of this kind2 it is introducing into the mind the new conception which is indicated by the phenomenon already present4 ,nature consists of many entities towards the apprehension of which we strive4 ,if for a time we break down the bounds which we have set up1 and unify vast fields of observation under one common law1 it is that the conceptions we formed at first are inadequate1 and must be replaced by greater ones4 ,but it is always the case1 that1 to understand nature1 a conception must be formed in the mind4 ,this process of growth in the mental history is hidden2 but it is the really important one4 ,the new conception satisfies more facts than the old ones1 is truer phenomenally2 and the arguments for it are its simplicity1 its power of accounting for many facts4 ,but the conception has to be formed first4 ,and the real history of advance lies in the growth of the new conceptions which every now and then come to light4 ,when the weather-wise savage looked at the sky at night1 he saw many specks of yellow light1 like fire-flies1 sprinkled amidst whitish fleece2 and sometimes the fleece remained1 the fire-spots went1 and rain came2 sometimes the fire-spots remained1 and the night was fine4 ,he did not see that the fire-points were ever the same1 the clouds different2 but by feeling dimly1 he knew enough for his purpose4 ,but when the thinking mind turned itself on these appearances1 there sprang up1--not all at once1 but gradually1--the knowledge of the sublime existences of the distant heavens1 and all the lore of the marvellous forms of water1 of air1 and the movements of the earth4 ,surely these realities1 in which lies a wealth of embodied poetry1 are well worth the delighted sensuous apprehension of the savage as he gazed4 ,perhaps something is lost1 but in the realities1 of which we know1 there is compensation4 ,and so1 when we learn to understand the meaning of these mysterious changes1 this course of natural events1 we shall find in the greater realities amongst which we move a fair exchange for the instinctive reverence1 which they now awaken in us4 ,in this book the task is taken up of forming the most simple and elementary of the great conceptions that are about us4 ,in the works of the poets1 and still more in the pages of religious thinkers1 lies an untold wealth of conception1 the organization of which in our every-day intellectual life is the work of the practical educator4 ,but none is capable of such simple demonstration and absolute presentation as this of higher space1 and none so immediately opens our eyes to see the world as a different place4 ,and1 indeed1 it is very instructive2 for when the new conception is formed1 it is found to be quite simple and natural4 ,we ask ourselves what we have gained2 and we answer3 ,nothing2 we have simply removed an obvious limitation4 ,and this is universally true2 it is not that we must rise to the higher by a long and laborious process4 ,we may have a long and laborious process to go through1 but1 when we find the higher1 it is this3 we discover our true selves1 our essential being1 the fact of our lives4 ,in this case1 we pass from the ridiculous limitation1 to which our eyes and hands seem to be subject1 of acting in a mere section of space1 to the fuller knowledge and feeling of space as it is4 ,how do we pass to this truer intellectual life8 ,simply by observing1 by laying aside our intellectual powers1 and by looking at what is4 ,we take that which is easiest to observe1 not that which is easiest to define2 we take that which is the most definitely limited real thing1 and use it as our touchstone whereby to explore nature4 ,as it seems to me1 ,kant made the great and fundamental statement in philosophy when he exploded all previous systems1 and all physics were reft from off the perceiving soul4 ,but what he did once and for all1 was too great to be a practical means of intellectual work4 ,the dynamic form of his absolute insight had to be found2 and it is in other works that the practical instances of the ,kantian method are to be found4 ,for1 instead of looking at the large foundations of knowledge1 the ultimate principles of experience1 late writers turned to the details of experience1 and tested every phenomenon1 not with the question1 ,what is this8 but with the question1 8,what makes me perceive thus80 ,and surely the question1 as so put1 is more capable of an answer2 for it is only the percipient1 as a subject of thought1 about which we can speak4 ,the absolute soul1 since it is the thinker1 can never be the subject of thought2 but1 as physically conditioned1 it can be thought about4 ,thus we can never1 without committing a ludicrous error1 think of the mind of man except as a material organ of some kind2 and the path of discovery lies in investigating what the devious line of his thought history is due to1 which winds between two domains of physics--the unknown conditions which affect the perceiver1 the partially known physics which constitute what we call the external world4 ,it is a pity to spend time over these reflections2 if they do not seem tame and poor compared to the practical apprehension which comes of working with the models1 then there is nothing in the whole subject4 ,if in the little real objects which the reader has to handle and observe does not lie to him a poetry of a higher kind than any expressed thought1 then all these words are not only useless1 but false4 ,if1 on the other hand1 there is true work to be done with them1 then these suggestions will be felt to be but mean and insufficient apprehensions4 ,for1 in the simplest apprehension of a higher space lies a knowledge of a reality which is1 to the realities we know1 as spirit is to matter2 and yet to this new vision all our solid facts and material conditions are but as a shadow is to that which casts it4 ,in the awakening light of this new apprehension1 the flimsy world quivers and shakes1 rigid solids flow and mingle1 all our material limitations turn into graciousness1 and the new field of possibility waits for us to look and behold4 ,,chapter ,,xi4 ,,space ,,the ,,scientific ,,basis ,,of ,,altruism ,,and ,,religion4 ,the reader will doubtless ask for some definite result corresponding to these words--something not of the nature of an hypothesis or a might-be4 ,and in that ,i can only satisfy him after my own powers4 ,my only strength is in detail and patience2 and if he will go through the practical part of the book1 it will assuredly dawn upon him that here is the beginning of an answer to his request4 ,i only study the blocks and stones of the higher life4 ,but here they are definite enough4 ,and the more eager he is for personal and spiritual truth1 the more eagerly do ,i urge him to take up the practical work1 for the true good comes to us through those who1 aspiring greatly1 still submit their aspirations to fact1 and who1 desiring to apprehend spirit1 still are willing to manipulate matter4 ,the particular problem at which ,i have worked for more than ten years1 has been completely solved4 ,it is possible for the mind to acquire a conception of higher space as adequate as that of our three-dimensional space1 and to use it in the same manner4 ,there are two distinct ways of studying space--our familiar space at present in use4 ,one is that of the analyst1 who treats space relations by his algebra1 and discovers marvellous relations4 ,the other is that of the observer or mechanician1 who studies the shapes of things in space directly4 ,a practical designer of machines would not find the knowledge of geometrical analysis of immediate help to him2 and an artist or draughtsman still less so4 ,now1 my inquiry was1 whether it was possible to get the same power of conception of four-dimensional space1 as the designer and draughtsman have of three-dimensional space4 ,it is possible4 ,and with this power it is possible for us to design machines in higher space1 and to conceive objects in this space1 just as a draughtsman or artist does4 ,analytical skill is not of much use in designing a statue or inventing a machine1 or in appreciating the detail of either a work of art or a mechanical contrivance4 ,and hitherto the study of four-dimensional space has been conducted by analysis4 ,here1 for the first time1 the fact of the power of conception of four-dimensional space is demonstrated1 and the means of educating it are given4 ,and ,i propose a complete system of work1 of which the volume on four space#b is the first instalment4 #b 8,science ,romance10 ,no4 ,i41 by ,c4 ,h4 ,hinton4 ,published by ,swan ,sonnenschein ,co4 ,i shall bring forward a complete system of four-dimensional thought--mechanics1 science1 and art4 ,the necessary condition is1 that the mind acquire the power of using four-dimensional space as it now does three-dimensional4 ,and there is another condition which is no less important4 ,we can never see1 for instance1 four-dimensional pictures with our bodily eyes1 but we can with our mental and inner eye4 ,the condition is1 that we should acquire the power of mentally carrying a great number of details4 ,if1 for instance1 we could think of the human body right down to every minute part in its right position1 and conceive its aspect1 we should have a four-dimensional picture which is a solid structure4 ,now1 to do this1 we must form the habit of mental painting1 that is1 of putting definite colours in definite positions1 not with our hands on paper1 but with our minds in thought1 so that we can recall1 alter1 and view complicated arrangements of colour existing in thought with the same ease with which we can paint on canvas4 ,this is simply an affair of industry2 and the mental power latent in us in this direction is simply marvellous4 ,in any picture1 a stroke of the brush put on without thought is valueless4 ,the artist is not conscious of the thought process he goes through4 ,for our purpose it is necessary that the manipulation of colour and form which the artist goes through unconsciously1 should become a conscious power1 and that1 at whatever sacrifice of immediate beauty1 the art of mental painting should exist beside our more unconscious art4 ,all that ,i mean is this--that in the course of our campaign it is necessary to take up the task of learning pictures by heart1 so that1 just as an artist thinks over the outlines of a figure he wants to draw1 so we think over each stroke in our pictures4 ,the means by which this can be done will be given in a future volume4 ,we throw ourselves on an enterprise in which we have to leave altogether the direct presentation to the senses4 ,we must acquire a sense-perception and memory of so keen and accurate a kind that we can build up mental pictures of greater complexity than any which we can see4 ,we have a vast work of organization1 but it is merely organization4 ,the power really exists and shows itself when it is looked for4 ,much fault may be found with the system of organization which ,i have adopted1 but it is the survivor of many attempts2 and although ,i could better it in parts1 still ,i think it is best to use it until1 the full importance of the subject being realized1 it will be the lifework of men of science to reorganize the methods4 ,the one thing on which ,i must insist is this--that knowledge is of no value1 it does not exist unless it comes into the mind4 ,to know that a thing must be is no use at all4 ,it must be clearly realized1 and in detail as it is1 before it can be used4 ,a whole world swims before us1 the apprehension of which simply demands a patient cultivation of our powers2 and then1 when the faculty is formed1 we shall recognize what the universe in which we are is like4 ,we shall learn about ourselves and pass into a new domain4 ,and ,i would speak to some minds who1 like myself1 share to a large extent the feeling of unsettledness and unfixedness of our present knowledge4 ,religion has suffered in some respects from the inaccuracy of its statements2 and it is not always seen that it consists of two parts--one a set of rules as to the management of our relations to the physical world about us1 and to our own bodies2 another1 a set of rules as to our relationship to beings higher than ourselves4 ,now1 on the former of these subjects1 on physical facts1 on the laws of health1 science has a fair standing ground of criticism1 and can correct the religious doctrines in many important respects4 ,but on the other part of the subject matter1 as to our relationship to beings higher than ourselves1 science has not yet the materials for judging4 ,the proposition which underlies this book is1 that we should begin to acquire the faculties for judging4 ,to judge1 we must first appreciate2 and how far we are from appreciating with science the fundamental religious doctrines ,i leave to any one to judge4 ,there is absolutely no scientific basis for morality1 using morality in the higher sense of other than a code of rules to promote the greatest physical and mental health and growth of a human being4 ,science does not give us any information which is not equally acceptable to the most selfish and most generous man2 it simply tells him of means by which he may attain his own ends1 it does not show him ends4 ,the prosecution of science is an ennobling pursuit2 but it is of scientific knowledge that ,i am now speaking in itself4 ,we have no scientific knowledge of any existences higher than ourselves--at least1 not recognized as higher4 ,but we have abundant knowledge of the actions of beings less developed than ourselves1 from the striking unanimity with which all inorganic beings tend to move towards the earth's centre1 to the almost equally uniform modes of response in elementary organized matter to different stimuli4 ,the question may be put3 ,in what way do we come into contact with these higher beings at present8 ,and evidently the answer is1 ,in those ways in which we tend to form organic unions--unions in which the activities of individuals coalesce in a living way4 ,the coherence of a military empire or of a subjugated population1 presenting no natural nucleus of growth1 is not one through which we should hope to grow into direct contact with our higher destinies4 ,but in friendship1 in voluntary associations1 and above all1 in the family1 we tend towards our greater life4 ,and it seems that the instincts of women are much more relative to this1 the most fundamental and important side of life1 than are those of men4 ,in fact1 until we know1 the line of advance had better be left to the feeling of women1 as they organize the home and the social life spreading out therefrom4 ,it is difficult1 perhaps1 for a man to be still and perceive2 but if he is so1 he finds that what1 when thwarted1 are meaningless caprices and empty emotionalities1 are1 on the part of woman1 when allowed to grow freely and unchecked1 the first beginnings of a new life--the shadowy filaments1 as it were1 by which an organism begins to coagulate together from the medium in which it makes its appearance4 ,in very many respects men have to make the conditions1 and then learn to recognize4 ,how can we see the higher beings about us1 when we cannot even conceive the simplest higher shapes8 ,we may talk about space1 and use big words1 but1 after all1 the preferable way of putting our efforts is this3 let us look first at the simplest facts of higher existence1 and then1 when we have learnt to realize these1 ,we shall be able to see what the world presents4 ,and then1 also1 light will be thrown on the constituent organisms of our own bodies1 when we see in the thorough development of our social life a relation between ourselves and a larger organism similar to that which exists between us and the minute constituents of our frame4 ,the problem1 as it comes to me1 is this3 it is clearly demonstrated that self-regard is to be put on one side--and self-regard in every respect--not only should things painful and arduous be done1 but things degrading and vile1 so that they serve4 ,i am to sign any list of any number of deeds which the most foul imagination can suggest1 as things which ,i would do did the occasion come when ,i could benefit another by doing them2 and1 in fact1 there is to be no characteristic in any action which ,i would shrink from did the occasion come when it presented itself to be done for another's sake4 ,and ,i believe that the soul is absolutely unstained by the action1 provided the regard is for another4 ,but this is1 in truth1 a dangerous doctrine2 at one ,sweep it puts away all absolute commandments1 all absolute verdicts of right about things1 and leaves the agent to his own judgment4 ,it is a kind of rule of life which requires most absolute openness1 and demands that society should frame severe and insuperable regulations2 for otherwise1 with the motives of the individual thus liberated from absolute law1 endless varieties of conduct would spring forth1 and the wisdom of individual men is hardly enough to justify their irresponsible action4 ,still1 it does seem that1 as an ideal1 the absolute absence of self-regard is to be aimed at4 ,with a strong religious basis1 this would work no harm1 for the rules of life1 as laid down by religions1 would suffice4 ,but there are many who do not accept these rules as any absolute indication of the will of ,god1 but only as the regulations of good men1 which have a claim to respect and nothing more4 ,and thus it seems to me that altruism--thoroughgoing altruism--hands over those who regard it as an ideal1 and who are also of a sceptical turn of mind1 to the most absolute unfixedness of theory1 and1 very possibly1 to the greatest errors in life4 ,and here we come to the point where the study of space becomes so important4 ,for if this rule of altruism is the right one1 if it appeals with a great invitation to us1 we need not therefore try it with less precaution than we should use in other affairs of infinitely less importance4 ,when we want to know if a plank will bear1 we entrust it with a different load from that of a human body4 ,and if this law of altruism is the true one1 let us try it where failure will not mean the ruin of human beings4 ,now1 in knowledge1 pure altruism means so to bury the mind in the thing known that all particular relations of one's self pass away4 ,the altruistic knowledge of the heavens would be1 to feel that the stars were vast bodies1 and that ,i am moving rapidly4 ,it would be1 to know this1 not as a matter of theory1 but as a matter of habitual feeling4 ,whether this is possible1 ,i do not know2 but a somewhat similar attempt can be made with much simpler means4 ,in a different place ,i have described the process of acquiring an altruistic knowledge of a block of cubes2 and the results of the laborious processes involved are well worth the trouble4 ,for as a clearly demonstrable fact this comes before one4 ,to acquire an absolute knowledge of a block of cubes1 so that all self relations are cast out1 means that one has to take the view of a higher being4 ,it suddenly comes before one1 that the particular relations which are so fixed and important1 and seem so absolutely sure when one begins the process of learning1 are by no means absolute facts1 but marks of a singular limitation1 almost a degradation1 on one's own part4 ,in the determined attempt to know the most insignificant object perfectly and thoroughly1 there flashes before one's eyes an existence infinitely higher than one's own4 ,and with that vision there comes1--,i do not speak from my own experience only1--a conviction that our existence also is not what we suppose--that this bodily self of ours is but a limit too4 ,and the question of altruism1 as against self-regard1 seems almost to vanish1 for by altruism we come to know what we truly are4 8,what we truly are10 ,i do not mean apart from space and matter1 but what we really are as beings having a space existence2 for our way of thinking about existence is to conceive it as the relations of bodies in space4 ,to think is to conceive realities in space4 ,just as1 to explore the distant stars of the heavens1 a particular material arrangement is necessary which we call a telescope1 so to explore the nature of the beings who are higher than us1 a mental arrangement is necessary4 ,we must prepare our power of thinking as we prepare a more extended power of looking4 ,we want a structure developed inside the skull for the one purpose1 while an exterior telescope will do for the other4 ,and thus it seems that the difficulties which we first apprehended fall away4 ,to us1 looking with half-blinded eyes at merely our own little slice of existence1 our filmy all1 it seemed that altruism meant disorder1 vagary1 danger4 ,but when we put it into practice in knowledge1 we find that it means the direct revelation of a higher being and a call to us to participate ourselves too in a higher life--nay1 a consciousness comes that we are higher than we know4 ,and so with our moral life as with our intellectual life4 ,is it not the case that those1 who truly accept the rule of altruism1 learn life in new dangerous ways8 ,it is true that we must give up the precepts of religion as being the will of ,god2 but then we shall learn that the will of ,god shows itself partly in the religious precepts1 and comes to be more fully and more plainly known as an inward spirit4 ,and that difficulty1 too1 about what we may do and what we may not1 vanishes also4 ,for1 if it is the same about our fellow-creatures as it is about the block of cubes1 when we have thrown out the self-regard from our relationship to them1 we shall feel towards them as a higher being than man feels towards them1 we shall feel towards them as they are in their true selves1 not in their outward forms1 but as eternal loving spirits4 ,and then those instincts which humanity feels with a secret impulse to be sacred and higher than any temporary good will be justified--or fulfilled4 ,there are two tendencies--one towards the direct cultivation of the religious perceptions1 the other to reducing everything to reason4 ,it will be but just for the exponents of the latter tendency to look at the whole universe1 not the mere section of it which we know1 before they deal authoritatively with the higher parts of religion4 ,and those who feel the immanence of a higher life in us will be needed in this outlook on the wider field of reality1 so that they1 being fitted to recognize1 may tell us what lies ready for us to know4 ,the true path of wisdom consists in seeing that our intellect is foolishness--that our conclusions are absurd and mistaken1 not in speculating on the world as a form of thought projected from the thinking principle within us--rather to be amazed that our thought has so limited the world and hidden from us its real existences4 ,to think of ourselves as any other than things in space and subject to material conditions1 is absurd1 it is absurd on either of two hypotheses4 ,if we are really things in space1 then of course it is absurd to think of ourselves as if we were not so4 ,on the other hand1 if we are not things in space1 then conceiving in space is the mode in which that unknown which we are exists as a mind4 ,its mental action is space-conception1 and then to give up the idea of ourselves as in space1 is not to get a truer idea1 but to lose the only power of apprehension of ourselves which we possess4 ,and yet there is1 it must be confessed1 one way in which it may be possible for us to think without thinking of things in space4 ,that way is1 not to abandon the use of space-thought1 but to pass through it4 ,when we think of space1 we have to think of it as infinity extended1 and we have to think of it as of infinite dimensions4 ,now1 as ,i have shown in 8,the ,law of the ,valley10#c when we come upon infinity in any mode of our thought1 it is a sign that that mode of thought is dealing with a higher reality than it is adapted for1 and in struggling to represent it1 can only do so by an infinite number of terms4 ,now1 space has an infinite number of positions and turns1 and this may be due to the attempt forced upon us to think of things higher than space as in space4 ,if so1 then the way to get rid of space from our thoughts1 is1 not to go away from it1 but to pass through it--to think about larger and larger systems of space1 and space of more and more dimensions1 till at last we get to such a representation in space of what is higher than space1 that we can pass from the space-thought to the more absolute thought without that leap which would be necessary if we were to try to pass beyond space with our present very inadequate representation in it of what really is4 #c 8,science ,romances10 ,no4 ,,ii4 ,again and again has human nature aspired and fallen4 ,the vision has presented itself of a law which was love1 a duty which carried away the enthusiasm1 and in which the conflict of the higher and lower natures ceased because all was enlisted in one loving service4 ,but again and again have such attempts failed4 ,the common-sense view1 that man is subject to law1 external law1 remains--that there are fates whom he must propitiate and obey4 ,and there is a strong sharp curb1 which1 if it be not brought to bear by the will1 is soon pulled tight by the world1 and one more tragedy is enacted1 and the over-confident soul is brought low4 ,and the rock on which such attempts always split1 is in the indulgence of some limited passion4 ,some one object fills the soul with its image1 and in devotion to that1 other things are sacrificed1 until at last all comes to ruin4 ,but what does this mean8 ,surely it is simply this1 that where there should be knowledge there is ignorance4 ,it is not that there is too much devotion1 too much passion1 but that we are ignorant and blind1 and wander in error4 ,we do not know what it is we care for1 and waste our effort on the appearance4 ,there is no such thing as wrong love2 there is good love and bad knowledge1 and men who err1 clasp phantoms to themselves4 ,religion is but the search for realities2 and thought1 conscious of its own limitations1 is its best aid4 ,let a man care for any one object--let his regard for it be as concentrated and exclusive as you will1 there will be no danger if he truly apprehends that which he cares for4 ,its true being is bound up with all the rest of existence1 and1 if his regard is true to one1 then1 if that one is really known1 his regard is true to all4 ,there is a question sometimes asked1 which shows the mere formalism into which we have fallen4 ,we ask3 ,what is the end of existence8 ,a mere play on words6 ,for to conceive existence is to feel ends4 ,the knowledge of existence is the caring for objects1 the fear of dangers1 the anxieties of love4 ,immersed in these1 the triviality of the question1 what is the end of existence8 becomes obvious4 ,if1 however1 letting reality fade away1 we play with words1 some questions of this kind are possible2 but they are mere questions of words1 and all content and meaning has passed out of them4 ,the task before us is this3 we strive to find out that physical unity1 that body which men are parts of1 and in the life of which their true unity lies4 ,the existence of this one body we know from the utterances of those whom we cannot but feel to be inspired2 we feel certain tendencies in ourselves which cannot be explained except by a supposition of this kind4 ,and1 now1 we set to work deliberately to form in our minds the means of investigation1 the faculty of higher-space conception4 ,to our ordinary space-thought1 men are isolated1 distinct1 in great measure antagonistic4 ,but with the first use of the weapon of higher thought1 it is easily seen that all men may really be members of one body1 their isolation may be but an affair of limited consciousness4 ,there is1 of course1 no value as science in such a supposition4 ,but it suggests to us many possibilities2 it reveals to us the confined nature of our present physical views1 and stimulates us to undertake the work necessary to enable us to deal adequately with the subject4 ,the work is entirely practical and detailed2 it is the elaboration1 beginning from the simplest objects of an experience in thought1 of a higher-space world4 ,to begin it1 we take up those details of position and relation which are generally relegated to symbolism or unconscious apprehension1 and bring these waste products of thought into the central position of the laboratory of the mind4 ,we turn all our attention on the most simple and obvious details of our every-day experience1 and thence we build up a conception of the fundamental facts of position and arrangement in a higher world4 ,we next study more complicated higher shapes1 and get our space perception drilled and disciplined4 ,then we proceed to put a content into our framework4 ,the means of doing this are twofold--observation and inspiration4 ,as to observation1 it is hardly possible to describe the feelings of that investigator who shall distinctly trace in the physical world1 and experimentally demonstrate the existence of the higher-space facts which are so curiously hidden from us4 ,he will lay the first stone for the observation and knowledge of the higher beings to whom we are related4 ,as to the other means1 it is obvious1 surely1 that if there has ever been inspiration1 there is inspiration now4 ,inspiration is not a unique phenomenon4 ,it has existed in absolutely marvellous degree in some of the teachers of the ancient world2 but that1 whatever it was1 which they possessed1 must be present now1 and1 if we could isolate it1 be a demonstrable fact4 ,and ,i would propose to define inspiration as the faculty1 which1 to take a particular instance1 does the following3-- ,if a square penetrates a line cornerwise1 it marks out on the line a segment bounded by two points--that is1 we suppose a line drawn on a piece of paper1 and a square lying on the paper to be pushed so that its corner passes over the line4 ,then1 supposing the paper and the line to be in the same plane1 the line is interrupted by the square2 and1 of the square1 all that is observable in the line1 is a segment bounded by two points4 ,next1 suppose a cube to be pushed cornerwise through a plane1 and let the plane make a section of the cube4 ,the section will be a plane figure1 and it will be a triangle4 ,now1 first1 the section of a square by a line is a segment bounded by two points2 second1 the section of a cube by a plane is a triangle bounded by three lines4 ,hence1 we infer that the section of a figure in four dimensions analogous to a cube1 by three-dimensional space1 will be a tetrahedron--a figure bounded by four planes4 ,this is found to be true2 with a little familiarity with four-dimensional movements this is seen to be obvious4 ,but ,i would define inspiration as the faculty by which without actual experience this conclusion is formed4 ,how it is we come to this conclusion ,i am perfectly unable to say4 ,somehow1 looking at mere formal considerations1 there comes into the mind a conclusion about something beyond the range of actual experience4 ,we may call this reasoning from analogy2 but using this phrase does not explain the process4 ,it seems to me just as rational to say that the facts of the line and plane remind us of facts which we know already about four-dimensional figures--that they tend to bring these facts out into consciousness1 as ,plato shows with the boy's knowledge of the cube4 ,we must be really four-dimensional creatures1 or we could not think about four dimensions4 ,but whatever name we give to this peculiar and inexplicable faculty1 that we do possess it is certain2 and in our investigations it will be of service to us4 ,we must carefully investigate existence in a plane world1 and then1 making sure1 and impressing on our inward sense1 as we go1 every step we take with regard to a higher world1 we shall be reminded continually of fresh possibilities of our higher existence4 ,,part ,,ii4 ,,chapter ,i4 ,,three-,,space4 ,,genesis ,,of ,a ,,cube4 ,,appearances ,,of ,a ,,cube ,,to ,a ,,plane-,,being4 ,the models consist of a set of eight and a set of four cubes4 ,they are marked with different colours1 so as to show the properties of the figure in ,higher ,space1 to which they belong4 ,the simplest figure in one-dimensional space1 that is1 in a straight line1 is a straight line bounded at the two extremities4 ,the figure in this case consists of a length bounded by two points4 ,looking at ,cube #a1 and placing it so that the figure #a is uppermost1 we notice a straight line in contact with the table1 which is coloured ,orange4 ,it begins in a ,gold point and ends in a ,fawn point4 ,the ,orange extends to some distance on two faces of the ,cube2 but for our present purpose we suppose it to be simply a thin line4 ,this line we conceive to be generated in the following way4 ,let a point move and trace out a line4 ,let the point be the ,gold point1 and let it1 moving1 trace out the ,orange line and terminate in the ,fawn point4 ,thus the figure consists of the point at which it begins1 the point at which it ends1 and the portion between4 ,we may suppose the point to start as a ,gold point1 to change its colour to ,orange during the motion1 and when it stops to become ,fawn4 ,the motion we suppose from left to right1 and its direction we call ,x4 ,if1 now1 this ,orange line move away from us at right angles1 it will trace out a square4 ,let this be the ,black square1 which is seen underneath ,model #a4 ,the points1 which bound the line1 will during this motion trace out lines1 and to these lines there will be terminal points4 ,also1 the ,square will be terminated by a line on the opposite side4 ,let the ,gold point in moving away trace out a ,blue line and end in a ,buff point2 the ,fawn point a ,crimson line ending in a ,terracotta point4 ,the ,orange line1 having traced a ,black square1 ends in a ,green-grey line4 ,this direction1 away from the observer1 we call ,y4 ,now1 let the whole ,black square traced out by the ,orange line move upwards at right angles4 ,it will trace out a new figure1 a ,cube4 ,and the edges of the square1 while moving upwards1 will trace out squares4 ,bounding the cube1 and opposite to the ,black square1 will be another square4 ,let the ,orange line moving upwards trace a ,dark ,blue square and end in a ,reddish line4 ,the ,gold point traces a ,brown line2 the ,fawn point traces a ,french-grey line1 and these lines end in a ,light-blue and a ,dull-purple point4 ,let the ,blue line trace a ,vermilion square and end in a ,deep-yellow line4 ,let the ,buff point trace a ,green line1 and end in a ,red point4 ,the ,green-grey line traces a ,light-yellow square and ends in a ,leaden line2 the ,terracotta point traces a ,dark-slate line and ends in a ,deep-blue point4 ,the ,crimson line traces a ,blue-green square and ends in a ,bright-blue line4 ,finally1 the ,black square traces a ,cube1 the colour of which is invisible1 and ends in a white square4 ,we suppose the colour of the cube to be a ,light-buff4 ,the upward direction we call ,z4 ,thus we say3 ,the ,gold point moved ,z1 traces a ,brown line1 and ends in a ,light-blue point4 ,we can now clearly realize and refer to each region of the cube by a colour4 ,at the ,gold point1 lines from three directions meet1 the ,x line ,orange1 the ,y line ,blue1 the ,z line ,brown4 ,thus we began with a figure of one dimension1 a line1 we passed on to a figure of two dimensions1 a square1 and ended with a figure of three dimensions1 a cube4 99 99 99 99 99 ,the square represents a figure in two dimensions2 but if we want to realize what it is to a being in two dimensions1 we must not look down on it4 ,such a view could not be taken by a plane-being4 ,let us suppose a being moving on the surface of the table and unable to rise from it4 ,let it not know that it is upon anything1 but let it believe that the two directions and compounds of those two directions are all possible directions4 ,moreover1 let it not ask the question3 8,on what am ,i supported80 ,let it see no reason for any such question1 but simply call the smooth surface1 along which it moves1 ,space4 ,such a being could not tell the colour of the square traced by the ,orange line4 ,the square would be bounded by the lines which surround it1 and only by breaking through one of those lines could the plane-being discover the colour of the square4 ,in trying to realize the experience of a plane-being it is best to suppose that its two dimensions are upwards and sideways1 i4e41 ,z and ,x1 because1 if there be any matter in the plane-world1 it will1 like matter in the solid world1 exert attractions and repulsions4 ,the matter1 like the beings1 must be supposed very thin1 that is1 of so slight thickness that it is quite unnoticed by the being4 ,now1 if there be a very large mass of such matter lying on the table1 and a plane-being be free to move about it1 he will be attracted to it in every direction4 8,towards this huge mass0 would be 8,down10 and 8,away from it0 would be 8,up10 just as 8,towards the earth0 is to solid beings 8,down10 and 8,away from it0 is 8,up10 at whatever part of the globe they may be4 ,hence1 if we want to realize a plane-being's feelings1 we must keep the sense of up and down4 ,therefore we must use the ,z direction1 and it is more convenient to take ,z and ,x than ,z and ,y4 ,any direction lying between these is said to be compounded of the two2 for1 if we move slantwise for some distance1 the point reached might have been also reached by going a certain distance ,x1 and then a certain distance ,z1 or vice vers4 ,let us suppose the ,orange line has moved ,z1 and traced the ,dark-blue square ending in the ,reddish line4 ,if we now place a piece of stiff paper against the ,dark-blue square1 and suppose the plane-beings to move to and fro on that surface of the paper1 which touches the square1 we shall have means of representing their experience4 ,to obtain a more consistent view of their existence1 let us suppose the piece of paper extended1 so that it cuts through our earth and comes out at the antipodes1 thus cutting the earth in two4 ,then suppose all the earth removed away1 both hemispheres vanishing1 and only a very thin layer of matter left upon the paper on that side which touches the ,dark-blue square4 ,this represents what the world would be to a plane-being4 ,it is of some importance to get the notion of the directions in a plane-world1 as great difficulty arises from our notions of up and down4 ,we miss the right analogy if we conceive of a plane-world without the conception of up and down4 ,a good plan is1 to use a slanting surface1 a stiff card or book cover1 so placed that it slopes upwards to the eye4 ,then gravity acts as two forces4 ,it acts 7#a7 as a force pressing all particles upon the slanting surface into it1 and 7#b7 as a force of gravity along the plane1 making particles tend to slip down its incline4 ,we may suppose that in a plane-world there are two such forces1 one keeping the beings thereon to the plane1 the other acting between bodies in it1 and of such a nature that by virtue of it any large mass of plane-matter produces on small particles around it the same effects as the large mass of solid matter called our earth produces on small objects like our bodies situated around it4 ,in both cases the larger draws the smaller to itself1 and creates the sensations of up and down4 ,if we hold the cube so that its ,dark-blue side touches a sheet of paper held upwards to the eye1 and if we then look straight down along the paper1 confining our view to that which is in actual contact with the paper1 we see the same view of the cube as a plane-being would get4 ,we see a ,light-blue point1 a ,reddish line1 and a ,dull-purple point4 ,the plane-being only sees a line1 just as we only see a square of the cube4 ,the line where the paper rests on the table may be taken as representative of the surface of the plane-being's earth4 ,it would be merely a line to him1 but it would have the same property in relation to the plane-world1 as a square has in relation to a solid world2 in neither case can the notion of what in the latter is termed solidity be quite excluded4 ,if the plane-being broke through the line bounding his earth1 he would find more matter beyond it4 ,let us now leave out of consideration the question of 8up and down0 in a plane-world4 ,let us no longer consider it in the vertical1 or ,,zx1 position1 but simply take the surface 7,,xy7 of the table as that which supports a plane-world4 ,let us represent its inhabitants by thin pieces of paper1 which are free to move over the surface of the table1 but cannot rise from it4 ,also1 let the thickness 7i4e41 height above the surface7 of these beings be so small that they cannot discern it4 ,lastly let us premise there is no attraction in their world1 so that they have not any up and down4 ,placing ,cube #a in front of us1 let us now ask how a plane-being could apprehend such a cube4 ,the ,black face he could easily study4 ,he would find it bounded by ,gold point1 ,orange line1 ,fawn point1 ,crimson line1 and so on4 ,and he would discover it was ,black by cutting through any of these lines and entering it4 7,this operation would be equivalent to the mining of a solid being74 ,but of what came above the ,black square he would be completely ignorant4 ,let us now suppose a square hole to be made in the table1 so that the cube could pass through1 and let the cube fit the opening so exactly that no trace of the cutting of the table be visible to the plane-being4 ,if the cube began to pass through1 it would seem to him simply to change1 for of its motion he could not be aware1 as he would not know the direction in which it moved4 ,let it pass down till the ,white square be just on a level with the surface of the table4 ,the plane-being would then perceive a ,light-blue point1 a ,reddish line1 a ,dull-purple point1 a ,bright-blue line1 and so on4 ,these would surround a ,white square1 which belonged to the same body as that to which the ,black square belonged4 ,but in this body there would be a dimension1 which was not in the square4 ,our upward direction would not be apprehended by him directly4 ,motion from above downwards would only be apprehended as a change in the figure before him4 ,he would not say that he had before him different sections of a cube1 but only a changing square4 ,if he wanted to look at the upper square1 he could only do so when the ,black square had gone an inch below his plane4 ,to study the upper square simultaneously with the lower1 he would have to make a model of it1 and then he could place it beside the lower one4 ,looking at the cube1 we see that the ,reddish line corresponds precisely to the ,orange line1 and the ,deep-yellow to the ,blue line4 ,but if the plane-being had a model of the upper square1 and placed it on the right-hand side of the ,black square1 the ,deep-yellow line would come next to the ,crimson line of the ,black square4 ,there would be a discontinuity about it4 ,all that he could do would be to observe which part in the one square corresponded to which part in the other4 ,obviously too there lies something between the ,black square and the ,white4 ,the plane-being would notice that when a line moves in a direction not its own1 it traces out a square4 ,when the ,orange line is moved away1 it traces out the ,black square4 ,the conception of a new direction thus obtained1 he would understand that the ,orange line moving so would trace out a square1 and the ,blue line moving so would do the same4 ,to us these squares are visible as wholes1 the ,dark-blue1 and the ,vermilion4 ,to him they would be matters of verbal definition rather than ascertained facts4 ,however1 given that he had the experience of a cube being pushed through his plane1 he would know there was some figure1 whereof his square was part1 which was bounded by his square on one side1 and by a ,white square on another side4 ,we have supposed him to make models of these boundaries1 a ,black square and a ,white square4 ,the ,black square1 which is his solid matter1 is only one boundary of a figure in ,higher ,space4 ,but we can suppose the cube to be presented to him otherwise than by passing through his plane4 ,it can be turned round the ,orange line1 in which case the ,blue line goes out1 and1 after a time1 the ,brown line comes in4 ,it must be noticed that the ,brown line comes into a direction opposite to that in which the ,blue line ran4 ,these two lines are at right angles to each other1 and1 if one be moved upwards till it is at right angles to the surface of the table1 the other comes on to the surface1 but runs in a direction opposite to that in which the first ran4 ,thus1 by turning the cube about the ,orange line and the ,blue line1 different sides of it can be shown to a plane-being4 ,by combining the two processes of turning and pushing through the plane1 all the sides can be shown to the plane-being4 ,for instance1 if the cube be turned so that the ,dark-blue square be on the plane1 and it be then passed through1 the ,light-yellow square will come in4 ,now1 if the plane-being made a set of models of these different appearances and studied them1 he could form some rational idea of the ,higher ,solid which produced them4 ,he would become able to give some consistent account of the properties of this new kind of existence2 he could say what came into his plane space1 if the other space penetrated the plane edge-wise or corner-wise1 and could describe all that would come in as it turned about in any way4 ,he would have six models4 ,let us consider two of them--the ,black and the ,white squares4 ,we can observe them on the cube4 ,every colour on the one is different from every colour on the other4 ,if we now ask what lies between the ,orange line and the ,reddish line1 we know it is a square1 for the ,orange line moving in any direction gives a square4 ,and1 if the six models were before the plane-being1 he could easily select that which showed what he wanted4 ,for that which lies between ,orange line and ,reddish line must be bounded by ,orange and ,reddish lines4 ,he would search among the six models lying beside each other on his plane1 till he found the ,dark-blue square4 ,it is evident that only one other square differs in all its colours from the ,black square1 viz41 the ,white square4 ,for it is entirely separate4 ,the others meet it in one of their lines4 ,this total difference exists in all the pairs of opposite surfaces on the cube4 ,now1 suppose the plane-being asked himself what would appear if the cube turned round the ,blue line4 ,the cube would begin to pass through his space4 ,the ,crimson line would disappear beneath the plane and the ,blue-green square would cut it1 so that opposite to the ,blue line in the plane there would be a ,blue-green line4 ,the ,french-grey line and the ,dark-slate line would be cut in points1 and from the ,gold point to the ,french-grey point would be a ,dark-blue line2 and opposite to it would be a ,light-yellow line1 from the ,buff point to the ,dark-slate point4 ,thus the figure in the plane world would be an oblong instead of a square1 and the interior of it would be of the same ,light-buff colour as the interior of the cube4 ,it is assumed that the plane closes up round the passing cube1 as the surface of a liquid does round any object immersed4 ,illustration3 ,fig4 #a4 ,illustration3 ,fig4 #b4 ,illustration3 ,fig4 #c4 ,illustration3 ,fig4 #d4 ,illustration3 ,fig4 #e4 ,but1 in order to apprehend what would take place when this twisting round the ,blue line began1 the plane-being would have to set to work by parts4 ,he has no conception of what a solid would do in twisting1 but he knows what a plane does4 ,let him1 then1 instead of thinking of the whole ,black square1 think only of the ,orange line4 ,the ,dark-blue square stands on it4 ,as far as this square is concerned1 twisting round the ,blue line is the same as twisting round the ,gold point4 ,let him imagine himself in that plane at right angles to his plane-world1 which contains the ,dark-blue square4 ,let him keep his attention fixed on the line where the two planes meet1 viz41 that which is at first marked by the ,orange line4 ,we will call this line the line of his plane1 for all that he knows of his own plane is this line4 ,now1 let the ,dark-blue square turn round the ,gold point4 ,the ,orange line at once dips below the line of his plane1 and the ,dark-blue square passes through it4 ,therefore1 in his plane he will see a ,dark-blue line in place of the ,orange one4 ,and in place of the ,fawn point1 only further off from the ,gold point1 will be a ,french-grey point4 ,the ,diagrams 7#a71 7#b7 show how the cube appears as it is before and after the turning4 ,g is the ,gold1 ,f the ,fawn point4 ,in 7#b7 ,g is unmoved1 and the plane is cut by the ,french-grey line1 ,gr4 ,instead of imagining a direction he did not know1 the plane-being could think of the ,dark-blue square as lying in his plane4 ,but in this case the ,black square would be out off his plane1 and only the ,orange line would remain in it4 ,diagram 7#c7 shows the ,dark-blue square lying in his plane1 and ,diagram 7#d7 shows it turning round the ,gold point4 ,here1 instead of thinking about his plane and also that at right angles to it1 he has only to think how the square turning round the ,gold point will cut the line1 which runs left to right from ,g1 viz41 the dotted line4 ,the ,french-grey line is cut by the dotted line in a point4 ,to find out what would come in at other parts1 he need only treat a number of the plane sections of the cube perpendicular to the ,black square in the same manner as he had treated the ,dark-blue square4 ,every such section would turn round a point1 as the whole cube turned round the ,blue line4 ,thus he would treat the cube as a number of squares by taking parallel sections from the ,dark-blue to the ,light-yellow square1 and he would turn each of these round a corner of the same colour as the ,blue line4 ,combining these series of appearances1 he would discover what came into his plane as the cube turned round the ,blue line4 ,thus1 the problem of the turning of the cube could be settled by the consideration of the turnings of a number of squares4 ,as the cube turned1 a number of different appearances would be presented to the plane-being4 ,the ,black square would change into a ,light-buff oblong1 with ,dark-blue1 ,blue-green1 ,light-yellow1 and ,blue sides1 and would gradually elongate itself until it became as long as the diagonal of the square side of the cube2 and then the bounding line opposite to the ,blue line would change from ,blue-green to ,bright-blue1 the other lines remaining the same colour4 ,if the cube then turned still further1 the ,bright-blue line would become ,white1 and the oblong would diminish in length4 ,it would in time become a ,vermilion square1 with a ,deep-yellow line opposite to the ,blue line4 ,it would then pass wholly below the plane1 and only the ,blue line would remain4 ,if the turning were continued till half a revolution had been accomplished1 the ,black square would come in again4 ,but now it would come up into the plane from underneath4 ,it would appear as a ,black square exactly similar to the first2 but the ,orange line1 instead of running left to right from ,gold point1 would run right to left4 ,the square would be the same1 only differently disposed with regard to the ,blue line4 ,it would be the looking-glass image of the first square4 ,there would be a difference in respect of the lie of the particles of which it was composed4 ,if the plane-being could examine its thickness1 he would find that particles which1 in the first case1 lay above others1 now lay below them4 ,but1 if he were really a plane-being1 he would have no idea of thickness in his squares1 and he would find them both quite identical4 ,only the one would be to the other as if it had been pulled through itself4 ,in this phenomenon of symmetry he would apprehend the difference of the lie of the line1 which went in the1 to him1 unknown direction of up-and-down4 ,,chapter ,,ii4 ,,further ,,appearances ,,of ,a ,,cube ,,to ,a ,,plane-,,being4 ,before leaving the observation of the cube1 it is well to look at it for a moment as it would appear to a plane-being1 in whose world there was such a fact as attraction4 ,to do this1 let the cube rest on the table1 so that its ,dark-blue face is perpendicular in front of us4 ,now1 let a sheet of paper be placed in contact with the ,dark-blue square4 ,let up and sideways be the two dimensions of the plane-being1 and away the unknown direction4 ,let the line where the paper meets the table1 represent the surface of his earth4 ,then1 there is to him1 as all that he can apprehend of the cube1 a ,dark-blue square standing upright2 and1 when we look over the edge of the paper1 and regard merely the part in contact with the paper1 we see what the plane-being would see4 ,if the cube be turned round the up line1 the ,brown line1 the ,orange line will pass to the near side of the paper1 and the section made by the cube in the paper will be an oblong4 ,such an oblong can be cut out2 and when the cube is fitted into it1 it can be seen that it is bounded by a ,brown line and a ,blue-green line opposite thereto1 while the other boundaries are ,black and ,white lines4 ,next1 if we take a section half-way between the ,black and ,white squares1 we shall have a square cutting the plane of the aforesaid paper in a single line4 ,with regard to this section1 all we have to inquire is1 ,what will take the place of this line as the cube turns8 ,obviously1 the line will elongate4 ,from a ,dark-blue line it will change to a ,light-buff line1 the colour of the inside of the section1 and will terminate in a ,blue-green point instead of a ,french-grey4 ,again1 it is obvious that1 if the cube turns round the ,orange line1 it will give rise to a series of oblongs1 stretching upwards4 ,this turning can be continued till the cube is wholly on the near side of the paper1 and only the ,orange line remains4 ,and1 when the cube has made half a revolution1 the ,dark-blue square will return into the plane2 but it will run downwards instead of upwards as at first4 ,thereafter1 if the cube turn further1 a series of oblongs will appear1 all running downwards from the ,orange line4 ,hence1 if all the appearances produced by the revolution of the cube have to be shown1 it must be supposed to be raised some distance above the plane-being's earth1 so that those appearances may be shown which occur when it is turned round the ,orange line downwards1 as well as when it is turned upwards4 ,the unknown direction comes into the plane either upwards or downwards1 but there is no special connection between it and either of these directions4 ,if it come in upwards1 the ,brown line goes nearwards or -,y2 if it come in downwards1 or -,z1 the ,brown line goes away1 or ,y4 ,let us consider more closely the directions which the plane-being would have4 ,firstly1 he would have up-and-down1 that is1 away from his earth and towards it on the plane of the paper1 the surface of his earth being the line where the paper meets the table4 ,then1 if he moved along the surface of his earth1 there would only be a line for him to move in1 the line running right and left4 ,but1 being the direction of his movement1 he would say it ran forwards and backwards4 ,thus he would simply have the words up and down1 forwards and backwards1 and the expressions right and left would have no meaning for him4 ,if he were to frame a notion of a world in higher dimensions1 he must invent new words for distinctions not within his experience4 ,to repeat the observations already made1 let the cube be held in front of the observer1 and suppose the ,dark-blue square extended on every side so as to form a plane4 ,then let this plane be considered as independent of the ,dark-blue square4 ,now1 holding the ,brown line between finger and thumb1 and touching its extremities1 the ,gold and ,light-blue points1 turn the cube round the ,brown line4 ,the ,dark-blue square will leave the plane1 the ,orange line will tend towards the -,y direction1 and the ,blue line will finally come into the plane pointing in the ,x direction4 ,if we move the cube so that the line which leaves the plane runs ,y1 then the line which before ran ,y will come into the plane in the direction opposite to that of the line which has left the plane4 ,the ,blue line1 which runs in the unknown direction can come into either of the two known directions of the plane4 ,it can take the place of the ,orange line by turning the cube round the ,brown line1 or the place of the ,brown line by turning it round the ,orange line4 ,if the plane-being made models to represent these two appearances of the cube1 he would have identically the same line1 the ,blue line1 running in one of his known directions in the first model1 and in the other of his known directions in the second4 ,in studying the cube he would find it best to turn it so that the line of unknown direction ran in that direction in the positive sense4 ,in that case1 it would come into the plane in the negative sense of the known directions4 ,starting with the cube in front of the observer1 there are two ways in which the ,vermilion square can be brought into the imaginary plane1 that is the extension of the ,dark-blue square4 ,if the cube turn round the ,brown line so that the ,orange line goes away1 7i4e4 ,y71 the ,vermilion square comes in on the left of the ,brown line4 ,if it turn in the opposite direction1 the ,vermilion square comes in on the right of the ,brown line4 ,thus1 if we identify the plane-being with the ,brown line1 the ,vermilion square would appear either behind or before him4 ,these two appearances of the ,vermilion square would seem identical1 but they could not be made to coincide by any movement in the plane4 ,the diagram 7,fig4 #e47 shows the difference in them4 ,it is obvious that no turn in the plane could put one in the place of the other1 part for part4 ,thus the plane-being apprehends the reversal of the unknown direction by the disposition of his figures4 ,if a figure1 which lay on one side of a line1 changed into an identical figure on the other side of it1 he could be sure that a line of the figure1 which at first ran in the positive unknown direction1 now ran in the negative unknown direction4 ,we have dwelt at great length on the appearances1 which a cube would present to a plane-being1 and it will be found that all the points which would be likely to cause difficulty hereafter1 have been explained in this obvious case4 ,there is1 however1 one other way1 open to a plane-being of studying a cube1 to which we must attend4 ,this is1 by steady motion4 ,let the cube come into the imaginary plane1 which is the extension of the ,dark-blue square1 i4e4 let it touch the piece of paper which is standing vertical on the table4 ,then let it travel through this plane at right angles to it at the rate of an inch a minute4 ,the plane-being would first perceive a ,dark-blue square1 that is1 he would see the coloured lines bounding that square1 and enclosed therein would be what he would call a ,dark-blue solid4 ,in the movement of the cube1 however1 this ,dark-blue square would not last for more than a flash of time4 7,the edges and points on the models are made very large2 in reality they must be supposed very minute47 ,this ,dark-blue square would be succeeded by one of the colour of the cube's interior1 i4e4 by a ,light-buff square4 ,but this colour of the interior would not be visible to the plane-being4 ,he would go round the square on his plane1 and would see the bounding lines1 viz4 ,vermilion1 ,white1 ,blue-green1 ,black4 ,and at the corners he would see ,deep-yellow1 ,bright-blue1 ,crimson1 and ,blue points4 ,these lines and points would really be those parts of the faces and lines of the cube1 which were on the point of passing through his plane4 ,now1 there would be one difference between the ,dark-blue square and the ,light-buff with their respective boundaries4 ,the first only lasted for a flash2 the second would last for a minute or all but a minute4 ,consider the ,vermilion square4 ,it appears to the plane-being as a line4 ,the ,brown line also appears to him as a line4 ,but there is a difference between them4 ,the ,brown line only lasts for a flash1 whereas the ,vermilion line lasts for a minute4 ,hence1 in this mode of presentation1 we may say that for a plane-being a lasting line is the mode of apprehending a plane1 and a lasting plane 7which is a plane-being's solid7 is the mode of apprehending our solids4 ,in the same way1 the ,blue line1 as it passes through his plane1 gives rise to a point4 ,this point lasts for a minute1 whereas the ,gold point only lasted for a flash4 ,,chapter ,,iii4 ,,four-,,space4 ,,genesis ,,of ,a ,,tessaract4 ,,its ,,representation ,,in ,,three-,,space4 ,hitherto we have only looked at ,model #a4 ,this1 with the next seven1 represent what we can observe of the simplest body in ,higher ,space4 ,a few words will explain their construction4 ,a point by its motion traces a line4 ,a line by its motion traces either a longer line or an area2 if it moves at right angles to its own direction1 it traces a rectangle4 ,for the sake of simplicity1 we will suppose all movements to be an inch in length and at right angles to each other4 ,thus1 a point moving traces a line an inch long2 a line moving traces a square inch2 a square moving traces a cubic inch4 ,in these cases each of these movements produces something intrinsically different from what we had before4 ,a square is not a longer line1 nor a cube a larger square4 ,when the cube moves1 we are unable to see any new direction in which it can move1 and are compelled to make it move in a direction which has previously been used4 ,let us suppose there is an unknown direction at right angles to all our known directions1 just as a third direction would be unknown to a being confined to the surface of the table4 ,and let the cube move in this unknown direction for an inch4 ,we call the figure it traces a ,tessaract4 ,the models are representations of the appearances a ,tessaract would present to us if shown in various ways4 ,consider for a moment what happens to a square when moved to form a cube4 ,each of its lines1 moved in the new direction1 traces a square2 the square itself traces a new figure1 a cube1 which ends in another square4 ,now1 our cube1 moved in a new direction1 will trace a tessaract1 whereof the cube itself is the beginning1 and another cube the end4 ,these two cubes are to the tessaract as the ,black square and ,white square are to the cube4 ,a plane-being could not see both those squares at once1 but he could make models of them and let them both rest in his plane at once4 ,so also we can make models of the beginning and end of the tessaract4 ,model #a is the cube1 which is its beginning2 ,model #b is the cube which is its end4 ,it will be noticed that there are no two colours alike in the two models4 ,the ,silver point corresponds to the ,gold point1 that is1 the ,silver point is the termination of the line traced by the ,gold point moving in the new direction4 ,the sides correspond in the following manner3-- ,,sides4 ,model #a4 ,model #b4 ,black corresponds to ,bright-green ,white ,light-grey ,vermilion ,indian-red ,blue-green ,yellow-ochre ,dark-blue ,burnt-sienna ,light-yellow ,dun ,the two cubes should be looked at and compared long enough to ensure that the corresponding sides can be found quickly4 ,then there are the following correspondencies in points and lines4 ,,points4 ,model #a4 ,model #b4 ,gold corresponds to ,silver ,fawn ,turquoise ,terra-cotta ,earthen ,buff ,blue tint ,light-blue ,quaker-green ,dull-purple ,peacock-blue ,deep-blue ,orange-vermilion ,red ,purple ,,lines4 ,model #a4 ,model #b4 ,orange corresponds to ,leaf-green ,crimson ,dull-green ,green-grey ,dark-purple ,blue ,purple-brown ,brown ,dull-blue ,french-grey ,dark-pink ,dark-slate ,pale-pink ,green ,indigo ,reddish ,brown-green ,bright-blue ,dark-green ,leaden ,pale-yellow ,deep-yellow ,dark ,the colour of the cube itself is invisible1 as it is covered by its boundaries4 ,we suppose it to be ,sage-green4 ,these two cubes are just as disconnected when looked at by us as the black and white squares would be to a plane-being if placed side by side on his plane4 ,he cannot see the squares in their right position with regard to each other1 nor can we see the cubes in theirs4 ,let us now consider the vermilion side of ,model #a4 ,if it move in the ,x direction1 it traces the cube of ,model #a4 ,its ,gold point travels along the ,orange line1 and itself1 after tracing the cube1 ends in the ,blue-green square4 ,but if it moves in the new direction1 it will also trace a cube1 for the new direction is at right angles to the up and away directions1 in which the ,brown and ,blue lines run4 ,let this square1 then1 move in the unknown direction1 and trace a cube4 ,this cube we cannot see1 because the unknown direction runs out of our space at once1 just as the up direction runs out of the plane of the table4 ,but a plane-being could see the square1 which the ,blue line traces when moved upwards1 by the cube being turned round the ,blue line1 the ,orange line going upwards2 then the ,brown line comes into the plane of the table in the -,x direction4 ,so also with our cube4 ,as treated above1 it runs from the ,vermilion square out of our space4 ,but if the tessaract were turned so that the line which runs from the ,gold point in the unknown direction lay in our space1 and the ,orange line lay in the unknown direction1 we could then see the cube which is formed by the movement of the ,vermilion square in the new direction4 ,take ,model #e4 ,there is on it a ,vermilion square4 ,place this so that it touches the ,vermilion square on ,model #a4 ,all the marks of the two squares are identical4 ,this ,cube #e1 is the one traced by the ,vermilion square moving in the unknown direction4 ,in ,model #e1 the whole figure1 the tessaract1 produced by the movement of the cube in the unknown direction1 is supposed to be so turned that the ,orange line passes into the unknown direction1 and that the line which went in the unknown direction1 runs opposite to the old direction of the ,orange line4 ,looking at this new cube1 we see that there is a ,stone line running to the left from the ,gold point4 ,this line is that which the ,gold point traces when moving in the unknown direction4 ,it is obvious that1 if the ,tessaract turns so as to show us the side1 of which ,cube #e is a model1 then ,cube #a will no longer be visible4 ,the ,orange line will run in the unknown or fourth direction1 and be out of our sight1 together with the whole cube which the ,vermilion square generates1 when the ,gold point moves along the ,orange line4 ,hence1 if we consider these models as real portions of the tessaract1 we must not have more than one before us at once4 ,when we look at one1 the others must necessarily be beyond our sight and touch4 ,but we may consider them simply as models1 and1 as such1 we may let them lie alongside of each other4 ,in this case1 we must remember that their real relationships are not those in which we see them4 ,we now enumerate the sides of the new ,cube #e1 so that1 when we refer to it1 any colour may be recognised by name4 ,the square ,vermilion traces a ,pale-green cube1 and ends in an ,indian-red square4 7,the colour ,pale-green of this cube is not seen1 as it is entirely surrounded by squares and lines of colour47 ,each ,line traces a ,square and ends in a ,line4 ,the ,blue line ,light-brown square and,purple-brown line ,brown traces,yellow ends,dull-blue ,deep-yellow a ,light-red in ,dark ,green ,deep-crimson a ,indigo 4 ,each ,point traces a ,line and ends in a ,point4 ,the ,gold point ,stone line and,silver point ,buff traces,light-green ends,blue-tint ,light-blue a ,rich-red in ,quaker-green ,red ,emerald a ,purple 4 ,it will be noticed that besides the ,vermilion square of this cube another square of it has been seen before4 ,a moment's comparison with the experience of a plane-being will make this more clear4 ,if a plane-being has before him models of the ,black and ,white squares of the ,cube1 he sees that all the colours of the one are different from all the colours of the other4 ,next1 if he looks at a model of the ,vermilion square1 he sees that it starts from the ,blue line and ends in a line of the ,white square1 the ,deep-yellow line4 ,in this square he has two lines which he had before1 the ,blue line with ,gold and ,buff points1 the ,deep-yellow line with ,light-blue and ,red points4 ,to him the ,black and ,white squares are his ,models #a and #b1 and the ,vermilion square is to him as our ,model #e is to us4 ,the left-hand square of ,model #e is ,indian-red1 and is identical with that of the same colour on the left-hand side of ,model #b4 ,in fact1 ,model #e shows us what lies between the ,vermilion face of #a1 and the ,indian-red face of #b4 ,from the ,gold point we suppose four perfectly independent lines to spring forth1 each of them at right angles to all the others4 ,in our space there is only room for three lines mutually at right angles4 ,it will be found1 if we try to introduce a fourth at right angles to each of three1 that we fail2 hence1 of these four lines one must go out of the space we know4 ,the colours of these four lines are ,brown1 ,orange1 ,blue1 ,stone4 ,in ,model #a are shown the ,brown1 ,orange1 and ,blue4 ,in ,model #e are shown the ,brown1 ,blue1 and ,stone4 ,these lines might have had any directions at first1 but we chose to begin with the ,brown line going up1 or ,z1 the ,orange going ,x1 the ,blue going ,y1 and the ,stone line going in the unknown direction1 which we will call ,w4 ,consider for a moment the ,stone and the ,orange lines4 ,they can be seen together on ,model #g by looking at the lower face of it4 ,they are at right angles to each other1 and if the ,orange line be turned to take the place of the ,stone line1 the latter will run into the negative part of the direction previously occupied by the former4 ,this is the reason that the ,models #c1 #e1 and #g are made with the ,stone line always running in the reverse direction of that line of ,model #a1 which is wanting in each respectively4 ,it will now be easy to find out ,models #c and #g4 ,all that has to be done is1 to discover what faces they have in common with #a and #b1 and these faces will show from which planes of #a they are generated by motion in the unknown direction4 ,take ,model #g4 ,on one side of it there is a ,dark-blue square1 which is identical with the ,dark-blue square of ,model #a4 ,placing it so that it coincides with #a by this square line for line1 we see that the square nearest to us is ,burnt-sienna1 the same as the near square on ,model #b4 ,hence this cube is a model of what the ,dark-blue square traces on moving in the unknown direction4 ,here the unknown direction coincides with the negative away direction4 ,in fact1 to see this cube1 we have been obliged to suppose the ,blue line turned into the unknown direction1 for we cannot look at more than three of these rectangular lines at once in our space1 and in this ,model #g we have the ,brown1 ,orange1 and ,stone lines4 ,the faces1 lines1 and points of ,cube #g can be identified by the following list4 ,the ,dark-blue square traces a ,dark-stone cube 7whose interior is rendered invisible by the bounding squares71 and ends in a ,burnt-sienna square4 ,each ,line traces a ,square and ends in a ,line4 ,the ,orange line ,azure square and,leaf-green line ,brown traces,yellow ends,dull-blue ,french-grey an ,yellow-green in ,dark-pink ,reddish ,ochre a ,brown-green 4 ,each ,point traces a ,line and ends in a ,point4 ,the ,gold point ,stone line and,silver point ,fawn traces,smoke ends,turquoise ,light-blue a ,rich-red in ,quaker-green ,dull-purple ,green-blue a ,peacock-blue 4 ,if we now take ,model #c1 we see that it has a ,black square uppermost1 and has ,blue and ,orange lines4 ,hence1 it evidently proceeds from the ,black square in ,model #a2 and it has in it ,blue and ,orange lines1 which proceed from the ,gold point4 ,but besides these1 it has running downwards a ,stone line4 ,the line wanting is the ,brown line1 and1 as in the other cases1 when one of the three lines of ,model #a turns out into the unknown direction1 the ,stone line turns into the direction opposite to that from which the line has turned4 ,take this ,model #c and place it underneath ,model #a1 raising the latter so that the ,black squares on the two coincide line for line4 ,then we see what would come into our view if the ,brown line were to turn into the unknown direction1 and the ,stone line come into our space downwards4 ,looking at this cube1 we see that the following parts of the tessaract have been generated4 ,the ,black square traces a ,brick-red cube 7invisible because covered by its own sides and edges71 and ends in a ,bright-green square4 ,each ,line traces a ,square and ends in a ,line4 ,the ,orange line ,azure square and,leaf-green line ,crimson traces,rose ends,dull-green ,green-grey an ,sea-blue in ,dark-purple ,blue ,light-brown a ,purple-brown 4 ,each ,point traces a ,line and ends in a ,point4 ,the ,gold point ,stone line and,silver point ,fawn traces,smoke ends,turquoise ,terra-cotta a ,magenta in ,earthen ,buff ,light-green a ,blue-tint 4 ,this completes the enumeration of the regions of ,cube #c4 ,it may seem a little unnatural that it should come in downwards2 but it must be remembered that the new fourth direction has no more relation to up-and-down than to right-and-left or to near-and-far4 ,and if1 instead of thinking of a plane-being as living on the surface of a table1 we suppose his world to be the surface of the sheet of paper touching the ,dark-blue square of ,cube #a1 then we see that a turn round the ,orange line1 which makes the ,brown line go into the plane-being's unknown direction1 brings the ,blue line into his downwards direction4 ,there still remain to be described ,models #d1 #f1 and #h4 ,it will be shown that ,model #d is to ,model #c what ,model #b is to ,model #a4 ,that is1 if1 when #c is in our space1 it be moved so as to trace a tessaract1 #d will be the opposite cube in which the tessaract ends4 ,there is no colour common to #c and #d4 ,similarly1 #f is the opposite boundary of the tessaract generated by #e1 and #h of that by #g4 ,a little closer consideration will reveal several points4 ,looking at ,cube #e1 we see proceeding from the ,gold point a ,brown1 a ,blue1 and a ,stone line4 ,the ,orange line is wanting2 therefore1 it goes in the unknown direction4 ,if we want to discover what exists in the unknown direction from ,cube #e1 we can get help from ,cube #a4 ,for1 since the ,orange line lies in the unknown direction from ,cube #e1 the ,gold point will1 if moved along the ,orange line1 pass in the unknown direction4 ,so also1 the ,vermilion square1 if moved along in the direction of the ,orange line1 will proceed in the unknown direction4 ,looking at ,cube #a we see that the ,vermilion square thus moved ends in a ,blue-green square4 ,then1 looking at ,model #f1 on it1 corresponding to the ,vermilion square on ,cube #e1 is a ,blue-green square4 ,cube #f thus shows what exists an inch beyond #e in the unknown direction4 ,between the right-hand face on #e and the right-hand face on #f lies a cube1 the one which is shown in ,model #a4 ,model #a is traced by the ,vermilion square moving an inch along the direction of the ,orange line4 ,in ,model #e1 the ,orange line goes in the unknown direction2 and looking at ,model #f we see what we should get at the end of a movement of one inch in that direction4 ,we have still to enumerate the colours of ,cubes #d1 #f1 and #h1 and we do so in the following list4 ,in the first column is designated the part of the cube2 in the columns under #d1 #f1 #h1 come the colours which #d1 #f1 #h1 respectively have in the parts designated in the corresponding line in the first column4 ,cube itself3-- #d #f #h ,chocolate ,oak-yellow ,salmon ,squares3-- ,lower face ,light-grey ,rose ,sea-blue ,upper ,white ,deep-brown ,deep-green ,left-hand ,light-red ,yellow-ochre ,deep-crimson ,right-hand ,deep-brown ,blue-green ,dark-grey ,near ,ochre ,yellow-green ,dun ,far ,deep-green ,dark-grey ,light-yellow ,lines3-- ,on ground1 going round the square from left to right3-- #d #f #h #a4 ,brown-green ,smoke ,dark-purple #b4 ,dark-green ,crimson ,magenta #c4 ,pale-yellow ,magenta ,green-grey #d4 ,dark ,dull-green ,light-green ,vertical1 going round the sides from left to right3-- #a4 ,rich-red ,dark-pink ,indigo #b4 ,green-blue ,french-grey ,pale-pink #c4 ,sea-green ,dark-slate ,dark-slate #d4 ,emerald ,pale-pink ,green ,round upper face in same order3-- #a4 ,reddish ,green-blue ,pale-yellow #b4 ,bright-blue ,bright-blue ,sea-green #c4 ,leaden ,sea-green ,leaden #d4 ,deep-yellow ,dark-green ,emerald ,points3-- ,on lower face1 going from left to right3-- #a4 ,quaker-green ,turquoise ,blue-tint #b4 ,peacock-blue ,fawn ,earthen #c4 ,orange-vermilion ,terra-cotta ,terra-cotta #d4 ,purple ,earthen ,buff ,on upper face3-- #a4 ,light-blue ,peacock-blue ,purple #b4 ,dull-purple ,dull-purple ,orange-vermilion #c4 ,deep-blue ,deep-blue ,deep-blue #d4 ,red ,orange-vermilion ,red ,if any one of these cubes be taken at random1 it is easy enough to find out to what part of the ,tessaract it belongs4 ,in all of them1 except #b1 there will be one face1 which is a copy of a face on #a2 this face is1 in fact1 identical with the face on #a which it resembles4 ,and the model shows what lies in the unknown direction from that face4 ,this unknown direction is turned into our space1 so that we can see and touch the result of moving a square in it4 ,and we have sacrificed one of the three original directions in order to do this4 ,it will be found that the line1 which in #a goes in the #dth direction1 in the other models always runs in a negative direction4 ,let us take ,model #h1 for instance4 ,searching it for a face we know1 we come to a ,light-yellow face away from us4 ,we place this face parallel with the ,light-yellow face on ,cube #a1 and we see that it has a ,green line going up1 and a ,green-grey line going to the right from the ,buff point4 ,in these respects it is identical with the ,light-yellow face on ,cube #a4 ,but instead of a ,blue line coming towards us from the ,buff point1 there is a ,light-green line4 ,this ,light-green line1 then1 is that which proceeds in the unknown direction from the ,buff point4 ,the line is turned towards us in this ,model #h in the negative ,y direction2 and looking at the model1 we see exactly what is formed when in the motion of the whole cube in the unknown direction1 the ,light-yellow face is moved an inch in that direction4 ,it traces out a ,salmon cube 7v4 ,table on p4 #abg71 and it has ,sea-blue and ,deep-green sides below and above1 and ,deep-crimson and ,dark-grey sides left and right1 and ,dun and ,light-yellow sides near and far4 ,if we want to verify the correctness of any of these details1 we must turn to ,models #a and #b4 ,what lies an inch from the ,light-yellow square in the unknown direction8 ,model #b tells us1 a ,dun square4 ,now1 looking at #h1 we see that towards us lies a ,dun square4 ,this is what lies an inch in the unknown direction from the ,light-yellow square4 ,it is here turned to face us1 and we can see what lies between it and the ,light-yellow square4 ,,chapter ,,iv4 ,,tessaract ,,moving ,,through ,,three-,,space4 ,,models ,,of ,,the ,,sections4 ,in order to obtain a clear conception of the higher solid1 a certain amount of familiarity with the facts shown in these models is necessary4 ,but the best way of obtaining a systematic knowledge is shown hereafter4 ,what these models enable us to do1 is to take a general review of the subject4 ,in all of them we see simply the boundaries of the tessaract in our space2 we can no more see or touch the tessaract's solidity than a plane-being can touch the cube's solidity4 ,there remain the four models #i1 #aj1 #aa1 #ab4 ,model #i represents what lies between #a and #b4 ,if #a be moved an inch in the unknown direction1 it traces out the tessaract and ends in #b4 ,but1 obviously1 between #a and #b there must be an infinite number of exactly similar solid sections2 these are all like ,model #i4 ,take the case of a plane-being on the table4 ,he sees the ,black square1--that is1 he sees the lines round it1--and he knows that1 if it moves an inch in some mysterious direction1 it traces a new kind of figure1 the opposite boundary whereof is the ,white square4 ,if1 then1 he has models of the ,white and ,black squares1 he has before him the end and beginning of our cube4 ,but between these squares are any number of others1 the plane sections of the cube4 ,we can see what they are4 ,the interior of each is a ,light-buff 7the colour of the substance of the cube71 the sides are of the colours of the vertical faces of the cube1 and the points of the colours of the vertical lines of the cube1 viz41 ,dark-blue1 ,blue-green1 ,light-yellow1 ,vermilion lines1 and ,brown1 ,french-grey1 ,dark-slate1 ,green points4 ,thus1 the square1 in moving in the unknown direction1 traces out a succession of squares1 the assemblage of which makes the cube in layers4 ,so also the cube1 moving in the unknown direction1 will at any point of its motion1 still be a cube2 and the assemblage of cubes thus placed constitutes the tessaract in layers4 ,we suppose the cube to change its colour directly it begins to move4 ,its colour between #a and #b we can easily determine by finding what colours its different parts assume1 as they move in the unknown direction4 ,the ,gold point immediately begins to trace a ,stone-line4 ,we will look at ,cube #e to see what the ,vermilion face becomes2 we know the interior of that cube is ,pale-green 7v4 ,table1 p4 #abb74 ,hence1 as it moves in the unknown direction1 the ,vermilion square forms in its course a series of ,pale-green squares4 ,the ,brown line gives rise to a ,yellow square2 hence1 at every point of its course in the fourth direction1 it is a ,yellow line1 until1 on taking its final position1 it becomes a ,dull-blue line4 ,looking at ,cube #e1 we see that the ,deep yellow line becomes a ,light-red line1 the ,green line a ,deep ,crimson one1 the ,gold point a ,stone one1 the ,light-blue point a ,rich-red one1 the ,red point an ,emerald one1 and the ,buff point a ,light-green one4 ,now1 take the ,model #i4 ,looking at the left side of it1 we see exactly that into which the ,vermilion square is transformed1 as it moves in the unknown direction4 ,the left side is an exact copy of a section of ,cube #e1 parallel to the ,vermilion face4 ,but we have only accounted for one side of our ,model #i4 ,there are five other sides4 ,take the near side corresponding to the ,dark-blue square on ,cube #a4 ,when the ,dark-blue square moves1 it traces a ,dark-stone cube1 of which we have a copy in ,cube #g4 ,looking at #g 7v4 ,table1 p4 #abd71 we see that1 as soon as the ,dark-blue square begins to move1 it becomes of a ,dark-stone colour1 and has ,yellow1 ,ochre1 ,yellow-green1 and ,azure sides1 and ,stone1 ,rich-red1 ,green-blue1 ,smoke lines running in the unknown direction from it4 ,now1 the side of ,model #i1 which faces us1 has these colours the squares being seen as lines1 and the lines as points4 ,hence ,model #i is a copy of what the cube becomes1 so far as the ,vermilion and ,dark-blue sides are concerned1 when1 moving in the unknown direction1 it traces the tessaract4 ,we will now look at the lower square of our model4 ,it is a ,brick-red square1 with ,azure1 ,rose1 ,sea-blue1 and ,light-brown lines1 and with ,stone1 ,smoke1 ,magenta1 and ,light-green points4 ,this1 then1 is what the ,black square should change into1 as it moves in the unknown direction4 ,let us look at ,model #c4 ,here the ,stone line1 which is the line in the unknown direction1 runs downwards4 ,it is turned into the downwards direction1 so that the cube traced by the ,black square may be in our space4 ,the colour of this cube is ,brick-red2 the ,orange line has traced an ,azure1 the ,blue line a ,light-brown1 the ,crimson line a ,rose1 and the ,green-grey line a ,sea-blue square4 ,hence1 the lower square of ,model #i shows what the ,black square becomes1 as it traces the tessaract2 or1 in other words1 the section of ,model #c between the ,black and ,bright-green squares exactly corresponds to the lower face of ,model #i4 ,therefore1 it appears that ,model #i is a model of a section of the tessaract1 that it is to the tessaract what a square between the ,black and ,white squares is to the cube4 ,to prove the other sides correct1 we have to see what the ,white1 ,blue-green1 and ,light-yellow squares of ,cube #a become1 as the cube moves in the unknown direction4 ,this can be effected by means of the ,models #d1 #f1 #h4 ,each cube can be used as an index for showing the changes through which any side of the first model passes1 as it moves in the unknown direction till it becomes ,cube #b4 ,thus1 what becomes of the ,white square8 ,look at ,cube #d4 ,from the ,light-blue corner of its ,white square runs downwards the ,rich-red line in the unknown direction4 ,if we take a parallel section below the ,white square1 we have a square bounded by ,ochre1 ,deep-brown1 ,deep-green1 and ,light-red lines2 and by ,rich-red1 ,green-blue1 ,sea-green1 and ,emerald points4 ,the colour of the cube is ,chocolate1 and therefore its section is ,chocolate4 ,this description is exactly true of the upper surface of ,model #i4 ,there still remain two sides1 those corresponding to the ,light-yellow and ,blue-green of ,cube #a4 ,what the ,blue-green square becomes midway between ,cubes #a and #b can be seen on ,model #f4 ,the colour of the last-named is ,oak-yellow1 and a section parallel to its ,blue-green side is surrounded by ,yellow-green1 ,deep-brown1 ,dark-grey and ,rose lines and by ,green-blue1 ,smoke1 ,magenta1 and ,sea-green points4 ,this is exactly similar to the right side of ,model #i4 ,lastly1 that which becomes of the ,light-yellow side can be seen on ,model #h4 ,the section of the cube is a ,salmon square bounded by ,deep-crimson1 ,deep-green1 ,dark-grey and ,sea-blue lines and by ,emerald1 ,sea-green1 ,magenta1 and ,light-green points4 ,thus the models can be used to answer any question about sections4 ,for we have simply to take1 instead of the whole cube1 a plane1 and the relation of the whole tessaract to that plane can be told by looking at the model1 which1 starting with that plane1 stretches from it in the unknown direction4 ,we have not as yet settled the colour of the interior of ,model #i4 ,it is that part of the tessaract which is traced out by the interior of ,cube #a4 ,the unknown direction starts equally and simultaneously from every point of every part of ,cube #a1 just as the up direction starts equally and simultaneously from every point of a square4 ,let us suppose that the cube1 which is ,light-buff1 changes to a ,wood-colour directly it begins to trace the tessaract4 ,then the internal part of the section between #a and #b will be a ,wood-colour4 ,the sides of the ,model #i are of the greatest importance4 ,they are the colour of the six cubes1 #c1 #d1 #e1 #f1 #g1 and #h4 ,the colours of #a and #b are wanting1 viz4 ,light-buff and ,sage-green4 ,thus the section between #a and #b can be found by its wanting the colours of the ,cubes #a and #b4 ,looking at ,models #aj1 #aa1 and #ab in a similar manner1 the reader will find they represent the sections between ,cubes #c and #d1 ,cubes #e and #f1 and ,cubes #g and #h respectively4 ,,chapter ,v4 ,,representation ,,of ,,three-,,space ,,by ,,names1 ,,and ,,in ,a ,,plane4 ,we may now ask ourselves the best way of passing on to a clear comprehension of the facts of higher space4 ,something can be effected by looking at these models2 but it is improbable that more than a slight sense of analogy will be obtained thus4 ,indeed1 we have been trusting hitherto to a method which has something vicious about it--we have been trusting to our sense of what must be4 ,the plan adopted1 as the serious effort towards the comprehension of this subject1 is to learn a small portion of higher space4 ,if any reader feel a difficulty in the foregoing chapters1 or if the subject is to be taught to young minds1 it is far better to abandon all attempt to see what higher space must be1 and to learn what it is from the following chapters4 ,,naming ,a ,,piece ,,of ,,space4 ,the diagram 7,fig4 #f7 represents a block of #bg cubes1 which form ,set #a of the #ha cubes4 ,the cubes are coloured1 and it will be seen that the colours are arranged after the pattern of ,model #a of previous chapters1 which will serve as a key to the block4 ,in the diagram1 ,g4 denotes ,gold1 ,o4 ,orange1 ,f4 ,fawn1 ,br4 ,brown1 and so on4 ,we will give names to the cubes of this block4 ,they should not be learnt1 but kept for reference4 ,we will write these names in three sets1 the lowest consisting of the cubes which touch the table1 the next of those immediately above them1 and the third of those at the top4 ,thus the ,gold cube is called ,corvus1 the ,orange1 ,cuspis1 the ,fawn1 ,nug1 and the central one below1 ,syce4 ,the corresponding colours of the following set can easily be traced4 ,olus ,semita ,lama ,via ,mel ,iter ,ilex ,callis ,sors ,bucina ,murex ,daps ,alvus ,mala ,proes ,arctos ,mna ,far ,cista ,cadus ,crus ,dos ,syce ,bolus ,corvus ,cuspis ,nug ,thus the central or ,light-buff cube is called ,mala2 the middle one of the lower face is ,syce2 of the upper face ,mel2 of the right face1 ,proes2 of the left1 ,alvus2 of the front1 ,mna 7the ,dark-blue square of ,model #a72 and of the back1 ,murex 7the ,light-yellow square74 ,now1 if ,model #a be taken1 and considered as representing a block of #fd cubes1 the ,gold corner as one cube1 the ,orange line as two cubes1 the ,fawn point as one cube1 the ,dark-blue square as four cubes1 the ,light-buff interior as eight cubes1 and so on1 it will correspond to the diagram 7,fig4 #g74 ,this block differs from the last in the number of cubes1 but the arrangement of the colours is the same4 ,the following table gives the names which we will use for these cubes4 ,there are no new names2 they are only applied more than once to all cubes of the same colour4 ,olus ,semita ,semita ,lama ,fourth,via ,mel ,mel ,iter ,floor4,via ,mel ,mel ,iter ,ilex ,callis ,callis ,sors ,bucina ,murex ,murex ,daps ,third ,alvus ,mala ,mala ,proes ,floor4,alvus ,mala ,mala ,proes ,arctos ,mna ,mna ,far ,bucina ,murex ,murex ,daps ,second,alvus ,mala ,mala ,proes ,floor4,alvus ,mala ,mala ,proes ,arctos ,mna ,mna ,far ,cista ,cadus ,cadus ,crus ,first ,dos ,syce ,syce ,bolus ,floor4,dos ,syce ,syce ,bolus ,corvus ,cuspis ,cuspis ,nug ,illustration3 ,fig4 #f4 ,illustration3 ,fig4 #g4 ,illustration3 ,fig4 #h4 ,if we now consider ,model #a to represent a block1 five cubes each way1 built up of inch cubes1 and colour it in the same way1 that is1 with similar colours for the corner-cubes1 edge-cubes1 face-cubes1 and interior-cubes1 we obtain what is represented in the diagram 7,fig4 #h74 ,here we have nine ,dark-blue cubes called ,mna2 that is1 ,mna denotes the nine ,dark-blue cubes1 forming a layer on the front of the cube1 and filling up the whole front except the edges and points4 ,cuspis denotes three ,orange1 ,dos three ,blue1 and ,arctos three ,brown cubes4 ,now1 the block of cubes can be similarly increased to any size we please4 ,the corners will always consist of single cubes2 that is1 ,corvus will remain a single cubic inch1 even though the block be a hundred inches each way4 ,cuspis1 in that case1 will be #ih inches long1 and consist of a row of #ih cubes2 ,arctos1 also1 will be a long thin line of cubes standing up4 ,mna will be a thin layer of cubes almost covering the whole front of the block2 the number of them will be #ih times #ih4 ,syce will be a similar square layer of cubes on the ground1 so also ,mel1 ,alvus1 ,proes1 and ,murex in their respective places4 ,mala1 the interior of the cube1 will consist of #ih times #ih times #ih inch cubes4 ,illustration3 ,fig4 #i ,now1 if we continued in this manner till we had a very large block of thousands of cubes in each side ,corvus would1 in comparison to the whole block1 be a minute point of a cubic shape1 and ,cuspis would be a mere line of minute cubes1 which would have length1 but very small depth or height4 ,next1 if we suppose this much sub-divided block to be reduced in size till it becomes one measuring an inch each way1 the cubes of which it consists must each of them become extremely minute1 and the corner cubes and line cubes would be scarcely discernible4 ,but the cubes on the faces would be just as visible as before4 ,for instance1 the cubes composing ,mna would stretch out on the face of the cube so as to fill it up4 ,they would form a layer of extreme thinness1 but would cover the face of the cube 7all of it except the minute lines and points74 ,thus we may use the words ,corvus and ,nug1 etc41 to denote the corner-points of the cube1 the words ,mna1 ,syce1 ,mel1 ,alvus1 ,proes1 ,murex1 to denote the faces4 ,it must be remembered that these faces have a thickness1 but it is extremely minute compared with the cube4 ,mala would denote all the cubes of the interior except those1 which compose the faces1 edges1 and points4 ,thus1 ,mala would practically mean the whole cube except the colouring on it4 ,and it is in this sense that these words will be used4 ,in the models1 the ,gold point is intended to be a ,corvus1 only it is made large to be visible2 so too the ,orange line is meant for ,cuspis1 but magnified for the same reason4 ,finally1 the #bg names of cubes1 with which we began1 come to be the names of the points1 lines1 and faces of a cube1 as shown in the diagram 7,fig4 #i74 ,with these names it is easy to express what a plane-being would see of any cube4 ,let us suppose that ,mna is only of the thickness of his matter4 ,we suppose his matter to be composed of particles1 which slip about on his plane1 and are so thin that he cannot by any means discern any thickness in them4 ,so he has no idea of thickness4 ,but we know that his matter must have some thickness1 and we suppose ,mna to be of that degree of thickness4 ,if the cube be placed so that ,mna is in his plane1 ,corvus1 ,cuspis1 ,nug1 ,far1 ,sors1 ,callis1 ,ilex and ,arctos will just come into his apprehension2 they will be like bits of his matter1 while all that is beyond them in the direction he does not know1 will be hidden from him4 ,thus a plane-being can only perceive the ,mna or ,syce or some one other face of a cube2 that is1 he would take the ,mna of a cube to be a solid in his plane-space1 and he would see the lines ,cuspis1 ,far1 ,callis1 ,arctos4 ,to him they would bound it4 ,the points ,corvus1 ,nug1 ,sors1 and ,ilex1 he would not see1 for they are only as long as the thickness of his matter1 and that is so slight as to be indiscernible to him4 ,we must now go with great care through the exact processes by which a plane-being would study a cube4 ,for this purpose we use square slabs which have a certain thickness1 but are supposed to be as thin as a plane-being's matter4 ,now1 let us take the first set of #ha cubes again1 and build them from #a to #bg4 ,we must realize clearly that two kinds of blocks can be built4 ,it may be built of #bg cubes1 each similar to ,model #a1 in which case each cube has its regions coloured1 but all the cubes are alike4 ,or it may be built of #bg differently coloured cubes like ,set #a1 in which case each cube is coloured wholly with one colour in all its regions4 ,if the latter set be used1 we can still use the names ,mna1 ,alvus1 etc4 to denote the front1 side1 etc41 of any one of the cubes1 whatever be its colour4 ,when they are built up1 place a piece of card against the front to represent the plane on which the plane-being lives4 ,the front of each of the cubes in the front of the block touches the plane4 ,in previous chapters we have supposed ,mna to be a ,blue square4 ,but we can apply the name to the front of a cube of any colour4 ,let us say the ,mna of each front cube is in the plane2 the ,mna of the ,gold cube is ,gold1 and so on4 ,to represent this1 take nine slabs of the same colours as the cubes4 ,place a stiff piece of cardboard 7or a book-cover7 slanting from you1 and put the slabs on it4 ,they can be supported on the incline so as to prevent their slipping down away from you by a thin book1 or another sheet of cardboard1 which stands for the surface of the plane-being's earth4 ,we will now give names to the cubes of ,block #a of the #ha ,set4 ,we call each one ,mala1 to denote that it is a cube4 ,they are written in the following list in floors or layers1 and are supposed to run backwards or away from the reader4 ,thus1 in the first layer1 ,frenum ,mala is behind or farther away than ,urna ,mala2 in the second layer1 ,ostrum is in front1 ,uncus behind it1 and ,ala behind ,uncus4 ,third1 or ,mars ,mala ,merces ,mala ,tyro ,mala ,top ,spicula ,mala ,mora ,mala ,oliva ,mala ,floor4 ,comes ,mala ,tibicen ,mala ,vestis ,mala ,second1 or,ala ,mala ,cortis ,mala ,aer ,mala ,middle ,uncus ,mala ,pallor ,mala ,tergum ,mala ,floor4 ,ostrum ,mala ,bidens ,mala ,scena ,mala ,first1 or ,sector ,mala ,hama ,mala ,remus ,mala ,bottom ,frenum ,mala ,plebs ,mala ,sypho ,mala ,floor4 ,urna ,mala ,moles ,mala ,saltus ,mala ,these names should be learnt so that the different cubes in the block can be referred to quite easily and immediately by name4 ,they must be learnt in every order1 that is1 in each of the three directions backwards and forwards1 e4g4 ,urna to ,saltus1 ,urna to ,sector1 ,urna to ,comes2 and the same reversed1 viz41 ,comes to ,urna1 ,sector to ,urna1 etc4 ,only by so learning them can the mind identify any one individually without even a momentary reference to the others around it4 ,it is well to make it a rule not to proceed from one cube to a distant one without naming the intermediate cubes4 ,for1 in ,space we cannot pass from one part to another without going through the intermediate portions4 ,and1 in thinking of ,space1 it is well to accustom our minds to the same limitations4 ,urna ,mala is supposed to be solid ,gold an inch each way2 so too all the cubes are supposed to be entirely of the colour which they show on their faces4 ,thus any section of ,moles ,mala will be ,orange1 of ,plebs ,mala ,black1 and so on4 ,illustration3 ,fig4 #aj4 ,let us now draw a pair of lines on a piece of paper or cardboard like those in the diagram 7,fig4 #aj74 ,in this diagram the top of the page is supposed to rest on the table1 and the bottom of the page to be raised and brought near the eye1 so that the plane of the diagram slopes upwards to the reader4 ,let ,z denote the upward direction1 and ,x the direction from left to right4 ,let us turn the ,block of cubes with its front upon this slope i4e4 so that ,urna fits upon the square marked ,urna4 ,moles will be to the right and ,ostrum above ,urna1 i4e4 nearer the eye4 ,we might leave the block as it stands and put the piece of cardboard against it2 in this case our plane-world would be vertical4 ,it is difficult to fix the cubes in this position on the plane1 and therefore more convenient if the cardboard be so inclined that they will not slip off4 ,but the upward direction must be identified with ,z4 ,now1 taking the slabs1 let us compose what a plane-being would see of the ,block4 ,he would perceive just the front faces of the cubes of the ,block1 as it comes into his plane2 these front faces we may call the ,moenas of the cubes4 ,let each of the slabs represent the ,moena of its corresponding cube1 the ,gold slab of the ,gold cube and so on4 ,they are thicker than they should be2 but we must overlook this and suppose we simply see the thickness as a line4 ,we thus build a square of nine slabs to represent the appearance to a plane-being of the front face of the ,block4 ,the middle one1 ,bidens ,moena1 would be completely hidden from him by the others on all its sides1 and he would see the edges of the eight outer squares4 ,if the ,block now begin to move through the plane1 that is1 to cut through the piece of paper at right angles to it1 it will not for some time appear any different4 ,for the sections of ,urna are all ,gold like the front face ,moena1 so that the appearance of ,urna at any point in its passage will be a ,gold square exactly like ,urna ,moena1 seen by the plane-being as a line4 ,thus1 if the speed of the ,block's passage be one inch a minute1 the plane-being will see no change for a minute4 ,in other words1 this set of slabs lasting one minute will represent what he sees4 ,when the ,block has passed one inch1 a different set of cubes appears4 ,remove the front layer of cubes4 ,there will now be in contact with the paper nine new cubes1 whose names we write in the order in which we should see them through a piece of glass standing upright in front of the ,block3 ,spicula ,mala ,mora ,mala ,oliva ,mala ,uncus ,mala ,pallor ,mala ,tergum ,mala ,frenum ,mala ,plebs ,mala ,sypho ,mala ,we pick out nine slabs to represent the ,moenas of these cubes1 and placed in order they show what the plane-being sees of the second set of cubes as they pass through4 ,similarly the third wall of the ,block will come into the plane1 and looking at them similarly1 as it were through an upright piece of glass1 we write their names3 ,mars ,mala ,merces ,mala ,tyro ,mala ,ala ,mala ,cortis ,mala ,aer ,mala ,sector ,mala ,hama ,mala ,remus ,mala ,now1 it is evident that these slabs stand at different times for different parts of the cubes4 ,we can imagine them to stand for the ,moena of each cube as it passes through4 ,in that case1 the first set of slabs1 which we put up1 represents the ,moenas of the front wall of cubes2 the next set1 the ,moenas of the second wall4 ,thus1 if all the three sets of slabs be together on the table1 we have a representation of the sections of the cube4 ,for some purposes it would be better to have four sets of slabs1 the fourth set representing the ,murex of the third wall2 for the three sets only show the front faces of the cubes1 and therefore would not indicate anything about the back faces of the ,block4 ,for instance1 if a line passed through the ,block diagonally from the point ,corvus 7,gold7 to the point ,lama 7,deep-blue71 it would be represented on the slabs by a point at the bottom left-hand corner of the ,gold slab1 a second point at the same corner of the ,light-buff slab1 and a third at the same corner of the ,deep-blue slab4 ,thus1 we should have the points mapped at which the line entered the fronts of the walls of cubes1 but not the point in ,lama at which it would leave the ,block4 ,let the ,diagrams #a1 #b1 #c 7,fig4 #aa71 be the three sets of slabs4 ,to see the diagrams properly1 the reader must set the top of the page on the table1 and look along the page from the bottom of it4 ,the line in question1 which runs from the bottom left-hand near corner to the top right-hand far corner of the ,block will be represented in the three sets of slabs by the points ,a1 ,b1 ,c4 ,to complete the diagram of its course1 we need a fourth set of slabs for the ,murex of the third wall2 the same object might be attained1 if we had another ,block of #bg cubes behind the first ,block and represented its front or ,moenas by a set of slabs4 ,for the point1 at which the line leaves the first ,block is identical with that at which it enters the second ,block4 ,illustration3 ,fig4 #aa4 ,if we suppose a sheet of glass to be the plane-world1 the ,diagrams #a1 #b1 #c 7,fig4 #aa71 may be drawn more naturally to us as ,diagrams 1 1 7,fig4 #ab74 ,here represents the ,moenas of the first wall1 those of the second1 those of the third4 ,but to get the plane-being's view we must look over the edge of the glass down the ,z axis4 ,illustration3 ,fig4 #ab4 ,set #b of slabs represent the ,moenas of ,wall #b4 ,these ,moenas are in contact with the ,murex of ,wall #a4 ,thus ,set #b will show where the line issues from ,wall #a as well as where it enters ,wall #b4 ,the plane-being1 therefore1 could get an idea of the ,block of cubes by learning these slabs4 ,he ought not to call the ,gold slab ,urna ,mala1 but ,urna ,moena1 and so on1 because all that he learns are ,moenas1 merely the thin faces of the cubes4 ,by introducing the course of time1 he can represent the ,block more nearly4 ,for1 if he supposes it to be passing an inch a minute1 he may give the name ,urna ,mala to the ,gold slab enduring for a minute4 ,but1 when he has learnt the slabs in this position and sequence1 he has only a very partial view of the ,block4 ,let the ,block turn round the ,z axis1 as ,model #a turns round the ,brown line4 ,a different set of cubes comes into his plane1 and now they come in on the ,alvus faces4 7,alvus is here used to denote the left-hand faces of the cubes1 and is not supposed to be ,vermilion2 it is simply the thinnest slice on the left hand of the cube and of the same colour as the cube47 ,to represent this1 the plane-being should employ a fresh set of slabs1 for there is nothing common to the ,moena and ,alvus faces except an edge4 ,but1 since each cube is of the same colour throughout1 the same slab may be used for its different faces4 ,thus the ,alvus of ,urna ,mala can be represented by a ,gold slab4 ,only it must never be forgotten that it is meant to be a new slab1 and is not identical with the same slab used for ,moena4 ,illustration3 ,fig4 #ac4 ,illustration3 ,fig4 #ad4 ,now1 when the ,block of cubes has turned round the ,brown line into the plane1 it is clear that they will be on the side of the ,z axis opposite to that on which were the ,moena slabs4 ,the line1 which ran ,y1 now runs -,x4 ,thus the slabs will occupy the second quadrant marked by the axes1 as shown in the diagram 7,fig4 #ac74 ,each of these slabs we will name ,alvus4 ,in this view1 as before1 the book is supposed to be tilted up towards the reader1 so that the ,z axis runs from ,o to his eye4 ,then1 if the ,block be passed at right angles through the plane1 there will come into view the two sets of slabs represented in the ,diagrams 7,fig4 #ac74 ,in copying this arrangement with the slabs1 the cardboard on which they are arranged must slant upwards to the eye1 i4e41 ,,oz must run up to the eye1 and the sides of the slabs seen are in ,diagram #b 7,fig4 #ac71 the upper edges of ,tibicen1 ,mora1 ,merces2 in ,diagram #c1 the upper edges of ,vestis1 ,oliva1 ,tyro4 ,there is another view of the ,block possible to a plane-being4 ,if the ,block be turned round the ,x axis1 the lower face comes into the vertical plane4 ,this corresponds to turning ,model #a round the ,orange line4 ,on referring to the diagram 7,fig4 #ad71 we now see that the name of the faces of the cubes coming into the plane is ,syce4 ,here the plane-being looks from the extremity of the ,z axis and the squares1 which he sees run from him in the -,z direction4 7,as this turn of the ,block brings its ,syce into the vertical plane so that it extends three inches below the base line of its ,moena1 it is evident that the turn is only possible if the ,moena be originally at a height of at least three inches above the plane-being's earth line in the vertical plane47 ,next1 if the ,block be passed through the plane1 the sections shown in the ,diagrams #b and #c 7,fig4 #ad7 are brought into view4 ,thus1 there are three distinct ways of regarding the cubic ,block1 each of them equally primary2 and if the plane-being is to have a correct idea of the ,block1 he must be equally familiar with each view4 ,by means of the slabs each aspect can be represented2 but we must remember in each of the three cases1 that the slabs represent different parts of the cube4 ,when we look at the cube ,pallor ,mala in space1 we see that it touches six other cubes by its six faces4 ,but the plane-being could only arrive at this fact by comparing different views4 ,taking the three ,moena sections of the ,block1 he can see that ,pallor ,mala ,moena touches ,plebs ,moena1 ,mora ,moena1 ,uncus ,moena1 and ,tergum ,moena by lines4 ,and it takes the place of ,bidens ,moena1 and is itself displaced by ,cortis ,moena as the ,block passes through the plane4 ,next1 this same ,pallor ,mala can appear to him as an ,alvus4 ,in this case1 it touches ,plebs ,alvus1 ,mora ,alvus1 ,bidens ,alvus1 and ,cortis ,alvus by lines1 takes the place of ,uncus ,alvus1 and is itself displaced by ,tergum ,alvus as the ,block moves4 ,similarly he can observe the relations1 if the ,syce of the ,block be in his plane4 ,hence1 this unknown body ,pallor ,mala appears to him now as one plane-figure now as another1 and comes before him in different connections4 ,pallor ,mala is that which satisfies all these relations4 ,each of them he can in turn present to sense2 but the total conception of ,pallor ,mala itself can only1 if at all1 grow up in his mind4 ,the way for him to form this mental conception1 is to go through all the practical possibilities which ,pallor ,mala would afford him by its various movements and turns4 ,in our world these various relations are found by the most simple observations2 but a plane-being could only acquire them by considerable labour4 ,and if he determined to obtain a knowledge of the physical existence of a higher world than his own1 he must pass through such discipline4 99 99 99 99 99 ,illustration3 ,fig4 #ae4 ,illustration3 ,fig4 #af4 ,we will see what change could be introduced into the shapes he builds by the movements1 which he does not know in his world4 ,let us build up this shape with the cubes of the ,block3 ,urna ,mala1 ,moles ,mala1 ,bidens ,mala1 ,tibicen ,mala4 ,to the plane-being this shape would be the slabs1 ,urna ,moena1 ,moles ,moena1 ,bidens ,moena1 ,tibicen ,moena 7,fig4 #ae74 ,now let the ,block be turned round the ,z axis1 so that it goes past the position1 in which the ,alvus sides enter the vertical plane4 ,let it move until1 passing through the plane1 the same ,moena sides come in again4 ,the mass of the ,block will now have cut through the plane and be on the near side of it towards us2 but the ,moena faces only will be on the plane-being's side of it4 ,the diagram 7,fig4 #af7 shows what he will see1 and it will seem to him similar to the first shape 7,fig4 #ae7 in every respect except its disposition with regard to the ,z axis4 ,it lies in the direction -,x1 opposite to that of the first figure4 ,however much he turn these two figures about in the plane1 he cannot make one occupy the place of the other1 part for part4 ,hence it appears that1 if we turn the plane-being's figure about a line1 it undergoes an operation which is to him quite mysterious4 ,he cannot by any turn in his plane produce the change in the figure produced by us4 ,a little observation will show that a plane-being can only turn round a point4 ,turning round a line is a process unknown to him4 ,therefore one of the elements in a plane-being's knowledge of a space higher than his own1 will be the conception of a kind of turning which will change his solid bodies into their own images4 ,,chapter ,,vi4 ,,the ,,means ,,by ,,which ,a ,,plane-,,being ,,would ,,acquire ,a ,,conception ,,of ,,our ,,figures4 ,take the ,block of twenty-seven ,mala cubes1 and build up the following shape 7,fig4 #ah73-- ,urna ,mala1 ,moles ,mala1 ,plebs ,mala1 ,pallor ,mala1 ,mora ,mala4 ,if this shape1 passed through the vertical plane1 the plane-being would perceive3-- 7#a7 ,the squares ,urna ,moena and ,moles ,moena4 7#b7 ,the three squares ,plebs ,moena1 ,pallor ,moena1 ,mora ,moena1 and then the whole figure would have passed through his plane4 ,if the whole ,block were turned round the ,z axis till the ,alvus sides entered1 and the figure built up as it would be in that disposition of the cubes1 the plane-being would perceive during its passage through the plane3-- 7#a7 ,urna ,alvus2 7#b7 ,moles ,alvus1 ,plebs ,alvus1 ,pallor ,alvus1 ,mora ,alvus1 which would all enter on the left side of the ,z axis4 ,again1 if the ,block were turned round the ,x axis1 the ,syce side would enter1 and the cubes appear in the following order3-- 7#a7 ,urna ,syce1 ,moles ,syce1 ,plebs ,syce2 7#b7 ,pallor ,syce2 7#c7 ,mora ,syce4 ,illustration3 ,fig4 #ag4 ,illustration3 ,fig4 #ah4 ,a comparison of these three sets of appearances would give the plane-being a full account of the figure4 ,it is that which can produce these various appearances4 ,let us now suppose a glass plate placed in front of the ,block in its first position4 ,on this plate let the axes ,x and ,z be drawn4 ,they divide the surface into four parts1 to which we give the following names 7,fig4 #ag73-- ,z ,x that quarter defined by the positive ,z and positive ,x axis4 ,z ,x that quarter defined by the positive ,z and negative ,x axis 7which is called 8,z negative ,x074 ,z ,x that quarter defined by the negative ,z and negative ,x axis4 ,z ,x that quarter defined by the negative ,z and positive ,x axis4 ,the ,block appears in these different quarters or quadrants1 as it is turned round the different axes4 ,in ,z ,x the ,moenas appear1 in ,z ,x the ,alvus faces1 in ,z ,x the ,syces4 ,in each quadrant are drawn nine squares1 to receive the faces of the cubes when they enter4 ,for instance1 in ,z ,x we have the ,moenas of3-- ,z ,comes ,tibicen ,vestis ,ostrum ,bidens ,scena ,urna ,moles ,saltus --------------------------------------,x ,and in ,z ,x we have the ,alvus of3-- ,z ,mars ,spicula ,comes ,ala ,uncus ,ostrum ,sector ,frenum ,urna -,x------------------------------------- ,and in the ,z ,x we have the ,syces of3-- -----------------------------------,x ,urna ,moles ,saltus ,frenum ,plebs ,sypho ,sector ,hama ,remus -,z ,now1 if the shape taken at the beginning of this chapter be looked at through the glass1 and the distance of the second and third walls of the shape behind the glass be considered of no account--that is1 if they be treated as close up to the glass--we get a plane outline1 which occupies the squares ,urna ,moena1 ,moles ,moena1 ,bidens ,moena1 ,tibicen ,moena4 ,this outline is called a projection of the figure4 ,to see it like a plane-being1 we should have to look down on it along the ,z axis4 ,it is obvious that one projection does not give the shape4 ,for instance1 the square ,bidens ,moena might be filled by either ,pallor or ,cortis4 ,all that a square in the room of ,bidens ,moena denotes1 is that there is a cube somewhere in the ,y1 or unknown1 direction from ,bidens ,moena4 ,this view1 just taken1 we should call the front view in our space2 we are then looking at it along the negative ,y axis4 ,when the same shape is turned round on the ,z axis1 so as to be projected on the ,z ,x quadrant1 we have the squares--,urna ,alvus1 ,frenum ,alvus1 ,uncus ,alvus1 ,spicula ,alvus4 ,when it is turned round the ,x axis1 and projected on ,z ,x1 we have the squares1 ,urna ,syce1 ,moles ,syce1 ,plebs ,syce1 and no more4 ,this is what is ordinarily called the ground plan2 but we have set it in a different position from that in which it is usually drawn4 ,illustration3 ,fig4 #ai4 ,now1 the best method for a plane-being of familiarizing himself with shapes in our space1 would be to practise the realization of them from their different projections in his plane4 ,thus1 given the three projections just mentioned1 he should be able to construct the figure from which they are derived4 ,the projections 7,fig4 #ai7 are drawn below the perspective pictures of the shape 7,fig4 #ah74 ,from the front1 or ,moena view1 he would conclude that the shape was ,urna ,mala1 ,moles ,mala1 ,bidens ,mala1 ,tibicen ,mala2 or instead of these1 or also in addition to them1 any of the cubes running in the ,y direction from the plane4 ,that is1 from the ,moena projection he might infer the presence of all the following cubes 7the word ,mala is omitted for brevity73 ,urna1 ,frenum1 ,sector1 ,moles1 ,plebs1 ,hama1 ,bidens1 ,pallor1 ,cortis1 ,tibicen1 ,mora1 ,merces4 ,next1 the ,alvus view or projection might be given by the cubes 7the word ,mala being again omitted73 ,urna1 ,moles1 ,saltus1 ,frenum1 ,plebs1 ,sypho1 ,uncus1 ,pallor1 ,tergum1 ,spicula1 ,mora1 ,oliva4 ,lastly1 looking at the ground plan or ,syce view1 he would infer the possible presence of ,urna1 ,ostrum1 ,comes1 ,moles1 ,bidens1 ,tibicen1 ,plebs1 ,pallor1 ,mora4 ,now1 the shape in higher space1 which is usually there1 is that which is common to all these three appearances4 ,it can be determined1 therefore1 by rejecting those cubes which are not present in all three lists of cubes possible from the projections4 ,and by this process the plane-being could arrive at the enumeration of the cubes which belong to the shape of which he had the projections4 ,after a time1 when he had experience of the cubes 7which1 though invisible to him as wholes1 he could see part by part in turn entering his space71 the projections would have more meaning to him1 and he might comprehend the shape they expressed fragmentarily in his space4 ,to practise the realization from projections1 we should proceed in this way4 ,first1 we should think of the possibilities involved in the ,moena view1 and build them up in cubes before us4 ,secondly1 we should build up the cubes possible from the ,alvus view4 ,again1 taking the shape at the beginning of the chapter1 we should find that the shape of the ,alvus possibilities intersected that of the ,moena possibilities in ,urna1 ,moles1 ,frenum1 ,plebs1 ,pallor1 ,mora2 or1 in other words1 these cubes are common to both4 ,thirdly1 we should build up the ,syce possibilities1 and1 comparing their shape with those of the ,moena and ,alvus projections1 we should find ,urna1 ,moles1 ,plebs1 ,pallor1 ,mora1 of the ,syce view coinciding with the same cubes of the other views1 the only cube present in the intersection of the ,moena and ,alvus possibilities1 and not present in the ,syce view1 being ,frenum4 ,the determination of the figure denoted by the three projections1 may be more easily effected by treating each projection as an indication of what cubes are to be cut away4 ,taking the same shape as before1 we have in the ,moena projection ,urna1 ,moles1 ,bidens1 ,tibicen2 and the possibilities from them are ,urna1 ,frenum1 ,sector1 ,moles1 ,plebs1 ,hama1 ,bidens1 ,pallor1 ,cortis1 ,tibicen1 ,mora1 ,merces4 ,this may aptly be called the ,moena solution4 ,now1 from the ,syce projection1 we learn at once that those cubes1 which in it would produce ,frenum1 ,sector1 ,hama1 ,remus1 ,sypho1 ,saltus1 are not in the shape4 ,this absence of ,frenum and ,sector in the ,syce view proves that their presence in the ,moena solution is superfluous4 ,the absence of ,hama removes the possibility of ,hama1 ,cortis1 ,merces4 ,the absence of ,remus1 ,sypho1 ,saltus1 makes no difference1 as neither they nor any of their ,syce possibilities are present in the ,moena solution4 ,hence1 the result of comparison of the ,moena and ,syce projections and possibilities is the shape3 ,urna1 ,moles1 ,plebs1 ,bidens1 ,pallor1 ,tibicen1 ,mora4 ,this may be aptly called the ,moena-,syce solution4 ,now1 in the ,alvus projection we see that ,ostrum1 ,comes1 ,sector1 ,ala1 and ,mars are absent4 ,the absence of ,sector1 ,ala1 and ,mars has no effect on our ,moena-,syce solution2 as it does not contain any of their ,alvus possibilities4 ,but the absence of ,ostrum and ,comes proves that in the ,moena-,syce solution ,bidens and ,tibicen are superfluous1 since their presence in the original shape would give ,ostrum and ,comes in the ,alvus projection4 ,thus we arrive at the ,moena-,alvus-,syce solution1 which gives us the shape3 ,urna1 ,moles1 ,plebs1 ,pallor1 ,mora4 ,it will be obvious on trial that a shape can be instantly recognised from its three projections1 if the ,block be thoroughly well known in all three positions4 ,any difficulty in the realization of the shapes comes from the arbitrary habit of associating the cubes with some one direction in which they happen to go with regard to us4 ,if we remember ,ostrum as above ,urna1 we are not remembering the ,block1 but only one particular relation of the ,block to us4 ,that position of ,ostrum is a fact as much related to ourselves as to the ,block4 ,there is1 of course1 some information about the ,block implied in that position2 but it is so mixed with information about ourselves as to be ineffectual knowledge of the ,block4 ,it is of the highest importance to enter minutely into all the details of solution written above4 ,for1 corresponding to every operation necessary to a plane-being for the comprehension of our world1 there is an operation1 with which we have to become familiar1 if in our turn we would enter into some comprehension of a world higher than our own4 ,every cube of the ,block ought to be thoroughly known in all its relations4 ,and the ,block must be regarded1 not as a formless mass out of which shapes can be made1 but as the sum of all possible shapes1 from which any one we may choose is a selection4 ,in fact1 to be familiar with the ,block1 we ought to know every shape that could be made by any selection of its cubes2 or1 in other words1 we ought to make an exhaustive study of it4 ,in the ,appendix is given a set of exercises in the use of these names 7which form a language of shape71 and in various kinds of projections4 ,the projections studied in this chapter are not the only1 nor the most natural1 projections by which a plane-being would study higher space4 ,but they suffice as an illustration of our present purpose4 ,if the reader will go through the exercises in the ,appendix1 and form others for himself1 he will find every bit of manipulation done will be of service to him in the comprehension of higher space4 ,there is one point of view in the study of the ,block1 by means of slabs1 which is of some interest4 ,the cubes of the ,block1 and therefore also the representative slabs of their faces1 can be regarded as forming rows and columns4 ,there are three sets of them4 ,if we take the ,moena view1 they represent the views of the three walls of the ,block1 as they pass through the plane4 ,to form the ,alvus view1 we only have to rearrange the slabs1 and form new sets4 ,the first new set is formed by taking the first1 or left-hand1 column of each of the ,moena sets4 ,the second ,alvus set is formed by taking the second or middle columns of the three ,moena sets4 ,the third will consist of the remaining or right-hand columns of the ,moenas4 ,similarly1 the three ,syce sets may be formed from the three horizontal rows or floors of the ,moena sets4 ,hence1 it appears that the plane-being would study our space by taking all the possible combinations of the corresponding rows and columns4 ,he would break up the first three sets into other sets1 and the study of the ,block would practically become to him the study of these various arrangements4 ,,chapter ,,vii4 ,,four-,,space3 ,,its ,,representation ,,in ,,three-,,space4 ,we now come to the essential difficulty of our task4 ,all that has gone before is preliminary4 ,we have now to frame the method by which we shall introduce through our space-figures the figures of a higher space4 ,when a plane-being studies our shapes of cubes1 he has to use squares4 ,he is limited at the outset4 ,a cube appears to him as a square4 ,on ,model #a we see the various squares as which the cube can appear to him4 ,we suppose the plane-being to look from the extremity of the ,z axis down a vertical plane4 ,first1 there is the ,moena square4 ,then there is the square given by a section parallel to ,moena1 which he recognises by the variation of the bounding lines as soon as the cube begins to pass through his plane4 ,then comes the ,murex square4 ,next1 if the cube be turned round the ,z axis and passed through1 he sees the ,alvus and ,proes squares and the intermediate section4 ,so too with the ,syce and ,mel squares and the section between them4 ,now1 dealing with figures in higher space1 we are in an analogous position4 ,we cannot grasp the element of which they are composed4 ,we can conceive a cube2 but that which corresponds to a cube in higher space is beyond our grasp4 ,but the plane-being was obliged to use two-dimensional figures1 squares1 in arriving at a notion of a three-dimensional figure2 so also must we use three-dimensional figures to arrive at the notion of a four-dimensional4 ,let us call the figure which corresponds to a square in a plane and a cube in our space1 a tessaract4 ,model #a is a cube4 ,let us assume a tessaract generated from it4 ,let us call the tessaract ,urna4 ,the generating cube may then be aptly called ,urna ,mala4 ,we may use cubes to represent parts of four-space1 but we must always remember that they are to us1 in our study1 only what squares are to a plane-being with respect to a cube4 ,let us again examine the mode in which a plane-being represents a ,block of cubes with slabs4 ,take ,block #a of the #ha ,set of cubes4 ,the plane-being represents this by nine slabs1 which represent the ,moena face of the block4 ,then1 omitting the solidity of these first nine cubes1 he takes another set of nine slabs to represent the next wall of cubes4 ,lastly1 he represents the third wall by a third set1 omitting the solidity of both second and third walls4 ,in this manner1 he evidently represents the extension of the ,block upwards and sideways1 in the ,z and ,x directions2 but in the ,y direction he is powerless1 and is compelled to represent extension in that direction by setting the three sets of slabs alongside in his plane4 ,the second and third sets denote the height and breadth of the respective walls1 but not their depth or thickness4 ,now1 note that the ,block extends three inches in each of the three directions4 ,the plane-being can represent two of these dimensions on his plane2 but the unknown direction he has to represent by a repetition of his plane figures4 ,the cube extends three inches in the ,y direction4 ,he has to use #c sets of slabs4 ,the ,block is built up arbitrarily in this manner3 ,starting from ,urna ,mala and going up1 we come to a ,brown cube1 and then to a ,light-blue cube4 ,starting from ,urna ,mala and going right1 we come to an ,orange and a ,fawn cube4 ,starting from ,urna ,mala and going away from us1 we come to a ,blue and a ,buff cube4 ,now1 the plane-being represents the ,brown and ,orange cubes by squares lying next to the square which represents ,urna ,mala4 ,the ,blue cube is as close as the ,brown cube to ,urna ,mala1 but he can find no place in the plane where he can place a ,blue square so as to show this co-equal proximity of both cubes to the first4 ,so he is forced to put a ,blue square anywhere in his plane and say of it3 8,this ,blue square represents what ,i should arrive at1 if ,i started from ,urna ,mala and moved away1 that is in the ,y or unknown direction40 ,now1 just as there are three cubes going up1 so there are three going away4 ,hence1 besides the ,blue square placed anywhere on the plane1 he must also place a ,buff square beyond it1 to show that the ,block extends as far away as it does upwards and sideways4 7,each cube being a different colour1 there will be as many different colours of squares as of cubes47 ,it will easily be seen that not only the ,gold square1 but also the ,orange and every other square in the first set of slabs must have two other squares set somewhere beyond it on the plane to represent the extension of the ,block away1 or in the unknown ,y direction4 ,coming now to the representation of a four-dimensional block1 we see that we can show only three dimensions by cubic blocks1 and that the fourth can only be represented by repetitions of such blocks4 ,there must be a certain amount of arbitrary naming and colouring4 ,the colours have been chosen as now stated4 ,take the first ,block of the #ha ,set4 ,we are familiar with its colours1 and they can be found at any time by reference to ,model #a4 ,now1 suppose the ,gold cube to represent what we can see in our space of a ,gold tessaract2 the other cubes of ,block #a give the colours of the tessaracts which lie in the three directions ,x1 ,y1 and ,z from the ,gold one4 ,but what is the colour of the tessaract which lies next to the ,gold in the unknown direction1 ,w8 ,let us suppose it to be ,stone in colour4 ,taking out ,block #b of the #ha ,set and arranging it on the pattern of ,model #i1 we find in it a ,stone cube4 ,but1 just as there are three tessaracts in the ,x1 ,y1 and ,z directions1 as shown by the cubes in ,block #a1 so also must there be three tessaracts in the unknown direction1 ,w4 ,take ,block #c of the #ha ,set4 ,this ,block can be arranged on the pattern of ,model #b4 ,in it there is a ,silver cube where the ,gold cube lies in ,block #a4 ,hence1 we may say1 the tessaract which comes next to the ,stone one in the unknown direction from the ,gold1 is of a ,silver colour4 ,now1 a cube in all these cases represents a tessaract4 ,between the ,gold and ,stone cubes there is an inch in the unknown direction4 ,the ,gold tessaract is supposed to be ,gold throughout in all four directions1 and so also is the ,stone4 ,we may imagine it in this way4 ,suppose the set of three tessaracts1 the ,gold1 the ,stone1 and the ,silver to move through our space at the rate of an inch a minute4 ,we should first see the ,gold cube which would last a minute1 then the ,stone cube for a minute1 and lastly the ,silver cube a minute4 7,this is precisely analogous to the appearance of passing cubes to the plane-being as successive squares lasting a minute47 ,after that1 nothing would be visible4 ,now1 just as we must suppose that there are three tessaracts proceeding from the ,gold cube in the unknown direction1 so there must be three tessaracts extending in the unknown direction from every one of the #bg cubes of the first ,block4 ,the ,block of #bg cubes is not a ,block of #bg tessaracts1 but it represents as much of them as we can see at once in our space2 and they form the first portion or layer 7like the first wall of cubes to the plane-being7 of a set of eighty-one tessaracts1 extending to equal distances in all four directions4 ,thus1 to represent the whole ,block of tessaracts there are #ha cubes1 or three ,blocks of #bg each4 ,now1 it is obvious that1 just as a cube has various plane boundaries1 so a tessaract has various cube boundaries4 ,the cubes of the tessaract1 which we have been regarding1 have been those containing the ,x1 ,y1 and ,z directions1 just as the plane-being regarded the ,moena faces containing the ,x and ,z directions4 ,and1 as long as the tessaract is unchanged in its position with regard to our space1 we can never see any portion of it which is in the unknown direction4 ,similarly1 we saw that a plane-being could not see the parts of a cube which went in the third direction1 until the cube was turned round one of its edges4 ,in order to make it quite clear what parts of a cube came into the plane1 we gave distinct names to them4 ,thus1 the squares containing the ,z and ,x directions were called ,moena and ,murex2 those containing the ,z and ,y1 ,alvus and ,proes2 and those the ,x and ,y1 ,syce and ,mel4 ,now1 similarly with our four axes1 any three will determine a cube4 ,let the tessaract in its normal position have the cube ,mala determined by the axes ,z1 ,x1 ,y4 ,let the cube ,lar be that which is determined by ,x1 ,y1 ,w1 that is1 the cube which1 starting from the ,x ,y plane1 stretches one inch in the unknown or ,w direction4 ,let ,vesper be the cube determined by ,z1 ,y1 ,w1 and ,pluvium by ,z1 ,x1 ,w4 ,and let these cubes have opposite cubes of the following names3 ,mala has ,margo ,lar ,velum ,vesper ,idus ,pluvium ,tela ,another way of looking at the matter is this4 ,when a cube is generated from a square1 each of the lines bounding the square becomes a square1 and the square itself becomes a cube1 giving two squares in its initial and final positions4 ,when a cube moves in the new and unknown direction1 each of its planes traces a cube and it generates a tessaract1 giving in its initial and final positions two cubes4 ,thus there are eight cubes bounding the tessaract1 six of them from the six plane sides and two from the cube itself4 ,these latter two are ,mala and ,margo4 ,the cubes from the six sides are3 ,lar from ,syce1 ,velum from ,mel1 ,vesper from ,alvus1 ,idus from ,proes1 ,pluvium from ,moena1 ,tela from ,murex4 ,and just as a plane-being can only see the squares of a cube1 so we can only see the cubes of a tessaract4 ,it may be said that the cube can be pushed partly through the plane1 so that the plane-being sees a section between ,moena and ,murex4 ,similarly1 the tessaract can be pushed through our space so that we can see a section between ,mala and ,margo4 ,there is a method of approaching the matter1 which settles all difficulties1 and provides us with a nomenclature for every part of the tessaract4 ,we have seen how by writing down the names of the cubes of a block1 and then supposing that their number increases1 certain sets of the names come to denote points1 lines1 planes1 and solid4 ,similarly1 if we write down a set of names of tessaracts in a block1 it will be found that1 when their number is increased1 certain sets of the names come to denote the various parts of a tessaract4 ,for this purpose1 let us take the #ha ,set1 and use the cubes to represent tessaracts4 ,the whole of the #ha cubes make one single tessaractic set extending three inches in each of the four directions4 ,the names must be remembered to denote tessaracts4 ,thus1 ,corvus is a tessaract which has the tessaracts ,cuspis and ,nug to the right1 ,arctos and ,ilex above it1 ,dos and ,cista away from it1 and ,ops and ,spira in the fourth or unknown direction from it4 ,it will be evident at once1 that to write these names in any representative order we must adopt an arbitrary system4 ,we require them running in four directions2 we have only two on paper4 ,the ,x direction 7from left to right7 and the ,y 7from the bottom towards the top of the page7 will be assumed to be truly represented4 ,the ,z direction will be symbolized by writing the names in floors1 the upper floors always preceding the lower4 ,lastly1 the fourth1 or ,w1 direction 7which has to be symbolized in three-dimensional space by setting the solids in an arbitrary position7 will be signified by writing the names in blocks1 the name which stands in any one place in any block being next in the ,w direction to that which occupies the same position in the block before or after it4 ,thus1 ,ops is written in the same place in the ,second ,block1 ,spira in the ,third ,block1 as ,corvus occupies in the ,first ,block4 ,since there are an equal number of tessaracts in each of the four directions1 there will be three floors ,z when there are three ,x and ,y4 ,also1 there will be three ,blocks ,w4 ,if there be four tessaracts in each direction1 there will be four floors ,z1 and four blocks ,w4 ,thus1 when the number in each direction is enlarged1 the number of blocks ,w is equal to the number of tessaracts in each known direction4 ,on pp4 #acf1 #acg were given the names as used for a cubic block of #bg or #fd4 ,using the same and more names for a tessaractic ,set1 and remembering that each name now represents1 not a cube1 but a tessaract1 we obtain the following nomenclature1 the order in which the names are written being that stated above3 ,,third ,,block4 ,upper ,solia ,livor ,talus ,floor4 ,lensa ,lares ,calor ,felis ,tholus ,passer ----------------- ,middle ,lixa ,portica ,vena ,floor4 ,crux ,margo ,sal ,pagus ,silex ,onager ----------------- ,lower ,panax ,mensura ,mugil ,floor4 ,opex ,lappa ,mappa ,spira ,luca ,ancilla ,,second ,,block4 ,upper ,orsa ,mango ,libera ,floor4 ,creta ,velum ,meatus ,lucta ,limbus ,pator ----------------- ,middle ,camoena ,tela ,orca ,floor4 ,vesper ,tessaract ,idus ,pagina ,pluvium ,pactum ----------------- ,lower ,lis ,lorica ,offex ,floor4 ,lua ,lar ,olla ,ops ,lotus ,limus ,,first ,,block4 ,upper ,olus ,semita ,lama ,floor4 ,via ,mel ,iter ,ilex ,callis ,sors ----------------- ,middle ,bucina ,murex ,daps ,floor4 ,alvus ,mala ,proes ,arctos ,moena ,far ----------------- ,lower ,cista ,cadus ,crus ,floor4 ,dos ,syce ,bolus ,corvus ,cuspis ,nug ,it is evident that this set of tessaracts could be increased to the number of four in each direction1 the names being used as before for the cubic blocks on pp4 #acf1 #acg1 and in that case the ,second ,block would be duplicated to make the four blocks required in the unknown direction4 ,comparing such an #ha ,set and #bef ,set1 we should find that ,cuspis1 which was a single tessaract in the #ha ,set became two tessaracts in the #bef ,set4 ,and1 if we introduced a larger number1 it would simply become longer1 and not increase in any other dimension4 ,thus1 ,cuspis would become the name of an edge4 ,similarly1 ,dos would become the name of an edge1 and also ,arctos4 ,ops1 which is found in the ,middle ,block of the #ha ,set1 occurs both in the ,second and ,third ,blocks of the #bef ,set2 that is1 it also tends to elongate and not extend in any other direction1 and may therefore be used as the name of an edge of a tessaract4 ,looking at the cubes which represent the ,syce tessaracts1 we find that1 though they increase in number1 they increase only in two directions2 therefore1 ,syce may be taken to signify a square4 ,but1 looking at what comes from ,syce in the ,w direction1 we find in the ,middle ,block of the #ha ,set one ,lar1 and in the ,second and ,third ,blocks of the #bef ,set four ,lars each4 ,hence1 ,lar extends in three directions1 ,x1 ,y1 ,w1 and becomes a cube4 ,similarly1 ,moena is a plane2 but ,pluvium1 which proceeds from it1 extends not only sideways and upwards like ,moena1 but in the unknown direction also4 ,it occurs in both ,middle ,blocks of the #bef ,set4 ,hence1 it also is a cube4 ,we have now considered such parts of the ,sets as contain one1 two1 and three dimensions4 ,but there is one part which contains four4 ,it is that named ,tessaract4 ,in the #bef ,set there are eight such cubes in the ,second1 and eight in the ,third ,block2 that is1 they extend ,z1 ,x1 ,y1 and also ,w4 ,they may1 therefore1 be considered to represent that part of a tessaract or tessaractic ,set1 which is analogous to the interior of a cube4 ,the arrangement of colours corresponding to these names is seen on ,model #a corresponding to ,mala1 ,model #b to ,margo1 and ,model #i to the intermediate block4 ,when we take the view of the tessaract with which we commenced1 and in which ,arctos goes ,z1 ,cuspis ,x1 ,dos ,y1 and ,ops ,w1 we see ,mala in our space4 ,but when the tessaract is turned so that the ,ops line goes -,x1 while ,cuspis is turned ,w1 the other two remaining as they were1 then we do not see ,mala1 but that cube which1 in the original position of the tessaract1 contains the ,z1 ,y1 ,w1 directions1 that is1 the ,vesper cube4 ,a plane-being may begin to study a block of cubes by their ,syce squares2 but if the block be turned round ,dos1 he will have ,alvus squares in his space1 and he must then use them to represent the cubic ,block4 ,so1 when the tessaractic ,set is turned round1 ,mala cubes leave our space1 and ,vespers enter4 ,there are two ways which can be followed in studying the ,set of tessaracts4 ,i4 ,each tessaract of one inch every way can be supposed to be of the same colour throughout1 so that1 whichever way it be turned1 whichever of its edges coincide with our known axes1 it appears to us as a cube of one uniform colour4 ,thus1 if ,urna be the tessaract1 ,urna ,mala would be a ,gold cube1 ,urna ,vesper a ,gold cube1 and so on4 ,this method is1 for the most part1 adopted in the following pages4 ,in this case1 a whole ,set of #d #d #d #d tessaracts would in colours resemble a set composed of four cubes like ,models #a1 #i1 #i1 and #b4 ,but1 when any question about a particular tessaract has to be settled1 it is advantageous1 for the sake of distinctness1 to suppose it coloured in its different regions as the whole set is coloured4 ,,ii4 ,the other plan is1 to start with the cubic sides of the inch tessaract1 each coloured according to the scheme of the ,models #a to #h4 ,in this case1 the lines1 if shown at all1 should be very thin4 ,for1 in fact1 only the faces would be seen1 as the width of the lines would only be equal to the thickness of our matter in the fourth dimension1 which is indistinguishable to the senses4 ,if such completely coloured cubes be used1 less error is likely to creep in2 but it is a disadvantage that each cube in the several blocks is exactly like the others in that block4 ,if the reader make such a set to work with for a time1 he will gain greatly1 for the real way of acquiring a sense of higher space is to obtain those experiences of the senses exactly1 which the observation of a four-dimensional body would give4 ,these ,models #a-h are called sides of the tessaract4 ,to make the matter perfectly clear1 it is best to suppose that any tessaract or set of tessaracts which we examine1 has a duplicate exactly similar in shape and arrangement of parts1 but different in their colouring4 ,in the first1 let each tessaract have one colour throughout1 so that all its cubes1 apprehended in turn in our space1 will be of one and the same colour4 ,in the duplicate1 let each tessaract be so coloured as to show its different cubic sides by their different colours4 ,then1 when we have it turned to us in different aspects1 we shall see different cubes1 and when we try to trace the contacts of the tessaracts with each other1 we shall be helped by realizing each part of every tessaract in its own colour4 ,,chapter ,,viii4 ,,representation ,,of ,,four-,,space ,,by ,,name4 ,,study ,,of ,,tessaracts4 ,we have now surveyed all the preliminary ground1 and can study the masses of tessaracts without obscurity4 ,we require a scaffold or framework for this purpose1 which in three dimensions will consist of eight cubic spaces or octants assembled round one point1 as in two dimensions it consisted of four squares or quadrants round a point4 ,these eight octants lie between the three axes ,z1 ,x1 ,y1 which intersect at the given point1 and can be named according to their positions between the positive and negative directions of those axes4 ,thus the octant ,z1 ,x1 ,y1 is that which is contained by the positive portions of all three axes2 the octant ,z1 ,x1 ,y1 that which is to the left of ,z1 ,x1 ,y1 and between the positive parts of ,z and ,y and the negative of ,x4 ,to illustrate this quite clearly1 let us take the eight cubes--,urna1 ,moles1 ,plebs1 ,frenum1 ,uncus1 ,pallor1 ,bidens1 ,ostrum--and place them in the eight octants4 ,let them be placed round the point of intersection of the axes2 ,pallor ,corvus1 ,plebs ,ilex1 etc41 will be at that point4 ,their positions will then be3-- ,urna in the ,octant ,z ,x ,y ,moles ,z ,x ,y ,plebs ,z ,x ,y ,frenum ,z ,x ,y ,uncus ,z ,x ,y ,pallor ,z ,x ,y ,bidens ,z ,x ,y ,ostrum ,z ,x ,y ,the names used for the cubes1 as they are before us1 are as follows3-- ,,third ,,block4 ,third ,arcus ,mala ,ovis ,mala ,portio ,mala ,floor4 ,laurus ,mala ,tigris ,mala ,segmen ,mala ,axis ,mala ,troja ,mala ,aries ,mala ,second ,postis ,mala ,clipeus ,mala ,tabula ,mala ,floor4 ,orcus ,mala ,lacerta ,mala ,testudo ,mala ,verbum ,mala ,luctus ,mala ,anguis ,mala ,first ,telum ,mala ,nepos ,mala ,angusta ,mala ,floor4 ,polus ,mala ,penates ,mala ,vulcan ,mala ,cervix ,mala ,securis ,mala ,vinculum ,mala ,,second ,,block4 ,third ,ara ,mala ,vomer ,mala ,pluma ,mala ,floor4 ,praeda ,mala ,sacerdos ,mala ,hydra ,mala ,cortex ,mala ,mica ,mala ,flagellum ,mala ,second ,pilum ,mala ,glans ,mala ,colus ,mala ,floor4 ,ocrea ,mala ,tessera ,mala ,domitor ,mala ,cardo ,mala ,cudo ,mala ,malleus ,mala ,first ,agmen ,mala ,lacus ,mala ,arvus ,mala ,floor4 ,crates ,mala ,cura ,mala ,limen ,mala ,thyrsus ,mala ,vitta ,mala ,sceptrum ,mala ,,first ,,block4 ,third ,mars ,mala ,merces ,mala ,tyro ,mala ,floor4 ,spicula ,mala ,mora ,mala ,oliva ,mala ,comes ,mala ,tibicen ,mala ,vestis ,mala ,second ,ala ,mala ,cortis ,mala ,aer ,mala ,floor4 ,uncus ,mala ,pallor ,mala ,tergum ,mala ,ostrum ,mala ,bidens ,mala ,scena ,mala ,first ,sector ,mala ,hama ,mala ,remus ,mala ,floor4 ,frenum ,mala ,plebs ,mala ,sypho ,mala ,urna ,mala ,moles ,mala ,saltus ,mala ,their colours can be found by reference to the ,models #a1 #i1 #b1 which correspond respectively to the ,first1 ,second1 and ,third ,blocks4 ,thus1 ,urna ,mala is ,gold2 ,moles1 ,orange2 ,saltus1 ,fawn2 ,thyrsus1 ,stone2 ,cervix1 ,silver4 ,the cubes whose colours are not shown in the ,models1 are ,pallor ,mala1 ,tessera ,mala1 and ,lacerta ,mala1 which are equivalent to the interiors of the ,model cubes1 and are respectively ,light-buff1 ,wooden1 and ,sage-green4 ,these #ha cubes are the cubic sides and sections of the tessaracts of an #ha tessaractic ,set1 which measures three inches in every direction4 ,we suppose it to pass through our space4 ,let us call the positive unknown direction ,ana 7i4e41 ,w7 and the negative unknown direction ,kata 7-,w74 ,then1 as the whole tessaract moves ,kata at the rate of an inch a minute1 we see first the ,first ,block of #bg cubes for one minute1 then the ,second1 and lastly the ,third1 each lasting one minute4 ,now1 when the ,first ,block stands in the normal position1 the edges of the tessaract that run from the ,corvus corner of ,urna ,mala1 are3 ,arctos in ,z1 ,cuspis in ,x1 ,dos in ,y1 ,ops in ,w4 ,hence1 we denote this position by the following symbol3-- ,z ,x ,y ,w a c d o where a stands for ,arctos1 c for ,cuspis1 d for ,dos1 and o for ,ops1 and the other letters for the four axes in space4 a1 c1 d1 o are the axes of the tessaract1 and can take up different directions in space with regard to us4 99 99 99 99 99 ,let us now take a smaller four-dimensional set4 ,of the #ha ,set let us take the following3-- ,z ,x ,y ,w a c d o ,,second ,,block4 ,second ,floor4 ,ocrea ,mala ,tessera ,mala ,cardo ,mala ,cudo ,mala ,first ,floor4 ,crates ,mala ,cura ,mala ,thyrsus ,mala ,vitta ,mala ,,first ,,block4 ,second ,floor4 ,uncus ,mala ,pallor ,mala ,ostrum ,mala ,bidens ,mala ,first ,floor4 ,frenum ,mala ,plebs ,mala ,urna ,mala ,moles ,mala ,let the ,first ,block be put up before us in ,z ,x ,y1 7,urna ,corvus is at the junction of our axes ,z ,x ,y74 ,the ,second ,block is now one inch distant in the unknown direction2 and1 if we suppose the tessaractic ,set to move through our space at the rate of one inch a minute1 the ,second will enter in one minute1 and replace the first4 ,but1 instead of this1 let us suppose the tessaracts to turn so that ,ops1 which now goes ,w1 shall go -,x4 ,then we can see in our space that cubic side of each tessaract which is contained by the lines ,arctos1 ,dos1 and ,ops1 the cube ,vesper2 and we shall no longer have the ,mala sides but the ,vesper sides of the tessaractic ,set in our space4 ,we will now build it up in its ,vesper view 7as we built up the cubic ,block in its ,alvus view74 ,take the ,gold cube1 which now means ,urna ,vesper1 and place it on the left hand of its former position as ,urna ,mala1 that is1 in the octant ,z ,x ,y4 ,thyrsus ,vesper1 which previously lay just beyond ,urna ,vesper in the unknown direction1 will now lie just beyond it in the -,x direction1 that is1 to the left of it4 ,the tessaractic ,set is now in the position ,z ,x ,y ,w a d c 7the minus sign over the o meaning that ,ops runs in the negative direction71 and its ,vespers lie in the following order3-- ,,second ,,block4 ,second ,floor4 ,tessara ,pallor ,cudo ,bidens ,first ,floor4 ,cura ,plebs ,vitta ,moles ,,first ,,block4 ,second ,floor4 ,ocrea ,uncus ,cardo ,ostrum ,first ,floor4 ,crates ,frenum ,thyrsus ,urna ,the name ,vesper is left out in the above list for the sake of brevity1 but should be used in studying the positions4 ,illustration3 ,fig4 #bj4 ,on comparing the two lists of the ,mala view and ,vesper view1 it will be seen that the cubes presented in the ,vesper view are new sides of the tessaract1 and that the arrangement of them is different from that in the ,mala view4 7,this is analogous to the changes in the slabs from the ,moena to ,alvus view of the cubic ,block47 ,of course1 the ,vespers of all these tessaracts are not visible at once in our space1 any more than are the ,moenas of all three walls of a cubic ,block to a plane-being4 ,but if the tessaractic ,set be supposed to move through space in the unknown direction at the rate of an inch a minute1 the ,second ,block will present its ,vespers after the ,first ,block has lasted a minute4 ,the relative position of the ,mala ,block and the ,vesper ,block may be represented in our space as in the diagram1 ,fig4 #bj4 ,but it must be distinctly remembered that this arrangement is quite conventional1 no more real than a plane-being's symbolization of the ,moena ,wall and the ,alvus ,wall of the cubic ,block by the arrangement of their ,moena and ,alvus faces1 with the solidity omitted1 along one of his known directions4 ,the ,vespers of the ,first and ,second ,blocks cannot be in our space simultaneously1 any more than the ,moenas of all three walls in plane space4 ,to render their simultaneous presence possible1 the cubic or tessaractic ,block or ,set must be broken up1 and its parts no longer retain their relations4 ,this fact is of supreme importance in considering higher space4 ,endless fallacies creep in as soon as it is forgotten that the cubes are merely representative as the slabs were1 and the positions in our space merely conventional and symbolical1 like those of the slabs along the plane4 ,and these fallacies are so much fostered by again symbolizing the cubic symbols and their symbolical positions in perspective drawings or diagrams1 that the reader should surrender all hope of learning space from this book or the drawings alone1 and work every thought out with the cubes themselves4 ,if we want to see what each individual cube of the tessaractic faces presented to us in the last example is like1 we have only to consider each of the ,malas similar in its parts to ,model #a1 and each of the ,vespers to ,model #e4 ,and it must always be remembered that the cubes1 though used to represent both ,mala and ,vesper faces of the tessaract1 mean as great a difference as the slabs used for the ,moena and ,alvus faces of the cube4 ,if the tessaractic ,set move ,kata through our space1 when the ,vesper faces are presented to us1 we see the following parts of the tessaract ,urna 7and1 therefore1 also the same parts of the other tessaracts73 7#a7 ,urna ,vesper1 which is ,model #e4 7#b7 ,a parallel section between ,urna ,vesper and ,urna ,idus1 which is ,model #aa4 7#c7 ,urna ,idus1 which is ,model #f4 ,when ,urna ,idus has passed ,kata our space1 ,moles ,vesper enters it2 then a section between ,moles ,vesper and ,moles ,idus1 and then ,moles ,idus4 ,here we have evidently observed the tessaract more minutely2 as it passes ,kata through our space1 starting on its ,vesper side1 we have seen the parts which would be generated by ,vesper moving along ,cuspis--that is ,ana4 ,two other arrangements of the tessaracts have to be learnt besides those from the ,mala and ,vesper aspect4 ,one of them is the ,pluvium aspect4 ,build up the ,set in ,z ,x ,y1 letting ,arctos run ,z1 ,cuspis ,x1 and ,ops ,y4 ,in the common plane ,moena1 ,urna ,pluvium coincides with ,urna ,mala1 though they cannot be in our space together2 so too ,moles ,pluvium with ,moles ,mala1 ,ostrum ,pluvium with ,ostrum ,mala4 ,and lying towards us1 or ,y1 is now that tessaract which before lay in the ,w direction from ,urna1 viz41 ,thyrsus4 ,the order will therefore be the following 7a star denotes the cube whose corner is at point of intersection of the axes1 and the name ,pluvium must be understood to follow each of the names73 ,z ,x ,y ,w a c d ,,second ,,block4 ,second ,floor4 ,uncus ,pallor ,ocrea ,tessera ,first ,floor4 ,frenum ,plebs ,crates ,cura ,,first ,,block4 ,second ,floor4 ,ostrum ,bidens ,cardo ,cudo ,first ,floor4 99,urna ,moles ,thyrsus ,vitta ,thus the wall of cubes in contact with that wall of the ,mala position which contains the ,urna1 ,moles1 ,ostrum1 and ,bidens ,malas1 is a wall composed of the ,pluviums of ,urna1 ,moles1 ,ostrum1 and ,bidens4 ,the wall next to this1 and nearer to us1 is of ,thyrsus1 ,vitta1 ,cardo1 ,cudo1 ,pluviums4 ,the ,second ,block is one inch out of our ,space1 and only enters it if the ,block moves ,kata4 ,model #g shows the ,pluvium cube2 and each of the cubes of the tessaracts seen in the ,pluvium position is a ,pluvium4 ,if the tessaractic ,set moved ,kata1 we would see the ,section between ,pluvium and ,tela for all but a minute2 and then ,tela would enter our space1 and the ,tela of each tessaract would be seen4 ,model #ab shows the ,section from ,pluvium to ,tela4 ,model #h is ,tela4 ,tela only lasts for a flash1 as it has only the minutest magnitude in the unknown or ,ana direction4 ,then1 ,frenum ,pluvium takes the place of ,urna ,tela2 and1 when it passes through1 we see a similar section between ,frenum ,pluvium and ,frenum ,tela1 and lastly ,frenum ,tela4 ,then the tessaractic ,set passes out1 or ,kata1 our space4 ,a similar process takes place with every other tessaract1 when the ,set of tessaracts moves through our space4 ,there is still one more arrangement to be learnt4 ,if the line of the tessaract1 which in the ,mala position goes ,ana1 or ,w1 be changed into the ,z or downwards direction1 the tessaract will then appear in our space below the ,mala position2 and the side presented to us will not be ,mala1 but that which contains the lines ,dos1 ,cuspis1 and ,ops4 ,this side is ,model #c1 and is called ,lar4 ,underneath the place which was occupied by ,urna ,mala1 will come ,urna ,lar2 under the place of ,moles ,mala1 ,moles ,lar2 under the place of ,frenum ,mala1 ,frenum ,lar4 ,the tessaract1 which in the ,mala position was an inch out of our space ,ana1 or ,w1 from ,urna ,mala1 will now come into it an inch downwards1 or ,z1 below ,urna ,mala1 with its ,lar presented to us2 that is1 ,thyrsus ,lar will be below ,urna ,lar4 ,in the whole arrangement of them written below1 the highest floors are written first1 for now they stretch downwards instead of upwards4 ,the name ,lar is understood after each4 ,z ,x ,y ,w c d a ,,second ,,block4 ,second ,floor4 ,uncus ,pallor ,ostrum ,bidens ,first ,floor4 ,ocrea ,tessera ,cardo ,cudo ,,first ,,block4 ,second ,floor4 ,frenum ,plebs 99,urna ,moles ,first ,floor4 ,crates ,cura ,thyrsus ,vitta ,here it is evident that what was the lower floor of ,malas1 ,urna1 ,moles1 ,plebs1 ,frenum1 now appears as if carried downwards instead of upwards1 ,lars being presented in our space instead of ,malas4 ,this ,block of ,lars is what we see of the tessaract ,set when the ,arctos line1 which in the ,mala position goes up1 is turned into the ,ana1 or ,w1 direction1 and the ,ops line comes in downwards4 ,the rest of the tessaracts1 which consists of the cubes opposite to the four treated above1 and of the tessaractic space between them1 is all ,ana in our space4 ,if the tessaract be moved through our space--for instance1 when the ,lars are present in it--we see1 taking ,urna alone1 first the section between ,urna ,lar and ,urna ,velum 7,model #aj71 and then ,urna ,velum 7,model #d71 and similarly the sections and ,velums of each tessaract in the ,set4 ,when the ,first ,block has passed ,kata our space1 ,ostrum ,lar enters2 and the ,lars of the ,second ,block of tessaracts occupy the places just vacated by the ,velums of the ,first ,block4 ,then1 as the tessaractic ,set moves on ,kata1 the sections between ,velums and ,lars of the ,second ,block of tessaracts enter our space1 and finally their ,velums4 ,then the whole tessaractic ,set disappears from our space4 ,when we have learnt all these aspects and passages1 we have experienced some of the principal features of this small ,set of tessaracts4 ,,chapter ,,ix4 ,,further ,,study ,,of ,,tessaracts4 ,when the arrangement of a small set has been mastered1 the different views of the whole #ha ,set should be learnt4 ,it is now clear to us that1 in the list of the names of the eighty-one tessaracts given above1 those which lie in the ,w direction appear in different blocks1 while those that lie in the ,z1 ,x1 or ,y directions can be found in the same block4 ,therefore1 from the arrangement given1 which is denoted by ,z ,x ,y ,w 1 a c d o or more briefly by a c d o1 we can form any other arrangement4 ,to confirm the meaning of the symbol a c d o for position1 let us remember that the order of the axes known in our space will invariably be ,z ,x ,y1 and the unknown direction will be stated last1 thus3 ,z ,x ,y ,w4 ,hence1 if we write a d c1 we know that the position or aspect intended is that in which ,arctos 7a7 goes ,z1 ,ops 77 negative ,x1 ,dos 7d7 ,y1 and ,cuspis 7c7 ,w4 ,and such an arrangement can be made by shifting the nine cubes on the left side of the ,first ,block of the eighty-one tessaracts1 and putting them into the ,z ,x ,y octant1 so that they just touch their former position4 ,next to them1 to their left1 we set the nine of the left side of the ,second ,block of the #ha ,set2 and next to these again1 on their left1 the nine of the left side of the ,third ,block4 ,this ,block of twenty-seven now represents ,vesper ,cubes1 which have only one square identical with the ,mala cubes of the previous blocks1 from which they were taken4 ,similarly the ,block which is one inch ,ana1 can be made by taking the nine cubes which come vertically in the middle of each of the ,blocks in the first position1 and arranging them in a similar manner4 ,lastly1 the ,block which lies two inches ,ana1 can be made by taking the right sides of nine cubes each from each of the three original ,blocks1 and arranging them so that those in the ,second original ,block go to the left of those in the ,first1 and those in the ,third to their left4 ,in this manner we should obtain three new ,blocks1 which represent what we can see of the tessaracts1 when the direction in which ,urna1 ,moles1 ,saltus lie in the original ,set1 is turned ,w4 ,the ,pluvium ,block we can make by taking the front wall of each original ,block1 and setting each an inch nearer to us1 that is -,y4 ,the far sides of these cubes are ,moenas of ,pluviums4 ,by continuing this treatment of the other walls of the three original ,blocks parallel to the front wall1 we obtain two other ,blocks of tessaracts4 ,the three together are the tessaractic position a c d1 for in all of them ,ops lies in the -,y direction1 and ,dos has been turned ,w4 ,the ,lar position is more difficult to construct4 ,to put the ,lars of the ,blocks in their natural position in our space1 we must start with the original ,mala ,blocks1 at least three inches above the table4 ,the ,first ,lar ,block is made by taking the lowest floors of the three ,mala ,blocks1 and placing them so that that of the ,second is below that of the ,first1 and that of the ,third below that of the ,second4 ,the floor of cubes whose diagonal runs from ,urna ,lar to ,remus ,lar1 will be at the top of the ,block of ,lars2 and that whose diagonal goes from ,cervix ,lar to ,angusta ,lar1 will be at the bottom4 ,the next ,block of ,lars will be made by taking the middle horizontal floors of the three original ,blocks1 and placing them in a similar succession--the floor from ,ostrum ,lar to ,aer ,lar being at the top1 that from ,cardo ,lar to ,colus ,lar in the middle1 and ,verbum ,lar to ,tabula ,lar at the bottom4 ,the ,third ,lar ,block is composed of the top floor of the ,first ,block on the top--that is1 of ,comes ,lar to ,tyro ,lar1 of ,cortex ,lar to ,pluma ,lar in the middle1 and ,axis ,lar to ,portio ,lar at the bottom4 ,,chapter ,x4 ,,cyclical ,,projections4 ,let us denote the original position of the cube1 that wherein ,arctos goes ,z1 ,cuspis ,x1 and ,dos ,y1 by the expression1 ,z ,x ,y 7#a7 a c d ,if the cube be turned round ,cuspis1 ,dos goes ,z1 ,cuspis remains unchanged1 and ,arctos goes ,y1 and we have the position1 ,z ,x ,y d c a where ,z d means that ,dos runs in the negative direction of the ,z axis from the point where the axes intersect4 ,we might write ,z d but it is preferable to write ,z d4 ,if we next turn the cube round the line1 which runs ,y1 that is1 round ,arctos1 we obtain the position1 ,z ,x ,y 7#b7 c d a and by means of this double turn we have put c and d in the places of a and c4 ,moreover1 we have no negative directions4 ,this position we call simply c d a4 ,if from it we turn the cube round a1 which runs ,y1 we get ,z ,x ,y d c a1 and if1 then1 we turn it round ,dos we get ,z ,x ,y d a c or simply d a c4 ,this last is another position in which all the lines are positive1 and the projections1 instead of lying in different quadrants1 will be contained in one4 ,the arrangement of cubes in a c d we know4 ,that in c d a is3 ,vestis ,oliva ,tyro ,third ,scena ,tergum ,aer ,floor4 ,saltus ,sypho ,remus ,tibicen ,mora ,merces ,second ,bidens ,pallor ,cortis ,floor4 ,moles ,plebs ,hama ,comes ,spicula ,mars ,first ,ostrum ,uncus ,ala ,floor4 ,urna ,frenum ,sector ,it will be found that learning the cubes in this position gives a great advantage1 for thereby the axes of the cube become dissociated with particular directions in space4 ,the d a c position gives the following arrangement3 ,remus ,aer ,tyro ,hama ,cortis ,merces ,sector ,ala ,mars ,sypho ,tergum ,oliva ,plebs ,pallor ,mora ,frenum ,uncus ,spicula ,saltus ,scena ,vestis ,moles ,bidens ,tibicen ,urna ,ostrum ,comes ,the sides1 which touch the vertical plane in the first position1 are respectively1 in a c d ,moena1 in c d a ,syce1 in d a c ,alvus4 ,take the shape ,urna1 ,ostrum1 ,moles1 ,saltus1 ,scena1 ,sypho1 ,remus1 ,aer1 ,tyro4 ,this gives in a c d the projection3 ,urna ,moena1 ,ostrum ,moena1 ,moles ,moena1 ,saltus ,moena1 ,scena ,moena1 ,vestis ,moena4 7,if the different positions of the cube are not well known1 it is best to have a list of the names before one1 but in every case the block should also be built1 as well as the names used47 ,the same shape in the position c d a is1 of course1 expressed in the same words1 but it has a different appearance4 ,the front face consists of the ,syces of ,saltus ,sypho ,remus ,moles ,plebs ,hama ,urna ,frenum ,sector ,and taking the shape we find we have ,urna1 and we know that ,ostrum lies behind ,urna1 and does not come in2 next we have ,moles1 ,saltus1 and we know that ,scena lies behind ,saltus and does not come in2 lastly1 we have ,sypho and ,remus1 and we know that ,aer and ,tyro are in the ,y direction from ,remus1 and so do not come in4 ,hence1 altogether the projection will consist only of the ,syces of ,urna1 ,moles1 ,saltus1 ,sypho1 and ,remus4 ,next1 taking the position d a c1 the cubes in the front face have their ,alvus sides against the plane1 and are3 ,sector ,ala ,mars ,frenum ,uncus ,spicula ,urna ,ostrum ,comes ,and1 taking the shape1 we find ,urna1 ,ostrum2 ,moles and ,saltus are hidden by ,urna1 ,scena is behind ,ostrum1 ,sypho gives ,frenum1 ,remus gives ,sector1 ,aer gives ,ala1 and ,tyro gives ,mars4 ,all these are ,alvus sides4 ,let us now take the reverse problem1 and1 given the three cyclical projections1 determine the shape4 ,let the a c d projection be the ,moenas of ,urna1 ,ostrum1 ,bidens1 ,scena1 ,vestis4 ,let the c d a be the ,syces of ,urna1 ,frenum1 ,plebs1 ,sypho1 and the d a c be the ,alvus of ,urna1 ,frenum1 ,uncus1 ,spicula4 ,now1 from a c d we have ,urna1 ,frenum1 ,sector1 ,ostrum1 ,uncus1 ,ala1 ,bidens1 ,pallor1 ,cortis1 ,scena1 ,tergum1 ,aer1 ,vestis1 ,oliva1 ,tyro4 ,from c d a we have ,urna1 ,ostrum1 ,comes1 ,frenum1 ,uncus1 ,spicula1 ,plebs1 ,pallor1 ,mora1 ,sypho1 ,tergum1 ,oliva4 ,in order to see how these will modify each other1 let us consider the a c d solution as if it were a set of cubes in the c d a arrangement4 ,here1 those that go in the ,arctos direction1 go away from the plane of projection1 and must be represented by the ,syce of the cube in contact with the plane4 ,looking at the a c d solution we write down 7keeping those together which go away from the plane of projection73 ,urna and ,ostrum1 ,frenum and ,uncus1 ,sector and ,ala1 ,bidens1 ,pallor1 ,cortis1 ,scena and ,vestis1 ,tergum and ,oliva1 ,aer and ,tyro4 ,here we see that the whole c d a face is filled up in the projection1 as far as this solution is concerned4 ,but in the c d a solution we have only ,syces of ,urna1 ,frenum1 ,plebs1 ,sypho4 ,these ,syces only indicate the presence of a certain number of the cubes stated above as possible from the ,moena projection1 and those are ,urna1 ,ostrum1 ,frenum1 ,uncus1 ,pallor1 ,tergum1 ,oliva4 ,this is the result of a comparison of the ,moena projection with the ,syce projection4 ,now1 writing these last named as they come in the d a c projection1 we have ,urna1 ,ostrum1 ,frenum1 ,uncus and ,pallor and ,tergum1 ,oliva4 ,and of these ,ostrum ,alvus is wanting in the d a c projection as given above4 ,hence ,ostrum will be wanting in the final shape1 and we have as the final solution3 ,urna1 ,frenum1 ,uncus1 ,pallor1 ,tergum1 ,oliva4 ,,chapter ,,xi4 ,a ,,tessaractic ,,figure ,,and ,,its ,,projections4 ,we will now consider a fourth-dimensional shape composed of tessaracts1 and the manner in which we can obtain a conception of it4 ,the operation is precisely analogous to that described in chapter ,,vi41 by which a plane being could obtain a conception of solid shapes4 ,it is only a little more difficult in that we have to deal with one dimension or direction more1 and can only do so symbolically4 ,we will assume the shape to consist of a certain number of the #ha tessaracts1 whose names we have given on p4 #afh4 ,let it consist of the thirteen tessaracts3 ,urna1 ,moles1 ,plebs1 ,frenum1 ,pallor1 ,tessera1 ,cudo1 ,vitta1 ,cura1 ,penates1 ,polus1 ,orcus1 ,lacerta4 ,firstly1 we will consider what appearances or projections these tessaracts will present to us according as the tessaractic set touches our space with its 7a7 ,mala cubes1 7b7 ,vesper cubes1 7c7 ,pluvium cubes1 or 7d7 ,lar cubes4 ,secondly1 we will treat the converse question1 how the shape can be determined when the projections in each of those views are given4 ,let us build up in cubes the four different arrangements of the tessaracts according as they enter our space on their ,mala1 ,vesper1 ,pluvium or ,lar sides4 ,they can only be printed by symbolizing two of the directions4 ,in the following tabulations the directions ,y1 ,x will at once be understood4 ,the direction ,z 7expressed by the wavy line7 indicates that the floors of nine1 each printed nearer the top of the page1 lie above those printed nearer the bottom of it4 ,the direction ,w is indicated by the dotted line1 which shows that the floors of nine lying to the left or right are in the ,w direction 7,ana7 or the -,w direction 7,kata7 from those which lie to the right or left4 ,for instance1 in the arrangement of the tessaracts1 as ,malas 7,table ,a7 the tessaract ,tessara1 which is exactly in the middle of the eighty-one tessaracts has ,domitor on its right side or in the ,x direction4 ,ocrea on its left -,x ,glans away from us ,y ,cudo nearer to us -,y ,sacerdos above it ,z ,cura below it -,z ,lacerta in the ,ana or ,w ,pallor in the ,kata or -,w ,similarly ,cervix lies in the ,ana or ,w direction from ,urna1 with ,thyrsus between them4 ,and to take one more instance1 a journey from ,saltus to ,arcus would be made by travelling ,y to ,remus1 thence -,x to ,sector1 thence ,z to ,mars1 and finally ,w to ,arcus4 ,a line from ,saltus to ,arcus is therefore a diagonal of the set of #ha tessaracts1 because the full length of its side has been traversed in each of the four directions to reach one from the other1 i4e4 ,saltus to ,remus1 ,remus to ,sector1 ,sector to ,mars1 ,mars to ,arcus4 ,,table ,a4 ,mala presentation of #ha ,tessaracts4 ,z ,w-------------------------------------- ----------------- -,w ,y ,y ,y ,block ,a ,block ,b ,block ,c ---------------,x ---------------,x ---------------,x ,y ,y ,y ,block ,d ,block ,e ,block ,f ---------------,x ---------------,x ---------------,x ,y ,y ,y ,block ,g ,block ,h ,block ,i ---------------,x ---------------,x ---------------,x -,z ,block ,a3 ,arcus ,ovis ,portio ,laurus ,tigris ,segmen ,axis ,troja ,aries ,block ,b3 ,ara ,vomer ,pluma ,praeda ,sacerdos ,hydra ,cortex ,mica ,flagellum ,block ,c3 ,mars ,merces ,tyro ,spicula ,mora ,oliva ,comes ,tibicen ,vestis ,block ,d3 ,postis ,clipeus ,tabula ,orcus ,lacerta ,testudo ,verbum ,luctus ,anguis ,block ,e3 ,pilum ,glans ,coins ,ocrea ,tessera ,domitor ,cardo ,cudo ,malleus ,block ,f3 ,ala ,cortis ,aer ,uncus ,pallor ,tergum ,ostrum ,bidens ,scena ,block ,g3 ,telum ,nepos ,angusta ,polus ,penates ,vulcan ,cervix ,securis ,vinculum ,block ,h3 ,agmen ,lacus ,arvus ,crates ,cura ,limen ,thyrsus ,vitta ,sceptrum ,block ,i3 ,sector ,hama ,remus ,frenum ,plebs ,sypho ,urna ,moles ,saltus ,,table ,b4 ,vesper presentation of #ha ,tessaracts4 ,z ,w-------------------------------------- ---------------- -,w ,y ,y ,y ,block ,a ,block ,b ,block ,c -,x--------------- -,x--------------- -,x--------------- ,y ,y ,y ,block ,d ,block ,e ,block ,f -,x--------------- -,x--------------- -,x--------------- ,y ,y ,y ,block ,g ,block ,h ,block ,i -,x--------------- -,x--------------- -,x--------------- -,z ,block ,a3 ,portio ,pluma ,tyro ,segmen ,hydra ,oliva ,aries ,flagellum ,vestis ,block ,b3 ,ovis ,vomer ,merces ,tigris ,sacerdos ,mora ,troja ,mica ,tibicen ,block ,c3 ,arcus ,ara ,mars ,laurus ,praeda ,spicula ,axis ,cortex ,comes ,block ,d3 ,tabula ,colus ,aer ,testudo ,domitor ,tergum ,anguis ,malleus ,scena ,block ,e3 ,clipeus ,glans ,cortis ,lacerta99 ,tessera99 ,pallor99 ,luctus99 ,cudo99 ,bidens99 ,block ,f3 ,postis ,pilum ,ala ,orcus99 ,ocrea99 ,uncus99 ,verbum ,cardo ,ostrum ,block ,g3 ,angusta ,arvus ,remus ,vulcan ,limen ,sypho ,vinculum ,sceptrum ,saltus ,block ,h3 ,nepos ,lacus ,hama ,penates99 ,cura99 ,plebs99 ,securis99 ,vitta99 ,moles99 ,block ,i3 ,telum ,agmen ,sector ,polus99 ,crates99 ,frenum99 ,cervix99 ,thyrsus99 ,urna99 ,,table ,c4 ,pluvium presentation of #ha ,tessaracts4 ,z ,w-------------------------------------- ----------------- -,w ----------------,x ----------------,x ---------------,x ,block ,a ,block ,b ,block ,c -,y -,y -,y ----------------,x ----------------,x ---------------,x ,block ,d ,block ,e ,block ,f -,y -,y -,y ----------------,x ----------------,x ---------------,x ,block ,g ,block ,h ,block ,i -,y -,y -,y -,z ,block ,a3 ,mars ,merces ,tyro ,ara ,vomer ,pluma ,arcus ,ovis ,portio ,block ,b3 ,spicula ,mora ,oliva ,praeda ,sacerdos ,hydra ,laurus ,tigris ,segmen ,block ,c3 ,comes ,tibicen ,vestis ,cortex ,mica ,flagellum ,axis ,troja ,aries ,block ,d3 ,ala ,cortis ,aer ,pilum ,glans ,colus ,postis ,clipeus ,tabula ,block ,e3 ,uncus99 ,pallor99 ,tergum ,ocrea99 ,tessera99 ,domitor ,orcus99 ,lacerta99 ,testudo ,block ,f3 ,ostrum ,bidens ,scena ,cardo ,cudo99 ,malleus ,verbum ,luctus ,anguis ,block ,g3 ,sector ,hama ,remus ,agmen ,lacus ,arvus ,telum ,nepos ,angusta ,block ,h3 ,frenum99 ,plebs99 ,sypho ,crates99 ,cura99 ,limen ,polus99 ,penates99 ,vulcan ,block ,i3 ,urna99 ,moles99 ,saltus ,thyrsus99 ,vitta99 ,sceptrum ,cervix ,securis ,vinculum ,,table ,d4 ,lar presentation of #ha ,tessaracts4 ,z ,w-------------------------------------- ----------------- -,w ,y ,y ,y ,block ,a ,block ,a ,block ,a ---------------,x ---------------,x ---------------,x ,y ,y ,y ,block ,a ,block ,a ,block ,a ---------------,x ---------------,x ---------------,x ,y ,y ,y ,block ,a ,block ,a ,block ,a ---------------,x ---------------,x ---------------,x -,z ,block ,a3 ,mars ,merces ,tyro ,spicula ,mora ,oliva ,comes ,tibicen ,vestis ,block ,b3 ,ala ,cortis ,aer ,uncus ,pallor99 ,tergum ,ostrum ,bidens ,scena ,block ,c3 ,sector ,hama ,remus ,frenum99 ,plebs99 ,sypho ,urna99 ,moles99 ,saltus ,block ,d3 ,ara ,vomer ,pluma ,proeda ,sacerdos ,hydra ,cortex ,mica ,flagellum ,block ,e3 ,pilum ,glans ,colus ,ocrea ,tessera99 ,domitor ,cardo ,cudo99 ,malleus ,block ,f3 ,agmen ,laurus ,arvus ,crates ,cura99 ,limen ,thyrsus ,vitta99 ,sceptrum ,block ,g3 ,arcus ,ovis ,portio ,laurus ,tigris ,segmen ,axis ,troja ,aries ,block ,h3 ,postis ,clipeus ,tabula ,orcus99 ,lacerta99 ,testudo ,verbum ,luctus ,anguis ,block ,i3 ,telum ,nepos ,angusta ,polus99 ,penates99 ,vulcan ,cervix ,securis ,vinculum ,the relation between the four different arrangements shown in the tables ,a1 ,b1 ,c1 and ,d1 will be understood from what has been said in chapter ,,viii4 about a small set of sixteen tessaracts4 ,a glance at the lines1 which indicate the directions in each1 will show the changes effected by turning the tessaracts from the ,mala presentation4 ,in the ,vesper presentation3 ,the tessaracts-- 7e4g4 ,urna1 ,ostrum1 ,comes71 which ran ,z still run ,z4 7e4g4 ,urna1 ,frenum1 ,sector71 ,y ,y4 7e4g4 ,urna1 ,moles1 ,saltus71 ,x now run ,w4 7e4g4 ,urna1 ,thyrsus1 ,cervix71 ,w -,x4 ,in the ,pluvium presentation3 ,the tessaracts-- 7e4g4 ,urna1 ,ostrum1 ,comes71 which ran ,z still run ,z4 7e4g4 ,urna1 ,moles1 ,saltus71 ,x ,x4 7e4g4 ,urna1 ,frenum1 ,sector71 ,y now run ,w4 7e4g4 ,urna1 ,thyrsus1 ,cervix71 ,w -,y4 ,in the ,lar presentation3 ,the tessaracts-- 7e4g4 ,urna1 ,moles1 ,saltus71 which ran ,x still run ,x4 7e4g4 ,urna1 ,frenum1 ,sector71 ,y ,y4 7e4g4 ,urna1 ,ostrum1 ,comes71 ,z now run ,w4 7e4g4 ,urna1 ,thyrsus1 ,cervix71 ,w -,z4 ,this relation was already treated in chapter ,,ix41 but it is well to have it very clear for our present purpose4 ,for it is the apparent change of the relative positions of the tessaracts in each presentation1 which enables us to determine any body of them4 ,in considering the projections1 we always suppose ourselves to be situated ,ana or ,w towards the tessaracts1 and any movement to be ,kata or -,w through our space4 ,for instance1 in the ,mala presentation we have first in our space the ,malas of that block of tessaracts1 which is the last in the -,w direction4 ,thus1 the ,mala projection of any given tessaract of the set is that ,mala in the extreme -,w block1 whose place its 7the given tessaract's7 ,mala would occupy1 if the tessaractic set moved ,kata until the given tessaract reached our space4 ,or1 in other words1 if all the tessaracts were transparent except those which constitute the body under consideration1 and if a light shone through ,four-space from the ,ana 7,w7 side to the ,kata 7-,w7 side1 there would be darkness in each of those ,malas1 which would be occupied by the ,mala of any opaque tessaract1 if the tessaractic set moved ,kata4 ,let us look at the set of #ha tessaracts we have built up in the ,mala arrangements1 and trace the projections in the extreme -,w block of the thirteen of our shape4 ,the latter are printed in italics in ,table ,a1 and their projections are marked 4 ,thus the cube ,uncus ,mala is the projection of the tessaract ,orcus1 ,pallor ,mala of ,pallor and ,tessera and ,tacerta1 ,bidens ,mala of ,cudo1 ,frenum ,mala of ,frenum and ,polus1 ,plebs ,mala of ,plebs and ,cura and ,penates1 ,moles ,mala of ,moles and ,vitta1 ,urna ,mala of ,urna4 ,similarly1 we can trace the ,vesper projections 7,table ,b74 ,orcus ,vesper is the projection of the tessaracts ,orcus and ,lacerta1 ,ocrea ,vesper of ,tessera1 ,uncus ,vesper of ,pallor1 ,cardo ,vesper of ,cudo1 ,polus ,vesper of ,polus and ,penates1 ,crates ,vesper of ,cura1 ,frenum ,vesper of ,frenum and ,plebs1 ,urna ,vesper of ,urna and ,moles1 ,thyrsus ,vesper of ,vitta4 ,next in the ,pluvium presentation 7,table ,c7 we find that ,bidens ,pluvium is the projection of the tessaract ,pallor1 ,cudo ,pluvium of ,cudo and ,tessera1 ,luctus ,pluvium of ,lacerta1 ,verbum ,pluvium of ,orcus1 ,urna ,pluvium of ,urna and ,frenum1 ,moles ,pluvium of ,moles and ,plebs1 ,vitta ,pluvium of ,vitta and ,cura1 ,securis ,pluvium of ,penates1 ,cervix ,pluvium of ,polus4 ,lastly1 in the ,lar presentation 7,table ,d7 we observe that ,frenum ,lar is the projection of ,frenum1 ,plebs ,lar of ,plebs and ,pallor1 ,moles ,lar of ,moles1 ,urna ,lar of ,urna1 ,cura ,lar of ,cura and ,tessara1 ,vitta ,lar of ,vitta and ,cudo1 ,penates ,lar of ,penates and ,lacerta1 ,polur ,lar of ,polus and ,orcus4 ,secondly1 we will treat the converse problem1 how to determine the shape when the projections in each presentation are given4 ,looking back at the list just given above1 let us write down in each presentation the projections only4 ,mala projections3 ,uncus1 ,pallor1 ,bidens1 ,frenum1 ,plebs1 ,moles1 ,urna4 ,vesper projections3 ,orcus1 ,ocrea1 ,uncus1 ,cardo1 ,polus1 ,crates1 ,frenum1 ,urna1 ,thyrsus4 ,pluvium projections3 ,bidens1 ,cudo1 ,luctus1 ,verbum1 ,urna1 ,moles1 ,vitta1 ,securis1 ,cervix4 ,lar projections3 ,frenum1 ,plebs1 ,moles1 ,urna1 ,cura1 ,vitta1 ,polus1 ,penates4 ,now let us determine the shape indicated by these projections4 ,in now using the same tables we must not notice the italics1 as the shape is supposed to be unknown4 ,it is assumed that the reader is building the problem in cubes4 ,from the ,mala projections we might infer the presence of all or any of the tessaracts written in the brackets in the following list of the ,mala presentation4 7,uncus1 ,ocrea1 ,orcus72 7,pallor1 ,tessera1 ,lacerta72 7,bidens1 ,cudo1 ,luctus72 7,frenum1 ,crates1 ,polus72 7,plebs1 ,cura1 ,penates72 7,moles1 ,vitta1 ,securis72 7,urna1 ,thyrsus1 ,cervix74 ,let us suppose them all to be present in our shape1 and observe what their appearance would be in the ,vesper presentation4 ,we mark them all with an asterisk in ,table ,b4 ,in addition to those already marked we must mark 77 ,verbum1 ,cardo1 ,ostrum1 and then we see all the ,vesper projections1 which would be formed by all the tessaracts possible from the ,mala projections4 ,let us compare these ,vesper projections1 viz4 ,orcus1 ,ocrea1 ,uncus1 ,verbum1 ,cardo1 ,ostrum1 ,polus1 ,crates1 ,frenum1 ,cervix1 ,thyrsus1 ,urna1 with the given ,vesper projections4 ,we see at once that ,verbum1 ,ostrum1 and ,cervix are absent4 ,therefore1 we may conclude that all the tessaracts1 which would be implied as possible by their presence1 are absent1 and of the ,mala possibilities may exclude the tessaracts ,bidens1 ,luctus1 ,securis1 and ,cervix itself4 ,thus1 of the #ba tessaracts possible in the ,mala view1 there remain only #ag possible1 both in the ,mala and ,vesper views1 viz4 ,uncus1 ,ocrea1 ,orcus1 ,pallor1 ,tessera1 ,lacerta1 ,cudo1 ,frenum1 ,crates1 ,polus1 ,plebs1 ,cura1 ,penates1 ,moles1 ,vitta1 ,urna1 ,thyrsus4 ,this we call the ,mala-,vesper solution4 ,next let us take the ,pluvium presentation4 ,we again mark with an asterisk in ,table ,c the possibilities inferred from the ,mala-,vesper solution1 and take the projections those possibilities would produce4 ,the additional projections are again marked 774 ,there are twelve ,pluvium projections altogether1 viz4 ,bidens1 ,ostrum1 ,cudo1 ,cardo1 ,luctus1 ,verbum1 ,urna1 ,moles1 ,vitta1 ,thyrsus1 ,securis1 ,cervix4 ,again we compare these with the given ,pluvium projections1 and find three are absent1 viz4 ,ostrum1 ,cardo1 ,thyrsus4 ,hence the tessaracts implied by ,ostrum and ,cardo and ,thyrsus cannot be in our shape1 viz4 ,uncus1 ,ocrea1 ,crates1 nor ,thyrsus itself4 ,excluding these four from the seventeen possibilities of the ,mala-,vesper solution we have left the thirteen tessaracts3 ,orcus1 ,pallor1 ,tessera1 ,lacerta1 ,cudo1 ,frenum1 ,polus1 ,plebs1 ,cura1 ,penates1 ,moles1 ,vitta1 ,urna4 ,this we call the ,mala-,vesper-,pluvium solution4 ,lastly1 we have to consider whether these thirteen tessaracts are consistent with the given ,lar projections4 ,we mark them again on ,table ,d with an asterisk1 and we find that the projections are exactly those given1 viz4 ,frenum1 ,plebs1 ,moles1 ,urna1 ,cura1 ,vitta1 ,polus1 ,penates4 ,therefore1 we have not to exclude any of the thirteen1 and can infer that they constitute the shape1 which produces the four different given views or projections4 ,in fine1 any shape in space consists of the possibilities common to the projections of its parts upon the boundaries of that space1 whatever be the number of its dimensions4 ,hence the simple rule for the determination of the shape would be to write down all the possibilities of the sets of projections1 and then cancel all those possibilities which are not common to all4 ,but the process adopted above is much preferable1 as through it we may realize the gradual delimitation of the shape view by view4 ,for once more we must remind ourselves that our great object is1 not to arrive at results by symbolical operations1 but to realize those results piece by piece through realized processes4 ,,appendices4 ,,appendix ,a4 ,this set of #ajj names is useful for studying ,plane ,space1 and forms a square #aj #aj4 ,ain ,bios ,hupar ,neas ,kairos ,enos ,thlipsis ,cheimas ,theion ,epei ,itea ,hagios ,phaino ,geras ,tholos ,ergon ,pachs ,kin ,eris ,cleos ,loma ,ets ,trochos ,klazo ,lutron ,hds ,ischs ,paigma ,hedna ,demas ,numphe ,bathus ,pauo ,euthu ,holos ,para ,thuos ,kar ,pyl ,spareis ,ania ,en ,seranx ,mesoi ,dramo ,thallos ,akt ,ozo ,onos ,magos ,notos ,mnis ,lampas ,ornis ,thama ,eni ,pholis ,mala ,strizo ,rudon ,labo ,helor ,rupa ,rabdos ,doru ,epos ,theos ,idris d ,hepo ,sophos ,ichor ,kanen ,ephthra ,oxis ,luk ,blue ,helos ,peri ,thelus ,eunis ,limos ,keedo ,igde ,mat ,lukos ,pteris ,holmos ,oulo ,dokos ,aeido ,ias ,assa ,muzo ,hippeus ,es ,at ,akme r ,gua ,,appendix ,b4 ,the following list of names is used to denote cubic spaces4 ,it makes a cubic block of six floors1 the highest being the sixth4 ,f ,fons ,plectrum ,vulnus ,arena ,mensa ,terminus ,s l ,testa ,plausus ,uva ,collis ,coma ,nebula i o ,copia ,cornu ,solum ,munus ,rixum ,vitrum x o ,ars ,fervor ,thyma ,colubra ,seges ,cor t r4 ,lupus ,classis ,modus ,flamma ,mens ,incola h ,thalamus ,hasta ,calamus ,crinis ,auriga ,vallum ,f ,linteum ,pinnis ,puppis ,nuptia ,aegis ,cithara ,f l ,triumphus ,curris ,lux ,portus ,latus ,funis i o ,regnum ,fascis ,bellum ,capellus ,arbor ,custos f o ,sagitta ,puer ,stella ,saxum ,humor ,pontus t r4 ,nomen ,imago ,lapsus ,quercus ,mundus ,proelium h ,palaestra ,nuncius ,bos ,pharetra ,pumex ,tibia ,f ,f ,lignum ,focus ,ornus ,lucrum ,alea ,vox o l ,caterva ,facies ,onus ,silva ,gelu ,flumen u o ,tellus ,sol ,os ,arma ,brachium ,jaculum r o ,merum ,signum ,umbra ,tempus ,corona ,socius t r4 ,moena ,opus ,honor ,campus ,rivus ,imber h ,victor ,equus ,miles ,cursus ,lyra ,tunica ,f ,haedus ,taberna ,turris ,nox ,domus ,vinum ,t l ,pruinus ,chorus ,luna ,flos ,lucus ,agna h o ,fulmen ,hiems ,ver ,carina ,arator ,pratum i o ,oculus ,ignis ,aether ,cohors ,penna ,labor r r4 ,aes ,pectus ,pelagus ,notus ,fretum ,gradus d ,princeps ,dux ,ventus ,navis ,finis ,robur ,s ,f ,vultus ,hostis ,figura ,ales ,coelum ,aura e l ,humerus ,augur ,ludus ,clamor ,galea ,pes c o ,civis ,ferrum ,pugna ,res ,carmen ,nubes o o ,litus ,unda ,rex ,templum ,ripa ,amnis n r4 ,pannus ,ulmus ,sedes ,columba ,aequor ,dama d ,dexter ,urbs ,gens ,monstrum ,pecus ,mons ,f ,nemus ,sidus ,vertex ,nix ,grando ,arx ,f l ,venator ,cerva ,aper ,plagua ,hedera ,frons i o ,membrum ,aqua ,caput ,castrum ,lituus ,tuba r o ,fluctus ,rus ,ratis ,amphora ,pars ,dies s r4 ,turba ,ager ,trabs ,myrtus ,fibra ,nauta t ,decus ,pulvis ,meta ,rota ,palma ,terra ,,appendix ,c4 ,the following names are used for a set of #bef ,tessaracts4 ,,fourth ,,block4 ,,third ,,block4 ,fourth ,floor4 ,fourth ,floor4 ,dolium ,caballus ,python ,circaea ,charta ,cures ,quaestor ,cliens ,cussis ,pulsus ,drachma ,cordax ,frux ,pyra ,lena ,procella ,porrum ,consul ,diota ,dyka ,hera ,esca ,secta ,rug ,columen ,ravis ,corbis ,rapina ,eurus ,gloria ,socer ,sequela ,third ,floor4 ,third ,floor4 ,alexis ,planta ,corymbus ,lectrum ,arche ,agger ,cumulus ,cassis ,aestus ,labellum ,calathus ,nux ,arcus ,ovis ,portio ,mimus ,septum ,sepes ,turtur ,ordo ,laurus ,tigris ,segmen ,obolus ,morsus ,aestas ,capella ,rheda ,axis ,troja ,aries ,fuga ,second ,floor4 ,second ,floor4 ,corydon ,jugum ,tornus ,labrum ,ruina ,culmen ,fenestra ,aedes ,lac ,hibiscus ,donum ,caltha ,postis ,clipeus ,tabula ,lingua ,senex ,palus ,salix ,cespes ,orcus ,lacerta ,testudo ,scala ,amictus ,gurges ,otium ,pomum ,verbum ,luctus ,anguis ,dolus ,first ,floor4 ,first ,floor4 ,odor ,aprum ,pignus ,messor ,additus ,salus ,clades ,rana ,color ,casa ,cera ,papaver ,telum ,nepos ,angusta ,mucro ,spes ,lapis ,apis ,afrus ,polus ,penates ,vulcan ,ira ,vitula ,clavis ,fagus ,cornix ,cervix ,securis ,vinculum ,furor ,,second ,,block4 ,,first ,,block4 ,fourth ,floor4 ,fourth ,floor4 ,actus ,spadix ,sicera ,anser ,horreum ,fumus ,hircus ,erisma ,auspex ,praetor ,atta ,sonus ,anulus ,pluor ,acies ,naxos ,fulgor ,ardea ,prex ,aevum ,etna ,gemma ,alpis ,arbiter ,spina ,birrus ,acerra ,ramus ,alauda ,furca ,gena ,alnus ,third ,floor4 ,third ,floor4 ,machina ,lex ,omen ,artus ,fax ,venenum ,syrma ,ursa ,ara ,vomer ,pluma ,odium ,mars ,merces ,tyro ,fama ,proeda ,sacerdos ,hydra ,luxus ,spicula ,mora ,oliva ,conjux ,cortex ,mica ,flagellum ,mas ,comes ,tibicen ,vestis ,plenum ,second ,floor4 ,second ,floor4 ,ardor ,rupes ,pallas ,arista ,rostrum ,armiger ,premium ,tribus ,pilum ,glans ,colus ,pellis ,ala ,cortis ,aer ,fragor ,ocrea ,tessara ,domitor ,fera ,uncus ,pallor ,tergum ,reus ,cardo ,cudo ,malleus ,thorax ,ostrum ,bidens ,scena ,torus ,first ,floor4 ,first ,floor4 ,regina ,canis ,marmor ,tectum ,pardus ,rubor ,nurus ,hospes ,agmen ,lacus ,arvus ,rumor ,sector ,hama ,remus ,fortuna ,crates ,cura ,limen ,vita ,frenum ,plebs ,sypho ,myrrha ,thyrsus ,vitta ,sceptrum ,pax ,urna ,moles ,saltus ,acus ,,appendix ,d4 ,the following list gives the colours1 and the various uses for them4 ,they have already been used in the foregoing pages to distinguish the various regions of the ,tessaract1 and the different individual cubes or ,tessaracts in a block4 ,the other use suggested in the last column of the list has not been discussed2 but it is believed that it may afford great aid to the mind in amassing1 handling1 and retaining the quantities of formulae requisite in scientific training and work4 ,region of ,tessaract ,colour4 ,tessaract4 in #ha ,set4 ,symbol4 ,black ,syce ,plebs #j ,white ,mel ,mora #a ,vermilion ,alvus ,uncus #b ,orange ,cuspis ,moles #c ,light-yellow ,murex ,cortis #d ,bright-green ,lappa ,penates #e ,bright-blue ,iter ,oliva #f ,light-grey ,lares ,tigris #g ,indian-red ,crux ,orcus #h ,yellow-ochre ,sal ,testudo #i ,buff ,cista ,sector 7plus7 ,wood ,tessaract ,tessara - 7minus7 ,brown-green ,tholus ,troja 7plus or minus7 ,sage-green ,margo ,lacerta 7multiplied by7 ,reddish ,callis ,tibicen 7divided by7 ,chocolate ,velum ,sacerdos 7equal to7 ,french-grey ,far ,scena 7not equal to7 ,brown ,arctos ,ostrum 7greater than7 ,dark-slate ,daps ,aer 7less than7 ,dun ,portica ,clipeus signs ,orange-vermilion ,talus ,portio of proportion ,stone ,ops ,thyrsus 7decimal point7 ,quaker-green ,felis ,axis 7factorial7 ,leaden ,semita ,merces 7parallel7 ,dull-green ,mappa ,vulcan 7not parallel7 ,indigo ,lixa ,postis #b 7#ij7 7at right angles7 ,dull-blue ,pagus ,verbum log4 base #aj ,dark-purple ,mensura ,nepos sin4 7sine7 ,pale-pink ,vena ,tabula cos4 7cosine7 ,dark-blue ,moena ,bidens tan4 7tangent7 ,earthen ,mugil ,angusta 7infinity7 ,blue ,dos ,frenum a ,terracotta ,crus ,remus b ,oak ,idus ,domitor c ,yellow ,pagina ,cardo d ,green ,bucina ,ala e ,rose ,olla ,limen f ,emerald ,orsa ,ara g ,red ,olus ,mars h ,sea-green ,libera ,pluma i ,salmon ,tela ,glans j ,pale-yellow ,livor ,ovis k ,purple-brown ,opex ,polus l ,deep-crimson ,camoena ,pilum m ,blue-green ,proes ,tergum n ,light-brown ,lua ,crates o ,deep-blue ,lama ,tyro p ,brick-red ,lar ,cura q ,magenta ,offex ,arvus r ,green-grey ,cadus ,hama s ,light-red ,croeta ,praeda t ,azure ,lotus ,vitta u ,pale-green ,vesper ,ocrea v ,blue-tint ,panax ,telum w ,yellow-green ,pactum ,malleus x ,deep-green ,mango ,vomer y ,light-green ,lis ,agmen z ,light-blue ,ilex ,comes ,crimson ,bolus ,sypho ,ochre ,limbus ,mica ,purple ,solia ,arcus ,leaf-green ,luca ,securis ,turquoise ,ancilla ,vinculum ,dark-grey ,orca ,colus ,fawn ,nug ,saltus ,smoke ,limus ,sceptrum ,light-buff ,mala ,pallor ,dull-purple ,sors ,vestis ,rich-red ,lucta ,cortex ,green-blue ,pator ,flagellum ,burnt-sienna ,silex ,luctus ,sea-blue ,lorica ,lacus ,peacock-blue ,passer ,aries ,deep-brown ,meatus ,hydra ,dark-pink ,onager ,anguis ,dark ,lensa ,laurus ,dark-stone ,pluvium ,cudo ,silver ,spira ,cervix ,gold ,corvus ,urna ,deep-yellow ,via ,spicula ,dark-green ,calor ,segmen ,,appendix ,e4 ,a ,,theorem ,,in ,,four-,,space4 ,if a pyramid on a triangular base be cut by a plane which passes through the three sides of the pyramid in such manner that the sides of the sectional triangle are not parallel to the corresponding sides of the triangle of the base2 then the sides of these two triangles1 if produced in pairs1 will meet in three points which are in a straight line1 namely1 the line of intersection of the sectional plane and the plane of the base4 ,let ,a ,b ,c ,d be a pyramid on a triangular base ,a ,b ,c1 and let a b c be a section such that ,a ,b1 ,b ,c1 ,a ,c1 are respectively not parallel to a b1 b c1 a c4 ,it must be understood that a is a point on ,a ,d1 b is a point on ,b ,d1 and c a point on ,c ,d4 ,let1 ,a ,b and a b1 produced1 meet in m4 ,b ,c and b c1 produced1 meet in n2 and ,a ,c and a c1 produced1 meet in o4 ,these three points1 m1 n1 o1 are in the line of intersection of the two planes ,a ,b ,c and a b c4 ,now1 let the line a b be projected on to the plane of the base1 by drawing lines from a and b at right angles to the base1 and meeting it in a b2 the line a b1 produced1 will meet ,a ,b produced in m4 ,if the lines b c and a c be projected in the same way on to the base1 to the points b c and a c2 then ,b ,c and b c produced1 will meet in n1 and ,a ,c and a c produced1 will meet in o4 ,the two triangles ,a ,b ,c and a b c are such1 that the lines joining ,a to a1 ,b to b1 and ,c to c1 will1 if produced1 meet in a point1 namely1 the point on the base ,a ,b ,c which is the projection of ,d4 ,any two triangles which fulfil this condition are the possible base and projection of the section of a pyramid2 therefore the sides of such triangles1 if produced in pairs1 will meet 7if they are not parallel7 in three points which lie in one straight line4 ,a four-dimensional pyramid may be defined as a figure bounded by a polyhedron of any number of sides1 and the same number of pyramids whose bases are the sides of the polyhedron1 and whose apices meet in a point not in the space of the base4 ,if a four-dimensional pyramid on a tetrahedral base be cut by a space which passes through the four sides of the pyramid in such a way that the sides of the sectional figure be not parallel to the sides of the base2 then the sides of these two tetrahedra1 if produced in pairs1 will meet in lines which all lie in one plane1 namely1 the plane of intersection of the space of the base and the space of the section4 ,if now the sectional tetrahedron be projected on to the base 7by drawing lines from each point of the section to the base at right angles to it71 there will be two tetrahedra fulfilling the condition that the line joining the angles of the one to the angles of the other will1 if produced1 meet in a point1 which point is the projection of the apex of the four-dimensional pyramid4 ,any two tetrahedra which fulfil this condition1 are the possible base and projection of a section of a four-dimensional pyramid4 ,therefore1 in any two such tetrahedra1 where the sides of the one are not parallel to the sides of the other1 the sides1 if produced in pairs 7one side of the one with one side of the other71 will meet in four straight lines which are all in one plane4 ,,appendix ,f4 ,,exercises ,,on ,,shapes ,,of ,,three ,,dimensions4 ,the names used are those given in ,appendix ,b4 ,find the shapes from the following projections3 #a4 ,syce projections3 ,ratis1 ,caput1 ,castrum1 ,plagua4 ,alvus projections3 ,merum1 ,oculus1 ,fulmen1 ,pruinus4 ,moena projections3 ,miles1 ,ventus1 ,navis4 #b4 ,syce3 ,dies1 ,tuba1 ,lituus1 ,frons4 ,alvus3 ,sagitta1 ,regnum1 ,tellus1 ,fulmen1 ,pruinus4 ,moena3 ,tibia1 ,tunica1 ,robur1 ,finis4 #c4 ,syce3 ,nemus1 ,sidus1 ,vertex1 ,nix1 ,cerva4 ,alvus3 ,lignum1 ,haedus1 ,vultus1 ,nemus1 ,humerus4 ,moena3 ,dexter1 ,princeps1 ,equus1 ,dux1 ,urbs1 ,pullis1 ,gens1 ,monstrum1 ,miles4 #d4 ,syce3 ,amphora1 ,castrum1 ,myrtus1 ,rota1 ,palma1 ,meta1 ,trabs1 ,ratis4 ,alvus3 ,dexter1 ,princeps1 ,moena1 ,aes1 ,merum1 ,oculus1 ,littus1 ,civis1 ,fulmen4 ,moena3 ,gens1 ,ventus1 ,navis1 ,finis1 ,monstrum1 ,cursus4 #e4 ,syce3 ,castrum1 ,plagua1 ,nix1 ,vertex1 ,aper1 ,caput1 ,cerva1 ,venator4 ,alvus3 ,triumphus1 ,tellus1 ,caterva1 ,lignum1 ,haedus1 ,pruinus1 ,fulmen1 ,civis1 ,humerus1 ,vultus4 ,moena3 ,pharetra1 ,cursus1 ,miles1 ,equus1 ,dux1 ,navis1 ,monstrum1 ,gens1 ,urbs1 ,dexter4 ,,answers4 ,the shapes are3 #a4 ,umbra1 ,aether1 ,ver1 ,carina1 ,flos4 #b4 ,pontus1 ,custos1 ,jaculum1 ,pratum1 ,arator1 ,agna4 #c4 ,focus1 ,omus1 ,haedus1 ,tabema1 ,vultus1 ,hostis1 ,figura1 ,ales1 ,sidus1 ,augur4 #d4 ,tempus1 ,campus1 ,finis1 ,navis1 ,ventus1 ,pelagus1 ,notus1 ,cohors1 ,aether1 ,carina1 ,res1 ,templum1 ,rex1 ,gens1 ,monstrum4 #e4 ,portus1 ,arma1 ,sylva1 ,lucrum1 ,ornus1 ,onus1 ,os1 ,facies1 ,chorus1 ,carina1 ,flos1 ,nox1 ,ales1 ,clamor1 ,res1 ,pugna1 ,ludus1 ,figura1 ,augur1 ,humerus4 ,,further ,,exercises ,,in ,,shapes ,,of ,,three ,,dimensions4 ,the ,names used are those given in ,appendix ,c2 and this set of exercises forms a preparation for their use in space of four dimensions4 ,all are in the #bg ,block 7,urna to ,syrma74 #a4 ,syce3 ,moles1 ,frenum1 ,plebs1 ,sypho4 ,alvus3 ,urna1 ,frenum1 ,uncus1 ,spicula1 ,comes4 ,moena3 ,moles1 ,bidens1 ,tibicen1 ,comes1 ,saltus4 #b4 ,syce3 ,urna1 ,moles1 ,plebs1 ,hama1 ,remus4 ,alvus3 ,urna1 ,frenum1 ,sector1 ,ala1 ,mars4 ,moena3 ,urna1 ,moles1 ,saltus1 ,bidens1 ,tibicen4 #c4 ,syce3 ,moles1 ,plebs1 ,hama1 ,remus4 ,alvus3 ,uma1 ,ostrum1 ,comes1 ,spicula1 ,frenum1 ,sector4 ,moena3 ,moles1 ,saltus1 ,bidens1 ,tibicen4 #d4 ,syce3 ,frenum1 ,plebs1 ,sypho1 ,moles1 ,hama4 ,alvus3 ,urna1 ,frenum1 ,uncus1 ,sector1 ,spicula4 ,moena3 ,urna1 ,moles1 ,saltus1 ,scena1 ,vestis4 #e4 ,syce3 ,urna1 ,moles1 ,plebs1 ,hama1 ,remus1 ,sector4 ,alvus3 ,urna1 ,frenum1 ,sector1 ,uncus1 ,spicula1 ,comes1 ,mars4 ,moena3 ,urna1 ,moles1 ,saltus1 ,bidens1 ,tibicen1 ,comes4 #f4 ,syce3 ,uma1 ,moles1 ,saltus1 ,sypho1 ,remus1 ,hama1 ,sector4 ,alvus3 ,comes1 ,ostrum1 ,uncus1 ,spicula1 ,mars1 ,ala1 ,sector4 ,moena3 ,urna1 ,moles1 ,saltus1 ,scena1 ,vestis1 ,tibicen1 ,comes1 ,ostrum4 #g4 ,syce3 ,sypho1 ,saltus1 ,moles1 ,urna1 ,frenum1 ,sector4 ,alvus3 ,urna1 ,frenum1 ,uncus1 ,spicula1 ,mars4 ,moena3 ,saltus1 ,moles1 ,urna1 ,ostrum1 ,comes4 #h4 ,syce3 ,moles1 ,plebs1 ,hama1 ,sector4 ,alvus3 ,ostrum1 ,frenum1 ,uncus1 ,spicula1 ,mars1 ,ala4 ,moena3 ,moles1 ,bidens1 ,tibicen1 ,ostrum4 #i4 ,syce3 ,moles1 ,saltus1 ,sypho1 ,plebs1 ,frenum1 ,sector4 ,alvus3 ,ostrum1 ,comes1 ,spicula1 ,mars1 ,ala4 ,moena3 ,ostrum1 ,comes1 ,tibicen1 ,bidens1 ,scena1 ,vestis4 #aj4 ,syce3 ,urna1 ,moles1 ,saltus1 ,sypho1 ,remus1 ,sector1 ,frenum4 ,alvus3 ,urna1 ,ostrum1 ,comes1 ,spicula1 ,mars1 ,ala1 ,sector4 ,moena3 ,urna1 ,ostrum1 ,comes1 ,tibicen1 ,vestis1 ,scena1 ,saltus4 #aa4 ,syce3 ,frenum1 ,plebs1 ,sypho1 ,hama4 ,alvus3 ,frenum1 ,sector1 ,ala1 ,mars1 ,spicula4 ,moena3 ,urna1 ,moles1 ,saltus1 ,bidens1 ,tibicen4 ,,answers4 ,the shapes are3 #a4 ,moles1 ,plebs1 ,sypho1 ,pallor1 ,mora1 ,tibicen1 ,spicula4 #b4 ,urna1 ,moles1 ,plebs1 ,hama1 ,cortis1 ,merces1 ,remus4 #c4 ,moles1 ,bidens1 ,tibicen1 ,mora1 ,plebs1 ,hama1 ,remus4 #d4 ,frenum1 ,plebs1 ,sypho1 ,tergum1 ,oliva1 ,moles1 ,hama4 #e4 ,urna1 ,moles1 ,plebs1 ,hama1 ,remus1 ,pallor1 ,mora1 ,tibicen1 ,mars1 ,merces1 ,comes1 ,sector4 #f4 ,ostrum1 ,comes1 ,tibicen1 ,vestis1 ,scena1 ,tergum1 ,oliva1 ,tyro1 ,aer1 ,remus1 ,hama1 ,sector1 ,merces1 ,mars1 ,ala4 #g4 ,sypho1 ,saltus1 ,moles1 ,urna1 ,frenum1 ,uncus1 ,spicula1 ,mars4 #h4 ,plebs1 ,pallor1 ,mora1 ,bidens1 ,merces1 ,cortis1 ,ala4 #i4 ,bidens1 ,tibicen1 ,vestis1 ,scena1 ,oliva1 ,mora1 ,spicula1 ,mars1 ,ala4 #aj4 ,urna1 ,ostrum1 ,comes1 ,spicula1 ,mars1 ,tibicen1 ,vestis1 ,oliva1 ,tyro1 ,aer1 ,remus1 ,sector1 ,ala1 ,saltus1 ,scena4 #aa4 ,frenum1 ,plebs1 ,sypho1 ,hama1 ,cortis1 ,merces1 ,mora4 ,,appendix ,g4 ,,exercises ,,on ,,shapes ,,of ,,four ,,dimensions4 ,the ,names used are those given in ,appendix ,c4 ,the first six exercises are in the #ha ,set1 and the rest in the #bef ,set4 #a4 ,mala projection3 ,urna1 ,moles1 ,plebs1 ,pallor1 ,cortis1 ,merces4 ,lar projection3 ,urna1 ,moles1 ,plebs1 ,cura1 ,penates1 ,nepos4 ,pluvium projection3 ,urna1 ,moles1 ,vitta1 ,cudo1 ,luctus1 ,troja4 ,vesper projection3 ,urna1 ,frenum1 ,crates1 ,ocrea1 ,orcus1 ,postis1 ,arcus4 #b4 ,mala3 ,urna1 ,frenum1 ,uncus1 ,pallor1 ,cortis1 ,aer4 ,lar3 ,urna1 ,frenum1 ,crates1 ,cura1 ,lacus1 ,arvus1 ,angusta4 ,pluvium3 ,urna1 ,thyrsus1 ,cardo1 ,cudo1 ,malleus1 ,anguis4 ,vesper3 ,urna1 ,frenum1 ,crates1 ,ocrea1 ,pilum1 ,postis4 #c4 ,mala3 ,comes1 ,tibicen1 ,mora1 ,pallor4 ,lar3 ,urna1 ,moles1 ,vitta1 ,cura1 ,penates4 ,pluvium3 ,comes1 ,tibicen1 ,mica1 ,troja1 ,luctus4 ,vesper3 ,comes1 ,cortex1 ,praeda1 ,laurus1 ,orcus4 #d4 ,mala3 ,vestis1 ,oliva1 ,tyro4 ,lar3 ,saltus1 ,sypho1 ,remus1 ,arvus1 ,angusta4 ,pluvium3 ,vestis1 ,flagellum1 ,aries4 ,vesper3 ,comes1 ,spicula1 ,mars1 ,ara1 ,arcus4 #e4 ,mala3 ,mars1 ,merces1 ,tyro1 ,aer1 ,tergum1 ,pallor1 ,plebs4 ,lar3 ,sector1 ,hama1 ,lacus1 ,nepos1 ,angusta1 ,vulcan1 ,penates4 ,pluvium3 ,comes1 ,tibicen1 ,mica1 ,troja1 ,aries1 ,anguis1 ,luctus1 ,securis4 ,vesper3 ,mars1 ,ara1 ,arcus1 ,postis1 ,orcus1 ,polus4 #f4 ,mala3 ,pallor1 ,mora1 ,oliva1 ,tyro1 ,merces1 ,mars1 ,spicula1 ,comes1 ,tibicen1 ,vestis4 ,lar3 ,plebs1 ,cura1 ,penates1 ,vulcan1 ,angusta1 ,nepos1 ,telum1 ,polus1 ,cervix1 ,securis1 ,vinculum4 ,pluvium3 ,bidens1 ,cudo1 ,luctus1 ,troja1 ,axis1 ,aries4 ,vesper3 ,uncus1 ,ocrea1 ,orcus1 ,laurus1 ,arcus1 ,axis4 #g4 ,mala3 ,hospes1 ,tribus1 ,fragor1 ,aer1 ,tyro1 ,mora1 ,oliva4 ,lar3 ,hospes1 ,tectum1 ,rumor1 ,arvus1 ,angusta1 ,cera1 ,apis1 ,lapis4 ,pluvium3 ,acus1 ,torus1 ,malleus1 ,flagellum1 ,thorax1 ,aries1 ,aestas1 ,capella4 ,vesper3 ,pardus1 ,rostrum1 ,ardor1 ,pilum1 ,ara1 ,arcus1 ,aestus1 ,septum4 #h4 ,mala3 ,pallor1 ,tergum1 ,aer1 ,tyro1 ,cortis1 ,syrma1 ,ursa1 ,fama1 ,naxos1 ,erisma4 ,lar3 ,plebs1 ,cura1 ,limen1 ,vulcan1 ,angusta1 ,nepos1 ,cera1 ,papaver1 ,pignus1 ,messor4 ,pluvium3 ,bidens1 ,cudo1 ,malleus1 ,anguis1 ,aries1 ,luctus1 ,capella1 ,rheda1 ,rapina4 ,vesper3 ,uncus1 ,ocrea1 ,orcus1 ,postis1 ,arcus1 ,aestus1 ,cussis1 ,dolium1 ,alexis4 #i4 ,mala3 ,fama1 ,conjux1 ,reus1 ,torus1 ,acus1 ,myrrha1 ,sypho1 ,plebs1 ,pallor1 ,mora1 ,oliva1 ,alpis1 ,acies1 ,hircus4 ,lar3 ,missale1 ,fortuna1 ,vita1 ,pax1 ,furor1 ,ira1 ,vulcan1 ,penates1 ,lapis1 ,apis1 ,cera1 ,pignus4 ,pluvium3 ,torus1 ,plenum1 ,pax1 ,thorax1 ,dolus1 ,furor1 ,vinculum1 ,securis1 ,clavis1 ,gurges1 ,aestas1 ,capella1 ,corbis4 ,vesper3 ,uncus1 ,spicula1 ,mars1 ,ocrea1 ,cardo1 ,thyrsus1 ,cervix1 ,verbum1 ,orcus1 ,polus1 ,spes1 ,senex1 ,septum1 ,porrum1 ,cussis1 ,dolium4 ,,answers4 ,the shapes are3 #a4 ,urna1 ,moles1 ,plebs1 ,cura1 ,tessara1 ,lacerta1 ,clipeus1 ,ovis4 #b4 ,urna1 ,frenum1 ,crates1 ,ocrea1 ,tessara1 ,glans1 ,colus1 ,tabula4 #c4 ,comes1 ,tibicen1 ,mica1 ,sacerdos1 ,tigris1 ,lacerta4 #d4 ,vestis1 ,oliva1 ,tyro1 ,pluma1 ,portio4 #e4 ,mars1 ,merces1 ,vomer1 ,ovis1 ,portio1 ,tabula1 ,testudo1 ,lacerta1 ,penates4 #f4 ,pallor1 ,tessara1 ,lacerta1 ,tigris1 ,segmen1 ,portio1 ,ovis1 ,arcus1 ,laurus1 ,axis1 ,troja1 ,aries4 #g4 ,hospes1 ,tribus1 ,arista1 ,pellis1 ,colus1 ,pluma1 ,portio1 ,calathus1 ,turtur1 ,sepes4 #h4 ,pallor1 ,tessara1 ,domitor1 ,testudo1 ,tabula1 ,clipeus1 ,portio1 ,calathus1 ,nux1 ,lectrum1 ,corymbus1 ,circaea1 ,cordax4 #i4 ,fama1 ,conjux1 ,reus1 ,fera1 ,thorax1 ,pax1 ,furor1 ,dolus1 ,scala1 ,ira1 ,vulcan1 ,penates1 ,lapis1 ,palus1 ,sepes1 ,turtur1 ,diota1 ,drachma1 ,python4 ,,appendix ,h4 ,,sections ,,of ,,cube ,,and ,,tessaract4 ,there are three kinds of sections of a cube4 #a4 ,the sectional plane1 which is in all cases supposed to be infinite1 can be taken parallel to two of the opposite faces of the cube2 that is1 parallel to two of the lines meeting in ,corvus1 and cutting the third4 #b4 ,the sectional plane can be taken parallel to one of the lines meeting in ,corvus and cutting the other two1 or one or both of them produced4 #c4 ,the sectional plane can be taken cutting all three lines1 or any or all of them produced4 ,take the first case1 and suppose the plane cuts ,dos half-way between ,corvus and ,cista4 ,since it does not cut ,arctos or ,cuspis1 or either of them produced1 it will cut ,via1 ,iter1 and ,bolus at the middle point of each2 and the figure1 determined by the intersection of the ,plane and ,mala1 is a square4 ,if the length of the edge of the cube be taken as the unit1 this figure may be expressed thus3 ,z ,x ,y #j 4 #j 4 showing that the ,z and ,x lines from ,corvus are not cut at all1 and that the ,y line is cut at half-a-unit from ,corvus4 ,sections taken ,z ,x ,y #j 4 #j 4 and ,z ,x ,y #j 4 #j 4 #a would also be squares4 ,take the second case4 ,let the plane cut ,cuspis and ,dos1 each at half-a-unit from ,corvus1 and not cut ,arctos or ,arctos produced2 it will also cut through the middle points of ,via and ,callis4 ,the figure produced1 is a rectangle which has two sides of one unit1 and the other two are each the diagonal of a half-unit squared4 ,if the plane cuts ,cuspis and ,dos1 each at one unit from ,corvus1 and is parallel to ,arctos1 the figure will be a rectangle which has two sides of one unit in length2 and the other two the diagonal of one unit squared4 ,if the plane passes through ,mala1 cutting ,dos produced and ,cuspis produced1 each at one-and-a-half unit from ,corvus1 and is parallel to ,arctos1 the figure will be a parallelogram like the one obtained by the section ,z ,x ,y #j 4 4 4 ,this set of figures will be expressed ,z ,x ,y ,z ,x ,y ,z ,x ,y #j 4 4 #j 4 #a 4 #a #j 4 #a 4 #a ,it will be seen that these sections are parallel to each other2 and that in each figure ,cuspis and ,dos are cut at equal distances from ,corvus4 ,we may express the whole set thus3-- ,z ,x ,y ,o 4 ,i 4 ,i it being understood that where ,roman figures are used1 the numbers do not refer to the length of unit cut off any given line from ,corvus1 but to the proportion between the lengths4 ,thus ,z ,x ,y ,o 4 ,i 4 ,,ii means that ,arctos is not cut at all1 and that ,cuspis and ,dos are cut1 ,dos being cut twice as far from ,corvus as is ,cuspis4 ,these figures will also be rectangles4 ,take the third case4 ,suppose ,arctos1 ,cuspis1 and ,dos are each cut half-way4 ,this figure is an equilateral triangle1 whose sides are the diagonal of a half-unit squared4 ,the figure ,z ,x ,y #a 4 #a 4 #a is also an equilateral triangle1 and the figure ,z ,x ,y #a 4 #a 4 #a is an equilateral hexagon4 ,it is easy for us to see what these shapes are1 and also1 what the figures of any other set would be1 as ,z ,x ,y ,i 4 ,,ii 4 ,,ii or ,z ,x ,y ,i 4 ,,ii 4 ,,iii but we must learn them as a two-dimensional being would1 so that we may see how to learn the three-dimensional sections of a tessaract4 ,it is evident that the resulting figures are the same whether we fix the cube1 and then turn the sectional plane to the required position1 or whether we fix the sectional plane1 and then turn the cube4 ,thus1 in the first case we might have fixed the plane1 and then so placed the cube that one plane side coincided with the sectional plane1 and then have drawn the cube half-way through1 in a direction at right angles to the plane1 when we should have seen the square first mentioned4 ,in the second case 7,z ,x ,y7 7,o 4 ,i 4 ,i7 we might have put the cube with ,arctos coinciding with the plane and with ,cuspis and ,dos equally inclined to it1 and then have drawn the cube through the plane at right angles to it until the lines 7,cuspis and ,dos7 were cut at the required distances from ,corvus4 ,in the third case we might have put the cube with only ,corvus coinciding with the plane and with ,cuspis1 ,dos1 and ,arctos equally inclined to it 7for any of the shapes in the set ,z ,x ,y7 ,i 4 ,i 4 ,i7 and then have drawn it through as before4 ,the resulting figures are exactly the same as those we got before2 but this way is the best to use1 as it would probably be easier for a two-dimensional being to think of a cube passing through his space than to imagine his whole space turned round1 with regard to the cube4 ,we have already seen 7p4 #aag7 how a two-dimensional being would observe the sections of a cube when it is put with one plane side coinciding with his space1 and is then drawn partly through4 ,now1 suppose the cube put with the line ,arctos coinciding with his space1 and the lines ,cuspis and ,dos equally inclined to it4 ,at first he would only see ,arctos4 ,if the cube were moved until ,dos and ,cuspis were each cut half-way1 ,arctos still being parallel to the plane1 ,arctos would disappear at once2 and to find out what he would see he would have to take the square sections of the cube1 and find on each of them what lines are given by the new set of sections4 ,thus he would take ,moena itself1 which may be regarded as the first section of the square set4 ,one point of the figure would be the middle point of ,cuspis1 and since the sectional plane is parallel to ,arctos1 the line of intersection of ,moena with the sectional plane will be parallel to ,arctos4 ,the required line then cuts ,cuspis half-way1 and is parallel to ,arctos1 therefore it cuts ,callis half-way4 ,illustration3 ,fig4 #ba4 ,next he would take the square section half-way between ,moena and ,murex4 ,he knows that the line ,alvus of this section is parallel to ,arctos1 and that the point ,dos at one of its ends is half-way between ,corvus and ,cista1 so that this line itself is the one he wants 7because the sectional plane cuts ,dos half-way between ,corvus and ,cista1 and is parallel to ,arctos74 ,in ,fig4 #ba the two lines thus found are shown4 a b is the line in ,moena1 and c d the line in the section4 ,he must now find out how far apart they are4 ,he knows that from the middle point of ,cuspis to ,corvus is half-a-unit1 and from the middle point of ,dos to ,corvus is half-a-unit1 and ,cuspis and ,dos are at right angles to each other2 therefore from the middle point of ,cuspis to the middle point of ,dos is the diagonal of a square whose sides are half-a-unit in length4 ,this diagonal may be written d 774 ,he would also see that from the middle point of ,callis to the middle point of ,via is the same length2 therefore the figure is a parallelogram1 having two of its sides1 each one unit in length1 and the other two each d 774 ,he could also see that the angles are right1 because the lines a c and b d are made up of the ,x and ,y directions1 and the other two1 a b and d1 are purely ,z1 and since they have no tendency in common1 they are at right angles to each other4 ,illustration3 ,fig4 #bb4 ,if he wanted the figure made by ,z ,x ,y #j 4 #a 4 #a it would be a little more difficult4 ,he would have to take ,moena1 a section halfway between ,moena and ,murex1 ,murex and another square which he would have to regard as an imaginary section half-a-unit further ,y than ,murex 7,fig4 #bb74 ,he might now draw a ground plan of the sections2 that is1 he would draw ,syce1 and produce ,cuspis and ,dos half-a-unit beyond ,nug and ,cista4 ,he would see that ,cadus and ,bolus would be cut half-way1 so that in the half-way section he would have the point a 7,fig4 #bc71 and in ,murex the point c4 ,in the imaginary section he would have g2 but this he might disregard1 since the cube goes no further than ,murex4 ,from the points c and a there would be lines going ,z1 so that ,iter and ,semita would be cut half-way4 ,illustration3 ,groundplan of ,sections shown in ,fig4 #bb4 ,fig4 #bc4 ,he could find out how far the two lines a b and c d 7,fig4 #bb7 are apart by referring d and b to ,lama1 and a and c to ,crus4 ,in taking the third order of sections1 a similar method may be followed4 ,illustration3 ,fig4 #bd4 ,suppose the sectional plane to cut ,cuspis1 ,dos1 and ,arctos1 each at one unit from ,corvus4 ,he would first take ,moena1 and as the sectional plane passes through ,ilex and ,nug1 the line on ,moena would be the diagonal passing through these two points4 ,then he would take ,murex1 and he would see that as the plane cuts ,dos at one unit from ,corvus1 all he would have is the point ,cista4 ,so the whole figure is the ,ilex to ,nug diagonal1 and the point ,cista4 ,now ,cista and ,ilex are each one inch from ,corvus1 and measured along lines at right angles to each other2 therefore1 they are d 7#a7 from each other4 ,by referring ,nug and ,cista to ,corvus he would find that they are also d 7#a7 apart2 therefore the figure is an equilateral triangle1 whose sides are each d 7#a74 ,suppose the sectional plane to pass through ,mala1 cutting ,cuspis1 ,dos1 and ,arctos each at unit from ,corvus4 ,to find the figure1 the plane-being would have to take ,moena1 a section half-way between ,moena and ,murex1 ,murex1 and an imaginary section half-a-unit beyond ,murex 7,fig4 #bd74 ,he would produce ,arctos and ,cuspis to points half-a-unit from ,ilex and ,nug1 and by joining these points1 he would see that the line passes through the middle points of ,callis and ,far 7a1 b1 ,fig4 #bd74 ,in the last square1 the imaginary section1 there would be the point m2 for this is #a unit from ,corvus measured along ,dos produced4 ,there would also be lines in the other two squares1 the section and ,murex1 and to find these he would have to make many observations4 ,he found the points a and b 7,fig4 #bd7 by drawing a line from r to s1 r and s being each #a unit from ,corvus1 and simply seeing that it cut ,callis and ,far at the middle point of each4 ,he might now imagine a cube ,mala turned about ,arctos1 so that ,alvus came into his plane2 he might then produce ,arctos and ,dos until they were each unit long1 and join their extremities1 when he would see that ,via and ,bucina are each cut half-way4 ,again1 by turning ,syce into his plane1 and producing ,dos and ,cuspis to points #a unit from ,corvus and joining the points1 he would see that ,bolus and ,cadus are cut half-way4 ,he has now determined six points on ,mala1 through which the plane passes1 and by referring them in pairs to ,ilex1 ,olus1 ,cista1 ,crus1 ,nug1 ,sors1 he would find that each was d 77 from the next2 so he would know that the figure is an equilateral hexagon4 ,the angles he would not have got in this observation1 and they might be a serious difficulty to him4 ,it should be observed that a similar difficulty does not come to us in our observation of the sections of a tessaract3 for1 if the angles of each side of a solid figure are determined1 the solid angles are also determined4 ,there is another1 and in some respects a better1 way by which he might have found the sides of this figure4 ,if he had noticed his plane-space much1 he would have found out that1 if a line be drawn to cut two other lines which meet1 the ratio of the parts of the two lines cut off by the first line1 on the side of the angle1 is the same for those lines1 and any other two that are parallel to them4 ,thus1 if a b and a c 7,fig4 #be7 meet1 making an angle at a1 and b c crosses them1 and also crosses a b and a c1 these last two being parallel to a b and a c1 then a b a c a b a c4 ,illustration3 ,fig4 #be ,if the plane-being knew this1 he would rightly assume that if three lines meet1 making a solid angle1 and a plane passes through them1 the ratio of the parts between the plane and the angle is the same for those three lines1 and for any other three parallel to them4 ,in the case we are dealing with he knows that from ,ilex to the point on ,arctos produced where the plane cuts1 it is half-a-unit2 and as the ,z1 ,x1 and ,y lines are cut equally from ,corvus1 he would conclude that the ,x and ,y lines are cut the same distance from ,ilex as the ,z line1 that is half-a-unit4 ,he knows that the ,x line is cut at #a units from ,corvus2 that is1 half-a-unit from ,nug3 so he would conclude that the ,z and ,y lines are cut half-a-unit from ,nug4 ,he would also see that the ,z and ,x lines from ,cista are cut at half-a-unit4 ,he has now six points on the cube1 the middle points of ,callis1 ,via1 ,bucina1 ,cadus1 ,bolus1 and ,far4 ,now1 looking at his square sections1 he would see on ,moena a line going from middle of ,far to middle of ,callis1 that is1 a line d 77 long4 ,on the section he would see a line from middle of ,via to middle of ,bolus d 7#a7 long1 and on ,murex he would see a line from middle of ,cadus to middle of ,bucina1 d 77 long4 ,of these three lines a b1 c d1 e f1 7,fig4 #bd7--;a b and e f are sides1 and c d is a section of the required figure4 ,he can find the distances between a and c by reference to ,ilex1 between b and d by reference to ,nug1 between c and e by reference to ,olus1 and between d and f by reference to ,crus2 and he will find that these distances are each d 774 ,thus1 he would know that the figure is an equilateral hexagon with its sides d 77 long1 of which two of the opposite points 7c and d7 are d 7#a7 apart1 and the only figure fulfilling all these conditions is an equilateral and equiangular hexagon4 ,enough has been said about sections of a cube1 to show how a plane-being would find the shapes in any set as in ,z ,x ,y ,i 4 ,,ii 4 ,,ii or ,z ,x ,y ,i 4 ,i 4 ,,ii4 ,he would always have to bear in mind that the ratio of the lengths of the ,z1 ,x1 and ,y lines is the same from ,corvus to the sectional plane as from any other point to the sectional plane4 ,thus1 if he were taking a section where the plane cuts ,arctos and ,cuspis at one unit from ,corvus and ,dos at one-and-a-half1 that is where the ratio of ,z and of ,x to ,y is as two to three1 he would see that ,dos itself is not cut at all2 but from ,cista to the point on ,dos produced is half-a-unit2 therefore from ,cista1 the ,z and ,x lines will be cut at of unit from ,cista4 ,it is impossible in writing to show how to make the various sections of a tessaract2 and even if it were not so1 it would be unadvisable2 for the value of doing it is not in seeing the shapes themselves1 so much as in the concentration of the mind on the tessaract involved in the process of finding them out4 ,any one who wishes to make them should go carefully over the sections of a cube1 not looking at them as he himself can see them1 or determining them as he1 with his three-dimensional conceptions1 can2 but he must limit his imagination to two dimensions1 and work through the problems which a plane-being would have to work through1 although to his higher mind they may be self-evident4 ,thus a three-dimensional being can see at a glance1 that if a sectional plane passes through a cube at one unit each way from ,corvus1 the resulting figure is an equilateral triangle4 ,if he wished to prove it1 he would show that the three bounding lines are the diagonals of equal squares4 ,this is all a two-dimensional being would have to do2 but it is not so evident to him that two of the lines are the diagonals of squares4 ,moreover1 when the figure is drawn1 we can look at it from a point outside the plane of the figure1 and can thus see it all at once2 but he who has to look at it from a point in the plane can only see an edge at a time1 or he might see two edges in perspective together4 ,then there are certain suppositions he has to make4 ,for instance1 he knows that two points determine a line1 and he assumes that three points determine a plane1 although he cannot conceive any other plane than the one in which he exists4 ,we assume that four points determine a solid space4 ,or rather1 we say that if this supposition1 together with certain others of a like nature1 are true1 we can find all the sections of a tessaract1 and of other four-dimensional figures by an infinite solid4 ,when any difficulty arises in taking the sections of a tessaract1 the surest way of overcoming it is to suppose a similar difficulty occurring to a two-dimensional being in taking the sections of a cube1 and1 step by step1 to follow the solution he might obtain1 and then to apply the same or similar principles to the case in point4 ,a few figures are given1 which1 if cut out and folded along the lines1 will show some of the sections of a tessaract4 ,but the reader is earnestly begged not to be content with looking at the shapes only4 ,that will teach him nothing about a tessaract1 or four-dimensional space1 and will only tend to produce in his mind a feeling that 8the fourth dimension0 is an unknown and unthinkable region1 in which any shapes may be right1 as given sections of its figures1 and of which any statement may be true4 ,while1 in fact1 if it is the case that the laws of spaces of two and three dimensions may1 with truth1 be carried on into space of four dimensions2 then the little our solidity 7like the flatness of a plane-being7 will allow us to learn of these shapes and relations1 is no more a matter of doubt to us than what we learn of two- and three-dimensional shapes and relations4 ,there are given also sections of an octa-tessaract1 and of a tetra-tessaract1 the equivalents in four-space of an octahedron and tetrahedron4 ,a tetrahedron may be regarded as a cube with every alternate corner cut off4 ,thus1 if ,mala have the corner towards ,corvus cut off as far as the points ,ilex1 ,nug1 ,cista1 and the corner towards ,sors cut off as far as ,ilex1 ,nug1 ,lama1 and the corner towards ,crus cut off as far as ,lama1 ,nug1 ,cista1 and the corner towards ,olus cut off as far as ,ilex1 ,lama1 ,cista1 what is left of the cube is a tetrahedron1 whose angles are at the points ,ilex1 ,nug1 ,cista1 ,lama4 ,in a similar manner1 if every alternate corner of a tessaract be cut off1 the figure that is left is a tetra-tessaract1 which is a figure bounded by sixteen regular tetrahedrons4 ,illustration3 7i7 ,fig4 #bf4 ,fig4 #bg4 ,fig4 #bg4 ,fig4 #bf4 ,illustration3 7ii7 ,fig4 #bh4 ,fig4 #bi4 ,fig4 #cj4 ,illustration3 7iii7 ,fig4 #ca4 ,fig4 #cb4 ,illustration3 7iv7 ,fig4 #cc4 ,fig4 #cd4 ,fig4 #ce4 ,illustration3 7v7 ,fig4 #cf4 ,fig4 #cg4 ,fig4 #ch4 ,illustration3 7vi7 ,fig4 #ci4 ,fig4 #dj4 ,fig4 #da4 ,the octa-tessaract is got by cutting off every corner of the tessaract4 ,if every corner of a cube is cut off1 the figure left is an octa-hedron1 whose angles are at the middle points of the sides4 ,the angles of the octa-tessaract are at the middle points of its plane sides4 ,a careful study of a tetra-hedron and an octa-hedron as they are cut out of a cube will be the best preparation for the study of these four-dimensional figures4 ,it will be seen that there is much to learn of them1 as--,how many planes and lines there are in each1 ,how many solid sides there are round a point in each4 ,a ,,description ,,of ,,figures #bf ,,to #da4 ,z ,x ,y ,w ,z ,x ,y ,w #bf is a section taken #a 4 #a 4 #a 4 #a ,i 4 ,i 4 ,i 4 ,i #bg #a 4 #a 4 #a 4 #a #bh #b 4 #b 4 #b 4 #b ,z ,x ,y ,w ,z ,x ,y ,w #bi is a section taken #a 4 #a 4 #a 4 ,,ii 4 ,,ii 4 ,,ii 4 ,i #cj #a 4 #a 4 #a 4 #ca #b 4 #b 4 #b 4 #a #cb #b 4 #b 4 #b 4 #a ,the above are sections of a tessaract4 ,figures #cc to #ce are of a tetra-tessaract4 ,the tetra-tessaract is supposed to be imbedded in a tessaract1 and the sections are taken through it1 cutting the ,z1 ,x and ,y lines equally1 and corresponding to the figures given of the sections of the tessaract4 ,figures #cf1 #cg1 and #ch are similar sections of an octa-tessaract4 ,figures #ci1 #dj1 and #da are the following sections of a tessaract4 ,z ,x ,y ,w ,z ,x ,y ,w #ci is a section taken #j 4 4 4 ,o 4 ,i 4 ,i 4 ,i #dj #j 4 #a 4 #a 4 #a #da #j 4 #a 4 #a 4 #a ,it is clear that there are four orders of sections of every four-dimensional figure2 namely1 those beginning with a solid1 those beginning with a plane1 those beginning with a line1 and those beginning with a point4 ,there should be little difficulty in finding them1 if the sections of a cube with a tetra-hedron1 or an octa-hedron enclosed in it1 are carefully examined4 ,,appendix ,k4 ,illustration3 ,,model #a4 ,,mala4 ,,colours3 ,,mala1 ,,light-,,buff4 ,points3 ,corvus1 ,gold4 ,nug1 ,fawn4 ,crus1 ,terra-cotta4 ,cista1 ,buff4 ,ilex1 ,light-blue4 ,sors1 ,dull-purple4 ,lama1 ,deep-blue4 ,olus1 ,red4 ,lines3 ,cuspis1 ,orange4 ,bolus1 ,crimson4 ,cadus1 ,green-grey4 ,dos1 ,blue4 ,arctos1 ,brown4 ,far1 ,french-grey4 ,daps1 ,dark-slate4 ,bucina1 ,green4 ,callis1 ,reddish4 ,iter1 ,bright-blue4 ,semita1 ,leaden4 ,via1 ,deep-yellow4 ,surfaces3 ,moena1 ,dark-blue4 ,proes1 ,blue-green4 ,murex1 ,light-yellow4 ,alvus1 ,vermilion4 ,mel1 ,white4 ,syce1 ,black4 ,illustration3 ,,model #b4 ,,margo4 ,,colours3 ,,margo1 ,,sage-,,green4 ,points3 ,spira1 ,silver4 ,ancilla1 ,turquoise4 ,mugil1 ,earthen4 ,panax1 ,blue-tint4 ,felis1 ,quaker-green4 ,passer1 ,peacock-blue4 ,talus1 ,orange-vermilion4 ,solia1 ,purple4 ,lines3 ,luca1 ,leaf-green4 ,mappa1 ,dull-green4 ,mensura1 ,dark-purple4 ,opex1 ,purple-brown4 ,pagus1 ,dull-blue4 ,onager1 ,dark-pink4 ,vena1 ,pale-pink4 ,lixa1 ,indigo4 ,tholus1 ,brown-green4 ,calor1 ,dark-green4 ,livor1 ,pale-yellow4 ,lensa1 ,dark4 ,surfaces3 ,silex1 ,burnt-sienna4 ,sal1 ,yellow-ochre4 ,portica1 ,dun4 ,crux1 ,indian-red4 ,lares1 ,light-grey4 ,lappa1 ,bright-green4 ,illustration3 ,,model #c4 ,,lar4 ,,colours3 ,,lar1 ,,brick-,,red4 ,points3 ,spira1 ,silver4 ,ancilla1 ,turquoise4 ,mugil1 ,earthen4 ,panax1 ,blue-tint4 ,corvus1 ,gold4 ,nug1 ,fawn4 ,crus1 ,terra-cotta4 ,cista1 ,buff4 ,lines3 ,luca1 ,leaf-green4 ,mappa1 ,dull-green4 ,mensura1 ,dark-purple4 ,opex1 ,purple-brown4 ,ops1 ,stone4 ,limus1 ,smoke4 ,offex1 ,magenta4 ,lis1 ,light-green4 ,cuspis1 ,orange4 ,bolus1 ,crimson4 ,cadus1 ,green-grey4 ,dos1 ,blue4 ,surfaces3 ,lotus1 ,azure4 ,olla1 ,rose4 ,lorica1 ,sea-blue4 ,lua1 ,bright-brown4 ,syce1 ,black4 ,lappa1 ,bright-green4 ,illustration3 ,,model #d4 ,,velum4 ,,colours3 ,,velum1 ,,chocolate4 ,points3 ,felis1 ,quaker-green4 ,passer1 ,peacock-blue4 ,talus1 ,orange-vermilion4 ,solia1 ,purple4 ,ilex1 ,light-blue4 ,sors1 ,dull-purple4 ,lama1 ,deep-blue4 ,olus1 ,red4 ,lines3 ,tholus1 ,brown-green4 ,calor1 ,dark-green4 ,livor1 ,pale-yellow4 ,lensa1 ,dark4 ,lucta1 ,rich-red4 ,pator1 ,green-blue4 ,libera1 ,sea-green4 ,orsa1 ,emerald4 ,callis1 ,reddish4 ,iter1 ,bright-blue4 ,semita1 ,leaden4 ,via1 ,deep-yellow4 ,surfaces3 ,limbus1 ,ochre4 ,meatus1 ,deep-brown4 ,mango1 ,deep-green4 ,croeta1 ,light-red4 ,mel1 ,white4 ,lares1 ,light-grey4 ,illustration3 ,,model #e4 ,,vesper4 ,,colours3 ,,vesper1 ,,pale-,,green4 ,points3 ,spira1 ,silver4 ,corvus1 ,gold4 ,cista1 ,buff4 ,panax1 ,blue-tint4 ,felis1 ,quaker-green4 ,ilex1 ,light-blue4 ,olus1 ,red4 ,solia1 ,purple4 ,lines3 ,ops1 ,stone4 ,dos1 ,blue4 ,lis1 ,light-green4 ,opex1 ,purple-brown4 ,pagus1 ,dull-blue4 ,arctos1 ,brown4 ,bucina1 ,green4 ,lixa1 ,indigo4 ,lucta1 ,rich-red4 ,via1 ,deep-yellow4 ,orsa1 ,emerald4 ,lensa1 ,dark4 ,surfaces3 ,pagina1 ,yellow4 ,alvus1 ,vermilion4 ,camoena1 ,deep-crimson4 ,crux1 ,indian-red4 ,croeta1 ,light-red4 ,lua1 ,light-brown4 ,illustration3 ,,model #f4 ,,idus4 ,,colours3 ,,idus1 ,,oak4 ,points3 ,ancilla1 ,turquoise4 ,nug1 ,fawn4 ,crus1 ,terra-cotta4 ,mugil1 ,earthen4 ,passer1 ,peacock-blue4 ,sors1 ,dull-purple4 ,lama1 ,deep-blue4 ,talus1 ,orange-vermilion4 ,lines3 ,limus1 ,smoke4 ,bolus1 ,crimson4 ,offex1 ,magenta4 ,mappa1 ,dull-green4 ,onager1 ,dark-pink4 ,far1 ,french-grey4 ,daps1 ,dark-slate4 ,vena1 ,pale-pink4 ,pator1 ,green-blue4 ,iter1 ,bright-blue4 ,libera1 ,sea-green4 ,calor1 ,dark-green4 ,surfaces3 ,pactum1 ,yellow-green4 ,proes1 ,blue-green4 ,orca1 ,dark-grey4 ,sal1 ,yellow-ochre4 ,meatus1 ,deep-brown4 ,olla1 ,rose4 ,illustration3 ,,model #g4 ,,pluvium4 ,,colours3 ,,pluvium1 ,,dark-,,stone4 ,points3 ,spira1 ,silver4 ,ancilla1 ,turquoise4 ,nug1 ,fawn4 ,corvus1 ,gold4 ,felis1 ,quaker-green4 ,passer1 ,peacock-blue4 ,sors1 ,dull-purple4 ,ilex1 ,light-blue4 ,lines3 ,luca1 ,leaf-green4 ,limus1 ,smoke4 ,cuspis1 ,orange4 ,ops1 ,stone4 ,pagus1 ,dull-blue4 ,onager1 ,dark-pink4 ,far1 ,french-grey4 ,arctos1 ,brown4 ,tholos1 ,brown-green4 ,pator1 ,green-blue4 ,callis1 ,reddish4 ,lucta1 ,rich-red4 ,surfaces3 ,silex1 ,burnt-,sienna4 ,pactum1 ,yellow-green4 ,moena1 ,dark-blue4 ,pagina1 ,yellow4 ,limbus1 ,ochre4 ,lotus1 ,azure4 ,illustration3 ,,model #h4 ,,tela4 ,,colours3 ,,tela1 ,,salmon4 ,points3 ,panax1 ,blue-tint4 ,mugil1 ,earthen4 ,crus1 ,terra-cotta4 ,cista1 ,buff4 ,solia1 ,purple4 ,talus1 ,orange-vermilion4 ,lama1 ,deep-blue4 ,olus1 ,red4 ,lines3 ,mensura1 ,dark-purple4 ,offex1 ,magenta4 ,cadus1 ,green-grey4 ,lis1 ,light-green4 ,lixa1 ,indigo4 ,vena1 ,pale-pink4 ,daps1 ,dark-slate4 ,bucina1 ,green4 ,livor1 ,pale-yellow4 ,libera1 ,sea-green4 ,semita1 ,leaden4 ,orsa1 ,emerald4 ,surfaces3 ,portica1 ,dun4 ,orca1 ,dark-grey4 ,murex1 ,light-yellow4 ,camoena1 ,deep-crimson4 ,mango1 ,deep-green4 ,lorica1 ,sea-blue4 ,illustration3 ,,model #i4 ,,section ,,between ,,mala ,,and ,,margo4 ,,colours3 ,,interior ,,or ,,tessaract1 ,,wood4 ,points 7,lines73 ,ops1 ,stone4 ,limus1 ,smoke4 ,offex1 ,magenta4 ,lis1 ,light-green4 ,lucta1 ,rich-red4 ,pator1 ,green-blue4 ,libera1 ,sea-green4 ,orsa1 ,emerald4 ,lines 7,surfaces73 ,lotus1 ,azure4 ,olla1 ,rose4 ,lorica1 ,sea-blue4 ,lua ,bright-brown4 ,pagina1 ,yellow4 ,pactum1 ,yellow-green4 ,orca1 ,dark-grey4 ,camoena1 ,deep-crimson4 ,limbus1 ,ochre4 ,meatus1 ,deep-brown4 ,mango1 ,deep-green4 ,croeta1 ,light red4 ,surfaces 7,solids73 ,pluvium1 ,dark-stone4 ,idus1 ,oak4 ,tela1 ,salmon4 ,vesper1 ,pale-green4 ,velum1 ,chocolate4 ,lar1 ,brick-red4 ,illustration3 ,,model #aj4 ,,section ,,between ,,lar ,,and ,,velum4 ,,colours3 ,,interior ,,or ,,tessaract1 ,,wood4 ,points 7,lines73 ,pagus1 ,dull-blue4 ,onager1 ,dark-pink4 ,vena1 ,pale-pink4 ,lixa1 ,indigo4 ,arctos1 ,brown4 ,far1 ,french-grey4 ,daps1 ,dark-slate4 ,bucina1 ,green4 ,lines 7,surfaces73 ,silex1 ,burnt-sienna4 ,sal1 ,yellow-ochre4 ,portica1 ,dun4 ,crux1 ,indian-red4 ,pagina1 ,yellow4 ,pactum1 ,yellow-green4 ,orca1 ,dark-grey4 ,camoena1 ,deep-crimson4 ,moena1 ,dark-blue4 ,proes1 ,blue-green4 ,murex1 ,light-yellow4 ,alvus1 ,vermilion4 ,surfaces 7,solids73 ,pluvium1 ,dark-stone4 ,idus1 ,oak4 ,tela1 ,salmon4 ,vesper1 ,pale-green4 ,mala1 ,light-buff4 ,margo1 ,sage-green4 ,illustration3 ,,model #aa4 ,,section ,,between ,,vesper ,,and ,,idus4 ,,colours3 ,,interior ,,or ,,tessaract1 ,,wood4 ,points 7,lines73 ,luca1 ,leaf-green4 ,cuspis1 ,orange4 ,cadus1 ,green-grey4 ,mensura1 ,dark-purple4 ,tholus1 ,brown-green4 ,callis1 ,reddish4 ,semita1 ,leaden4 ,livor1 ,pale-yellow4 ,lines 7,surfaces73 ,lotus1 ,azure4 ,syce1 ,black4 ,lorica1 ,sea-blue4 ,lappa1 ,bright-green4 ,silex1 ,burnt-sienna4 ,moena1 ,dark-blue4 ,murex1 ,light-yellow4 ,portica1 ,dun4 ,limbus1 ,ochre4 ,mel1 ,white4 ,mango1 ,deep-green4 ,lares1 ,light-grey4 ,surfaces 7,solids73 ,pluvium1 ,dark-stone4 ,mala1 ,light-buff4 ,tela1 ,salmon4 ,margo1 ,sage-green4 ,velum1 ,chocolate4 ,lar1 ,brick-red4 ,illustration3 ,,model #ab4 ,,section ,,between ,,pluvium ,,and ,,tela4 ,,colours3 ,,interior ,,or ,,tessaract1 ,,wood4 ,points 7,lines73 ,opex1 ,purple-brown4 ,mappa1 ,dull-green4 ,bolus1 ,crimson4 ,dos1 ,blue4 ,lensa1 ,dark4 ,calor1 ,dark-green4 ,iter1 ,bright-blue4 ,via1 ,deep-yellow4 ,lines 7,surfaces73 ,lappa1 ,bright-green4 ,olla1 ,rose4 ,syce1 ,black4 ,lua1 ,bright-brown4 ,crux1 ,indian-red4 ,sal1 ,yellow-ochre4 ,proes1 ,blue-green4 ,alvus1 ,vermilion4 ,lares1 ,light-grey4 ,meatus1 ,deep-brown4 ,mel1 ,white4 ,croeta1 ,light-red4 ,surfaces 7,solids73 ,margo1 ,sage-green4 ,idus1 ,oak4 ,mala1 ,light-buff4 ,vesper1 ,pale-green4 ,velum1 ,chocolate4 ,lar1 ,brick-red4 ,transcriber's ,notes ,lay-out and formatting have been optimised for browser html 7available at www4gutenberg4org72 some versions and narrow windows may not display all elements of the book as intended1 depending on the hard- and software used and their settings4 ,for this text file1 several diagrams and tables have been split or otherwise re-arranged in order to fit the available width4 ,wherever possible1 these splits and re-arrangements have been done so that the meaning of the lay-out has been retained4 ,the contents of ,tables ,a through ,d have been replaced with place holders2 their meanings are listed below the diagrams4 ,inconsistencies in spelling 7,mnas v4 ,moenas2 ,praeda v4 ,proeda71 hyphenation 7,deep-blue v4 ,deep blue1 etc47 have been retained4 ,page #aig1 row starting ,sophos3 the last letter of ,blue has been assumed4 ,changes made3 ,footnotes1 tables1 diagrams and illustrations have been moved outside text paragraphs4 ,indications for the location of illustrations 7,to face p4 4447 have been removed2 the illustrations concerned have been moved to where they are discussed4 ,some minor obvious typographical errors have been corrected silently4 ,page #db3 444 the flat1 being 444 changed to 444 the flat being 444 ,page #abg3 ,cube itself3 considered to be the table header rather than a table element ,page #age3 is all ,ana our space changed to is all ,ana in our space ,page #ahg3 ,clipens changed to ,clipeus2 legend ,y added to right-hand side grid axes ,page #bai3 ,part ,,ii4 ,appendix ,k4 changed to ,appendix ,k4 cf4 other ,appendices4