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If you are not located in the United States, you'll -have to check the laws of the country where you are located before using -this ebook. - - - -Title: A New Era of Thought - -Author: Charles Howard Hinton - -Release Date: November 1, 2019 [EBook #60607] - -Language: English - -Character set encoding: UTF-8 - -*** START OF THIS PROJECT GUTENBERG EBOOK A NEW ERA OF THOUGHT *** - - - - -Produced by Chris Curnow, Harry Lame and the Online -Distributed Proofreading Team at http://www.pgdp.net (This -file was produced from images generously made available -by The Internet Archive) - - - - - - - - Transcriber’s Notes - - Text printed in italics is represented between _underscores_, bold - face text between ~tildes~. Small capitals have been transcribed as - ALL CAPITALS. A letter in square brackets preceded by an equal sign, - as [=X], represents a letter with bar above. - - More Transcriber’s Notes may be found at the end of this text. - - - - -A NEW ERA OF THOUGHT. - - - - - SCIENTIFIC ROMANCES. - - By C. HOWARD HINTON, M.A. - - Crown 8vo, cloth gilt, 6_s._; or separately, 1_s._ each. - - 1. ~What is the Fourth Dimension?~ 1_s._ - - GHOSTS EXPLAINED. - - “A short treatise of admirable clearness. . . . Mr. Hinton brings - us, panting but delighted, to at least a momentary faith in the Fourth - Dimension, and upon the eye of this faith there opens a vista of - interesting problems. . . . His pamphlet exhibits a boldness of - speculation, and a power of conceiving and expressing even the - inconceivable, which rouses one’s faculties like a tonic.”--_Pall - Mall._ - - 2. ~The Persian King; or, The Law of the Valley~, 1_s._ - - THE MYSTERY OF PLEASURE AND PAIN. - - “A very suggestive and well-written speculation, by the inheritor of - an honoured name.”--_Mind._ - - “Will arrest the attention of the reader at once.”--_Knowledge._ - - 3. ~A Plane World~, 1_s._ - - 4. ~A Picture of our Universe~, 1_s._ - - 5. ~Casting out the Self~, 1_s._ - - - _SECOND SERIES._ - - 1. ~On the Education of the Imagination.~ - - 2. ~Many Dimensions~, 1_s._ - - LONDON: SWAN SONNENSCHEIN & CO. - - - - - _A New Era of Thought._ - - BY - - CHARLES HOWARD HINTON, M.A., OXON. - _Author of “What is the Fourth Dimension,” and other “Scientific - Romances.”_ - - [Illustration] - - London: - SWAN SONNENSCHEIN & CO., - PATERNOSTER SQUARE. - 1888. - - - BUTLER & TANNER, - THE SELWOOD PRINTING WORKS, - FROME, AND LONDON. - - - - -PREFACE. - - -The MSS. which formed the basis of this book were committed to us by the -author, on his leaving England for a distant foreign appointment. It was -his wish that we should construct upon them a much more complete -treatise than we have effected, and with that intention he asked us to -make any changes or additions we thought desirable. But long alliance -with him in this work has convinced us that his thought (especially that -of a general philosophical character) loses much of its force if -subjected to any extraneous touch. - -This feeling has induced us to print Part I. almost exactly as it came -from his hands, although it would probably have received much -rearrangement if he could have watched it through the press himself. - -Part II. has been written from a hurried sketch, which he considered -very inadequate, and which we have consequently corrected and -supplemented. Chapter XI. of this part has been entirely re-written by -us, and has thus not had the advantage of his supervision. This remark -also applies to Appendix E, which is an elaboration of a theorem he -suggested. Appendix H, and all the exercises have, in accordance with -his wish, been written solely by us. It will be apparent to the reader -that Appendix H is little more than a brief introduction to a very large -subject, which, being concerned with tessaracts and solids, is really -beyond treatment in writing and diagrams. - -This difficulty recalls us to the one great fact, upon which we feel -bound to insist, that the matter of this book _must_ receive objective -treatment from the reader, 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 consciousness. -Indeed, we consider that printing, as a method of spreading -space-knowledge, is but a “pis aller,” and we would go back to that -ancient and more fruitful method of the Greek geometers, and, while -describing figures on the sand, or piling up pebbles in series, would -communicate to others that spirit of learning and generalization -begotten in our consciousness by continuous contact with facts, and only -by continuous contact with facts vitally maintained. - - ALICIA BOOLE, - - H. JOHN FALK. - -N.B. Models.--It is unquestionably a most important part of the process -of learning space to construct these, and the reader should do so, -however roughly and hastily. But, if Models are required as patterns, -they may be ordered from Messrs. Swan Sonnenschein & Co., Paternoster -Square, London, and will be supplied as soon as possible, the -uncertainty as to demand for same not allowing us to have a large number -made in advance. Much of the work can be done with plain cubes by using -names without colours, but further on the reader will find colours -necessary to enable him to grasp and retain the complex series of -observations. Coloured models can easily be made by covering -Kindergarten cubes with white paper and painting them with water-colour, -and, if permanence be desired, dipping them in size and copal varnish. - - - - -TABLE OF CONTENTS. - - - PART I. - PAGE - INTRODUCTION 1-7 - - CHAPTER I. - Scepticism and Science. Beginning of Knowledge 8-13 - - CHAPTER II. - Apprehension of Nature. Intelligence. Study of Arrangement or - Shape 14-20 - - CHAPTER III. - The Elements of Knowledge 21-23 - - CHAPTER IV. - Theory and Practice 24-28 - - CHAPTER V. - Knowledge: Self-Elements 29-34 - - CHAPTER VI. - Function of Mind. Space against Metaphysics. Self-Limitation and - its Test. A Plane World 35-46 - - CHAPTER VII. - Self Elements in our Consciousness 47-50 - - CHAPTER VIII. - Relation of Lower to Higher Space. Theory of the Æther 51-60 - - CHAPTER IX. - Another View of the Æther. Material and Ætherial Bodies 61-66 - - CHAPTER X. - Higher Space and Higher Being. Perception and Inspiration 67-84 - - CHAPTER XI. - Space the Scientific Basis of Altruism and Religion 85-99 - - - PART II. - - CHAPTER I. - Three-space. Genesis of a Cube. Appearances of a Cube to a - Plane-being 101-112 - - CHAPTER II. - Further Appearances of a Cube to a Plane-being 113-117 - - CHAPTER III. - Four-space. Genesis of a Tessaract; its Representation in - Three-space 118-129 - - CHAPTER IV. - Tessaract moving through Three-space. Models of the Sections 130-134 - - CHAPTER V. - Representation of Three-space by Names and in a Plane 135-148 - - CHAPTER VI. - The Means by which a Plane-being would Acquire a Conception of - our Figures 149-155 - - CHAPTER VII. - Four-space: its Representation in Three-space 156-166 - - CHAPTER VIII. - Representation of Four-space by Name. Study of Tessaracts 167-176 - - CHAPTER IX. - Further Study of Tessaracts 177-179 - - CHAPTER X. - Cyclical Projections 180-183 - - CHAPTER XI. - A Tessaractic Figure and its Projections 184-194 - - - APPENDICES. - - A. 100 Names used for Plane Space 197 - - B. 216 Names used for Cubic Space 198 - - C. 256 Names used for Tessaractic Space 200-201 - - D. List of Colours, Names, and Symbols 202-203 - - E. A Theorem in Four-space 204-205 - - F. Exercises on Shapes of Three Dimensions 205-207 - - G. Exercises on Shapes of Four Dimensions 207-209 - - H. Sections of the Tessaract 209-217 - - K. Drawings of the Cubic Sides and Sections of the Tessaract - (Models 1-12) with Colours and Names 219-241 - - - - -INTRODUCTORY NOTE TO PART I. - - -At the completion of a work, or at the completion of the first part of a -work, the feelings are necessarily very different from those with which -the work was begun; and the meaning and value of the work itself bear a -very different appearance. It will therefore be the simplest and -shortest plan, if I tell the reader briefly what the work is to which -these pages are a guide, and what I consider to be its value when done. - -The task was to obtain a sense of the properties of higher space, or -space of four dimensions, in the same way as that by which we reach a -sense of our ordinary three-dimensional space. I now prefer to call the -task that of obtaining a familiarity with higher matter, which shall be -as intuitive to the mind as that of ordinary matter has become. The -expression “higher matter” is preferable to “higher space,” because it -is a somewhat hasty proceeding to split this concrete matter, which we -touch and feel, into the abstractions of extension and impenetrability. -It seems to me that I cannot think of space without matter, and -therefore, as no necessity compels me to such a course, I do not split -up the concrete object into subtleties, but I simply ask: “What is that -which is to a cube or block or shape of any kind as the cube is to a -square?” - -In entering upon this inquiry we find the task is twofold. Firstly, -there is the theoretical part, which is easy, viz. to set clearly before -us the relative conditions which would obtain if there were a matter -physically higher than this matter of ours, and to choose the best -means of liberating our minds from the limitations imposed on it by the -particular conditions under which we are placed. The second part of the -task is somewhat laborious, and consists of a constant presentation to -the senses of those appearances which portions of higher matter would -present, and of a continual dwelling on them, until the higher matter -becomes familiar. - -The reader must undertake this task, if he accepts it at all, as an -experiment. Those of us who have done it, are satisfied that there is -that in the results of the experiment which make it well worthy of a -trial. - -And in a few words I may state the general bearings of this work, for -every branch of work has its general bearings. It is an attempt, in the -most elementary and simple domain, to pass from the lower to the higher. -In pursuing it the mind passes from one kind of intuition to a higher -one, and with that transition the horizon of thought is altered. It -becomes clear that there is a physical existence transcending the -ordinary physical existence; and one becomes inclined to think that the -right direction to look is, not away from matter to spiritual -existences, but towards the discovery of conceptions of higher matter, -and thereby of those material existences whose definite relations to us -are apprehended as spiritual intuitions. Thus, “material” would simply -mean “grasped by the intellect, become known and familiar.” Our -apprehension of anything which is not expressed in terms of matter, is -vague and indefinite. To realize and live with that which we vaguely -discern, we need to apply the intuition of higher matter to the world -around us. And this seems to me the great inducement to this study. Let -us form our intuition of higher space, and then look out upon the world. - -Secondly, in this progress from ordinary to higher matter, as a general -type of progress from lower to higher, we make the following -observations. Firstly, we become aware that there are certain -limitations affecting our regard. Secondly, we discover by our reason -what those limitations are, and then force ourselves to go through the -experience which would be ours if the limitations did not affect us. -Thirdly, we become aware of a capacity within us for transcending those -limitations, and for living in the higher mode as we had lived in the -previous one. - -We may remark that this progress from the ordinary to the higher kind of -matter demands an absolute attention to details. It is only in the -retention of details that such progress becomes possible. And as, in -this question of matter, an absolute and unconventional examination -gives us the indication of a higher, so, doubtless, in other questions, -if we but come to facts without presupposition, we begin to know that -there is a higher and to discover indications of the way whereby we can -approach. That way lies in the fulness of detail rather than in the -generalization. - -Biology has shown us that there is a universal order of forms or -organisms, passing from lower to higher. Therein we find an indication -that we ourselves take part in this progress. And in using the little -cubes we can go through the process ourselves, and learn what it is in a -little instance. - -But of all the ways in which the confidence gained from this lesson can -be applied, the nearest to us lies in the suggestion it gives,--and more -than the suggestion, if inclination to think be counted for -anything,--in the suggestion of that which is higher than ourselves. We, -as individuals, are not the limit and end-all, but there is a higher -being than ours. What our relation to it is, we cannot tell, for that is -unlike our relation to anything we know. But, perhaps all that happens -to us is, could we but grasp it, our relation to it. - -At any rate, the discovery of it is the great object beside which all -else is as secondary as the routine of mere existence is to -companionship. And the method of discovery is full knowledge of each -other. Thereby is the higher being to be known. In as much as the least -of us knows and is known by another, in so much does he know the higher. -Thus, scientific prayer is when two or three meet together, and, in the -belief of one higher than themselves, mutually comprehend that vision of -the higher, which each one is, and, by absolute fulness of knowledge of -the facts of each other’s personality, strive to attain a knowledge of -that which is to each of their personalities as a higher figure is to -its solid sides. - - C. H. H. - - - - -A NEW ERA OF THOUGHT. - - - - -PART I. - - -INTRODUCTION. - -There are no new truths in this book, but it consists of an effort to -impress upon and bring home to the mind some of the more modern -developments of thought. A few sentences of Kant, a few leading ideas of -Gauss and Lobatschewski form the material out of which it is built up. - -It may be thought to be unduly long; but it must be remembered that in -these times there is a twofold process going on--one of discovery about -external nature, one of education, by which our minds are brought into -harmony with that which we know. In certain respects we find ourselves -brought on by the general current of ideas--we feel that matter is -permanent and cannot be annihilated, and it is almost an axiom in our -minds that energy is persistent, and all its transformations remains the -same in amount. But there are other directions in which there is need of -definite training if we are to enter into the thoughts of the time. - -And it seems to me that a return to Kant, the creator of modern -philosophy, is the first condition. Now of Kant’s enormous work only a -small part is treated here, but with the difference that should be found -between the work of a master and that of a follower. Kant’s statements -are taken as leading ideas, suggesting a field of work, and it is in -detail and manipulation merely that there is an opportunity for -workmanship. - -Of Kant’s work it is only his doctrine of space which is here -experimented upon. With Kant the perception of things as being in space -is not treated as it seems so obvious to do. We should naturally say -that there is space, and there are things in it. From a comparison of -those properties which are common to all things we obtain the properties -of space. But Kant says that this property of being in space is not so -much a quality of any definable objects, as the means by which we obtain -an apprehension of definable objects--it is the condition of our mental -work. - -Now as Kant’s doctrine is usually commented on, the negative side is -brought into prominence, the positive side is neglected. It is generally -said that the mind cannot perceive things in themselves, but can only -apprehend them subject to space conditions. And in this way the space -conditions are as it were considered somewhat in the light of -hindrances, whereby we are prevented from seeing what the objects in -themselves truly are. 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 -experience. - -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 nature. But there is no need to adopt this feeling. The -first postulate of this book is a full recognition of the fact, that it -is by means of space that we apprehend what is. Space is the instrument -of the mind. - -And here for the purposes of our work we can avoid all metaphysical -discussion. Very often a statement which seems to be very deep and -abstruse and hard to grasp, is simply the form into which deep thinkers -have thrown a very simple and practical observation. And for the present -let us look on Kant’s great doctrine of space from a practical point of -view, and it comes to this--it is important to develop the space sense, -for it is the means by which we think about real things. - -There is a doctrine which found much favour with the first followers of -Kant, that also affords us a simple and practical rule of work. It was -considered by Fichte that the whole external world was simply a -projection from the _ego_, and the manifold of nature was a recognition -by the spirit of itself. What this comes to as a practical rule is, that -we can only understand nature in virtue of our own activity; that there -is no such thing as mere passive observation, but every act of sight and -thought is an activity of our own. - -Now according to Kant the space sense, or the intuition of space, is the -most fundamental power of the mind. But I do not find anywhere a -systematic and thoroughgoing education of the space sense. In every -practical pursuit it is needed--in some it is developed. In geometry it -is used; but the great reason of failure in education is that, instead -of a systematic training of the space sense, it is left to be organized -by accident and is called upon to act without having been formed. -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 -developed. - -With regard to the education of the space sense, I must ask the -indulgence of the reader. It will seem obvious to him that any real -pursuit or real observation trains the space sense, and that it is going -out of the way to undertake any special discipline. - -To this I would answer that, according to my own experience, I was -perfectly ignorant of space relations myself before I actually worked at -the subject, and that directly I got a true view of space facts a whole -series of conceptions, which before I had known merely by repute and -grasped by an effort, became perfectly simple and clear to me. - -Moreover, to take one instance: in studying the relations of space we -always have to do with coloured objects, we always have the sense of -weight; for if the things themselves have no weight, there is always a -direction of up and down--which implies the sense of weight, and to get -rid of these elements requires careful sifting. 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 it, and the phraseology will serve him for the -considerations which come next. - -Amongst the followers of Kant, those who pursued one of the lines of -thought in his works have attracted the most attention and have been -considered as his successors. Fichte, Schelling, Hegel have developed -certain tendencies and have written remarkable books. But the true -successors of Kant are Gauss and Lobatchewski. - -For if our intuition of space is the means by which we apprehend, then -it follows that there may be different kinds of intuitions of space. Who -can tell what the absolute space intuition is? This intuition of space -must be coloured, so to speak, by the conditions of the being which uses -it. - -Now, after Kant had laid down his doctrine of space, 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 mind. - -The only way to investigate this is the practical way, and by a -remarkable analysis the great geometers above mentioned have shown that -space is not limited as ordinary experience would seem to inform us, but -that we are quite capable of conceiving different kinds of space. - -Our space as we ordinarily think of it is conceived as limited--not in -extent, but in a certain way which can only be realized when we think of -our ways of measuring space objects. It is found that there are only -three independent directions in which a body can be measured--it must -have height, length and breadth, but it has no more than these -dimensions. If any other measurement be taken in it, this new -measurement will be found to be compounded of the old measurements. 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 taken. - -But why should space be limited to three independent directions? - -Geometers have found that there is no reason why bodies should be thus -limited. As a matter of fact all the bodies which we can measure are -thus limited. So we come to this conclusion, that the space which we use -for conceiving ordinary objects in the world is limited to three -dimensions. But it might be possible for there to be beings living in a -world such that they would conceive a space of four dimensions. All that -we can say about such a supposition is, that it is not demanded by our -experience. 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 world. - -And this was the point at which about ten years ago I took up the -inquiry. - -It is possible to say a great deal about space of higher dimensions than -our own, and to work out analytically many problems which suggest -themselves. But can we conceive four-dimensional space in the same way -in which we can conceive our own space? 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 familiar? - -Now this question, as every other with which I am acquainted, can only -be answered by experiment. And I commenced a series of experiments to -arrive at a conclusion one way or the other. - -It is obvious that this is not a scientific inquiry--but one for the -practical teacher. - -And just as in experimental researches the skilful manipulator will -demonstrate a law of nature, the less skilled manipulator will fail; so -here, everything depended on the manipulation. I was not sure that this -power lay hidden in the mind, but to put the question fairly would -surely demand every resource of the practical art of education. - -And so it proved to be; for after many years of work, during which the -conception of four-dimensional bodies lay absolutely dark, at length, by -a certain change of plan, the whole subject of four-dimensional -existence became perfectly clear and easy to impart. - -There is really no more difficulty in conceiving four-dimensional -shapes, when we go about it the right way, than in conceiving the idea -of solid shapes, nor is there any mystery at all about it. - -When the faculty is acquired--or rather when it is brought into -consciousness, for it exists in every one in imperfect form--a new -horizon opens. The mind acquires a development of power, and in this use -of ampler space as a mode of thought, a path is opened by using that -very truth which, when first stated by Kant, seemed to close the mind -within such fast limits. Our perception is subject to the condition of -being in space. But space is not limited as we at first think. - -The next step after having formed this power of conception in ampler -space, is to investigate nature and see what phenomena are to be -explained by four-dimensional relations. - -But this part of the subject is hardly one for the same worker as the -one who investigates how to think in four-dimensional space. The work of -building up the power is the work of the practical educator, the work of -applying it to nature is the work of the scientific man. And it is not -possible to accomplish both tasks at the same time. Consequently the -crown is still to be won. Here the method is given of training the mind; -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 space. - -The thought of the past ages has used the conception of a -three-dimensional space, and by that means has classified many phenomena -and has obtained rules for dealing with matters of great practical -utility. The path which opens immediately before us in the future is -that of applying the conception of four-dimensional space to the -phenomena of nature, and of investigating what can be found out by this -new means of apprehension. - -In fact, what has been passed through may be called the -three-dimensional era; Gauss and Lobatchewski have inaugurated the -four-dimensional era. - - -CHAPTER I. - -SCEPTICISM AND SCIENCE. BEGINNING OF KNOWLEDGE. - -The following pages have for their object to induce the reader to apply -himself to the study, in the first place of Space, and then of Higher -Space; and, therefore, I have tried by indirect means to show forth -those thoughts and conceptions to which the practical work leads. - -And I feel that I have a great advantage in this project, inasmuch as -many of the thoughts which spring up in the mind of one who studies -higher space, and many of the conceptions to which he is driven, 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 world, but which are not for -that reason any the less valuable. - -And for my own part I cannot do more than put them forward in a very -feeble and halting manner. For I have come upon them, not in the way of -feeling or direct apprehension, 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 -good, but which simply asked for something to know. - -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 eat. All around me were the evidences of -knowledge--the arts, the sciences, interesting talk, useful -inventions--and yet I myself was profited nothing at all; for somehow, -amidst all this activity, I was left alone, I could get nothing which I -could know. - -The dialect was foreign to me--its inner meaning was hidden. If I would, -imitating the utterance of my fellows, say a few words, the effort was -forced, the whole result was an artificiality, and, if successful, would -be but a plausible imposture. - -The word “sceptical” has a certain unpleasant association attached to -it, for it has been used by so many people who are absolutely certain in -a particular line, and attack other people’s convictions. 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 companions. For to a mind that -inquires into what it really does know, it is hardly possible to -enunciate complete sentences, much less to put before it those complex -ideas which have so large a part in true human life. - -Every word we use has so wide and fugitive a meaning, and every -expression touches or rather grazes fact by so very minute a point, -that, if we wish to start with something which we do know, and thence -proceed in a certain manner, we are forced away from the study of -reality and driven to an artificial system, such as logic or -mathematics, which, starting from postulates and axioms, develops a body -of ideal truth which rather comes into contact with nature than is -nature. - -Scientific achievement is reserved for those who are content to absorb -into their consciousness, by any means and by whatever way they come, -the varied appearances of nature, whence and in which by reflection they -find floating as it were on the sea of the unknown, certain -similarities, certain resemblances and analogies, by means of which they -collect together a body of possible predictions and inferences; and in -nature they find correspondences which are actually verified. Hence -science exists, although the conceptions in the mind cannot be said to -have any real correspondence in nature. - -We form a set of conceptions in the mind, and the relations between -these conceptions give us relations which we find actually vibrating in -the world around us. But the conceptions themselves are essentially -artificial. - -We have a conception of atoms; but no one supposes that atoms actually -exist. We suppose a force varying inversely as the square of the -distance; but no one supposes such a mysterious thing to really be in -nature. And when we come to the region of descriptive science, when we -come to simple observation, we do not find ourselves any better provided -with a real knowledge of nature. If, for instance, we think of a plant, -we picture to ourselves a certain green shape, of a more or less -definite character. This green shape enables us to recognise the plant -we think of, and to describe it to a certain extent. But if we inquire -into our imagination of it, we find that our mental image very soon -diverges from the fact. If, for instance, we cut the plant in half, we -find cells and tissues of various kinds. If we examine our idea of the -plant, it has merely an external and superficial resemblance to the -plant itself. It is a mental drawing meeting the real plant in external -appearance; but the two things, the plant and our thought of it, come as -it were from different sides--they just touch each other as far as the -colour and shape are concerned, but as structures and as living -organisms they are as wide apart as possible. - -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 more. 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 nature, and whence -they can travel hand in hand. - -It almost seems as if, by sympathy and feeling, a human being was easier -to know than the simplest object. To know any object, however simple, by -the reason and observation requires an endless process of thought and -looking, building up the first vague impression into something like in -more and more respects. While, on the other hand, in dealing with human -beings there is an inward sympathy and capacity for knowing which is -independent of, though called into play by, the observation of the -actions and outward appearance of the human being. - -But for the purpose of knowing we must leave out these human -relationships. They are an affair of instinct and inherited unconscious -experience. The mind may some day rise to the level of these inherited -apprehensions, and be able to explain them; but at present it is far -more than overtasked to give an account of the simplest portions of -matter, and is quite inadequate to give an account of the nature of a -human being. - -Asking, then, what there was which I could know, I found no point of -beginning. There were plenty of ways of accumulating observations, but -none in which one could go hand in hand with nature. - -A child is provided in the early part of its life with a provision of -food adapted for it. But it seemed that our minds are left without a -natural subsistence, for on the one hand there are arid mathematics, and -on the other there is observation, and in observation there is, out of -the great mass of constructed mental images, but little which the mind -can assimilate. To the worker at science of course this crude and -omnivorous observation is everything; but if we ask for something which -we can know, it is like a vast mass of indigestible material with every -here and there a fibre or thread which we can assimilate. - -In this perplexity I was reduced to the last condition of mental -despair; and in default of finding anything which I could understand in -nature, I was sufficiently humbled to learn anything which seemed to -afford a capacity of being known. - -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 book. -For I found that the only assertion I could make about external objects, -without bringing in unknown and unintelligible relations, was this: I -could say how things were arranged. If a stone lay between two others, -that was a definite and intelligible fact, and seemed primary. As a -stone itself, it was an unknown somewhat which one could get more and -more information about the more one studied the various sciences. But -granting that there were some things there which we call stones, the way -they were arranged was a simple and obvious fact which could be easily -expressed and easily remembered. - -And so in despair of being able to obtain any other kind of mental -possession in the way of knowledge, I commenced to learn arrangements, -and I took as the objects to be arranged certain artificial objects of a -simple shape. I built up a block of cubes, and giving each a name I -learnt a mass of them. - -Now I do not recommend this as a thing to be done. 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 knowledge. It is perfectly definite and -certain. I could tell where each cube came and how it was related to -each of the others. As to the cube itself, I was profoundly ignorant of -that; but assuming that as a necessary starting-point, taking that as -granted, I had a definite mass of knowledge. - -But I do not wish to say that this is better than any kind of knowledge -which other people may find come home to them. All I want to do is to -take this humble beginning of knowledge and show how inevitably, by -devotion to it, it leads to marvellous and far-distant truths, and how, -by a strange path, it leads directly into the presence of some of the -highest conceptions which great minds have given us. - -I do not think it ought to be any objection to an inquiry, that it -begins with obvious and common details. In fact I do not think that it -is possible to get anything simpler, with less of hypothesis about it, -and more obviously a simple taking in of facts than the study of the -arrangement of a block of cubes. - -Many philosophers have assumed a starting point for their thought. I -want the reader to accept a very humble one and see what comes of it. If -this leads us to anything, no doubt greater results will come from more -ambitious beginnings. - -And now I feel that I have candidly exposed myself to the criticism of -the reader. If he will have the patience to go on, we will begin and -build up on our foundations. - - -CHAPTER II. - -APPREHENSION OF NATURE. INTELLIGENCE. STUDY OF ARRANGEMENT OR SHAPE. - -Nature is that which is around us. But it is by no means easy to get to -nature. The savage living we may say in the bosom of nature, is -certainly unapprehensive of it, 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 nature. - -Thus it is clear that it is not by mere passivity that we can comprehend -nature; it is the goal of an activity, not a free gift. - -And there are many ways of apprehending nature. There are the sounds and -sights of nature which delight the senses, and the involved harmonies -and the secret affinities which poetry makes us feel; then, moreover, -there is the definite knowledge of natural facts in which the memory and -reason are employed. - -Thus we may divide our means of coming into contact with nature into -three main channels: the senses, the imagination, and the mind. The -imagination is perhaps the highest faculty, but we leave it out of -consideration here, and ask: How can we bring our minds into contact -with nature? - -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 -faculties. - -And thus our power of understanding nature depends on our own -possession; it is in virtue of some mental activity of our own that we -can apprehend that outside activity which we call nature. And thus the -training to enable us to approach nature with our minds will be some -active process on our own part. - -In the course of my experience as a teacher I have often been struck by -the want of the power of reason displayed by pupils; they are not able -to put two and two together, as the saying goes, and I have been at some -pains to investigate wherein this curious deficiency lies, and how it -can be supplied. 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 methods, it is a something which most children obtain -in the natural and unsupervised education of their first contact with -the world, and lies before any recognised mode of distinction. They can -only understand in virtue of an activity of their own, and they have not -had sufficient exercise in this activity. - -In the present state of education it is impossible to diverge from the -ordinary routine. But it is always possible to experiment on children -who are out of the common line of education. And I believe I am amply -justified by the result of my experiments. - -I have seen that the same activity which I have found makes that habit -of mind which we call intelligence in a child, is the source of our -common and everyday rational intellectual work, and that just as the -faculties of a child can be called forth by it, so also the powers of a -man are best prepared by the same means, but on an ampler scale. - -A more detailed development of the practical work of Part II., would be -the best training for the mind of a child. An extension of the work of -that Part would be the training which, hand in hand with observation -and recapitulation, would best develop a man’s thought power. - -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 “understand” in any phenomenon of nature which has become clear -to us. - -When we look at a bright object it seems very different from a dull one. -A piece of bright steel hardly looks like the same substance as a piece -of dull steel. But the difference of appearance in the two is easily -accounted for by the different nature of the surface in the two cases; -in the one all the irregularities are done away with, and the rays of -light which fall on it are sent off again without being dispersed and -broken up. In the case of the dull iron the rays of light are broken up -and divided, so that they are not transmitted with intensity in any one -direction, but flung off in all sorts of directions. - -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 light. - -Again, with light itself the differences of colour are explained as -being the effect on us of rays of different rates of vibration. Now a -vibration is essentially this, a series of arrangements of matter which -follow each other in a closed order, so that when the set has been run -through, the first arrangement follows again. The whole theory of light -is an account of arrangements of the particles in the transmitting -medium, only the arrangements alter--are not permanent in any one -characteristic, but go through a complete cycle of varieties. - -Again, when the movements of the heavenly bodies are deduced from the -theory of universal gravitation, what we primarily do is to take -account of arrangement; for the law of gravity connects the movements -which the attracted bodies tend to make with their distances, that is, -it shows how their movements depend on their arrangement. And if gravity -as a force is to be explained itself, the suppositions which have been -put forward resolve it into the effect of the movements of small bodies; -that is to say, gravity, if explained at all, is explained as the result -of the arrangement and altering arrangements of small particles. - -Again, to take the idea which proceeding from Goethe casts such an -influence on botanical observation. Goethe (and also Wolf) 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 carpel, so unlike a leaf in form and -function. - -Now the essential value in this conception is, that it enables us to -look, 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 form. We can trace -correspondences between them, and are led to possible explanations of -their growth. And all this is done by getting rid of pistil and stamen -as separate entities, and looking on them as leaves, and their parts due -to different arrangement of the leaf structure. We have reduced these -diverse objects to a common element, we have found the unit by whose -arrangements the whole is produced. And in this department of thought, -as also to take another instance, in chemistry, to understand is -practically this: we find units (leaves or atoms) combinations of which -account for the results which we see. Thus we see that that which the -mind essentially apprehends is arrangement. - -And this holds over the whole range of mental work, from the simplest -observation to the most complex theory. When the eye takes in the form -of an external object there is something more than a sense impression, -something more than a sensation of greenness and light and dark. The -mind works as well as the sense, and these sense impressions are -definitely grouped in what we call the shape of the object. The -essential act of perceiving lies in the apprehension of a shape, and a -shape is an arrangement of parts. It does not matter what these parts -are; 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 -degree. If we want to make our representation still more like, we must -treat each of the dots as in themselves arrangements, we must compose -each of them by many strokes and dots of the brush. But even in this -case we have not got anything else besides arrangement. The ultimate -element, the small items of light and shade or of colour, are in -themselves meaningless; it is in their arrangement that the likeness of -the representation consists. - -Thus, from a drawing to our notion of the planetary system, all our -contact with nature lies in this, in an appreciation of arrangement. - -Hence to prepare ourselves for the understanding of nature, we must -“arrange.” In virtue of our activity in making arrangements we prepare -ourselves to do what is called understand nature. Or we may say, that -which we call understanding nature is to discern something similar in -nature to that which we do when we arrange elements into compounded -groups. - -Now if we study arrangement in the active way, we must have something to -arrange; and the things we work with may be either all alike, or each of -them varying from every other. - -If the elements are not alike then we are not studying pure arrangement; -but our knowledge is affected by the compound nature of that with which -we deal. If the elements are all alike, we have what we call units. -Hence the discipline preparatory for the understanding of nature is the -active arrangement of like units. - -And this is very much the case with all educational processes; only the -things chosen to arrange are in general words, which are so complicated -and carry such a train of association that, unless the mind has already -acquired a knowledge of arrangement, it is puzzled and hampered, and -never gets a clear apprehension of what its work is. - -Now what shall we choose for our units? Any unit would do; but it ought -to be a real thing--it ought to be something which can be touched and -seen, not something which no one has ever touched or seen, and which is -even incapable of definition, like a “number.” - -I would divide studies into two classes: those which create the faculty -of arrangement, and those which use it and exercise it. Mathematics -exercises it, but I do not think it creates it; and unfortunately, in -mathematics as it is now often taught, the pupil is launched , into a -vast system of symbols--the whole use and meaning of symbols (namely, as -means to acquire a clear grasp of facts) is lost to him. - -Of the possible units which will serve, I take the cube; and I have -found that whenever I took any other unit I got wrong, puzzled and lost -my way. With the cube one does not get along very fast, but everything -is perfectly obvious and simple, and builds up into a whole of which -every part is evident. - -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 nature, 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 way. Let him take -nothing which is not perfectly clear, patent and evident, demonstrable -to his senses, a simple repetition of obvious fact. - -Our work will then be this: a study, by means of cubes, of the facts of -arrangement. And the process of learning will be an active one of -actually putting up the cubes. In this way we do for the mind what -Wordsworth does for the imagination--we bring it into contact with -nature. - - -CHAPTER III. - -THE ELEMENTS OF KNOWLEDGE. - -There are two elements which enter into our knowledge with respect to -any phenomenon. - -If, for instance, we take the sun, and ask ourselves what we observe, we -notice that it is a bright, moving body; and of these two qualities, the -brightness and the movement, each seems equally predicable of the sun. -It does move, and it is bright. - -Now further study discloses to us that there is a difference between -these two affirmations. The motion of the sun in its diurnal course -round the earth is only apparent; but it is really a bright, hot body. - -Now of these two assertions which the mind naturally makes about the -sun, one--that it is moving--depends on the relation of the beholder to -the sun, the other is true about the sun itself. The observed motion -depends on a fact affecting oneself and having nothing to do with the -sun, while the brightness is really a quality of the sun itself. - -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 it, the self elements; those facts about it which -are independent of the observer’s particular relationship we will call -the residual element. Thus the sun’s motion is a self element in our -thought of the sun, its brightness is a residual element. - -It is not, of course, possible to draw a line distinctly between the -self elements and the residual elements. For instance, some people have -denied that brightness is a quality of things, but that it depends on -the capacity of the being for receiving sensations; 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 light. - -But there is no object in pursuing the discussion further. The main -distinction is sufficiently obvious. And it is important to separate the -self elements involved in our knowledge as far as possible, so that the -residual elements may be kept for our closer attention. By getting rid -of the self elements we put ourselves in a position in which we can -propound sensible questions. By getting rid of the notion of its -circular motion round the earth we prepare our way to study the sun as -it really is. We get the subject clear of complications and extraneous -considerations. - -It would hardly be worth while to dwell on this consideration were it -not of importance in our study of arrangement. But the fact is that -directly a subject has been cleared of the self elements, it seems so -absurd to have had them introduced at all that the great difficulty -there was in getting rid of them is forgotten. - -With regard to the knowledge we have at the present day about scientific -matters, there do not seem to be any self elements present. But the -worst about a self element is, that its presence is never dreamed of -till it is got rid of; to know that it is there is to have done away -with it. And thus our body of knowledge is like a fluid which keeps -clear, not because there are no substances in solution, but because -directly they become evident they fall down as precipitates. - -Now one of our serious pieces of work will be to get rid of the self -elements in the knowledge of arrangement. - -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 phenomena. In the large subjects which generally occupy the mind -the things thought of are so complicated that every detail cannot -possibly be considered. The principles of the whole are realized, and -then at any required time the principles can be worked out. Thus, with -regard to a knowledge of the planetary system, it is said to be known if -the law of movement of each of the planets is recognized, and their -positions at any one time committed to memory. It is not our habit to -remember their relative positions with regard to one another at many -intervals, so as to have an exhaustive catalogue of them in our minds. -But with regard to the elements of knowledge with which we shall work, -the subject is so simple that we may justly demand of ourselves that we -will know every detail. - -And the knowledge we shall acquire will be much more one of the sense -and feeling than of the reason. We do not want to have a rule in our -minds by which we can recall the positions of the different cubes, but -we want to have an immediate apprehension of them. It was Kant who first -pointed out how much of thought there was embodied in the sense -impressions; and it is this embodied thought which we wish to form. - - -CHAPTER IV. - -THEORY AND PRACTICE. - -Both in science and in morals there is an important distinction to be -drawn between theory and practice. A knowledge of chemistry does not -consist in the intellectual appreciation of different theories and -principles, but in being able to act in accordance with the facts of -chemical combination, so that by means of the appliances of chemistry -practical results are produced. 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 beings. - -Now the use of the word “learn” has been much restricted to merely -theoretic studies. It requires to be enlarged to the scientific meaning. -And to know, should include practice in actual manipulation. - -Let us take an instance. We all know what justice is, and any child can -be taught to tell the difference between acting justly and acting -unjustly. But it is a different thing to teach them to act with justice. -Something is done which affects them unpleasantly. They feel an impulse -to retaliate. In order to see what justice demands they have to put -their personal feeling on one side. They have to get rid of those -conditions under which they apprehended the effects of the action at -first, and they have to look on it from another point of view. Then they -have to act in accordance with this view. - -Now there are two steps--one an intellectual one of understanding, one a -practical one of carrying out the view. Neither is a moral step. One -demands intelligence, the other the formation of a habit, and this habit -can be inculcated by precept, by reward and punishment, by various -means. But as human nature is constituted, if the habit of justice is -inculcated it touches a part of the being. There is an emotional -response. We know but little of a human being, but we can safely say -that there are depths in it, beyond the feelings of momentary resentment -and the stimulus of pleasurable or painful sensation, to which justice -is natural. - -How little adequate is our physical knowledge of a human being as a -bodily frame to explain the fact of human life. Now and again we see one -of these isolated beings bound up in another, as if there was an -undiscovered physical bond between them. And in all there is this sense -of justice--a kind of indwelling verdict of the universal mind, if we -may use such an expression, in virtue of which a man feels not as a -single individual but as all men. - -With respect to justice, it is not only necessary to take the view of -one other person than oneself, but that of many. There may be justice -which is very good justice from the point of view of a party, but very -bad justice from the point of view of a nation. And if we suppose an -agency outside the human race, gifted with intelligence, and affecting -the race, in the way for instance of causing storms or disturbances of -the ground, in order to judge it with justice we should have to take a -standpoint outside the race of men altogether. We could not say that -this agency was bad. We should have to judge it with reference to its -effect on other sentient beings. - -There are some words which are often used in contrast with each -other--egoism and altruism; and each seems to me unmeaning except as -terms in a contrast. - -Let us take an instance. A boy has a bag of cakes, and is going to enjoy -them by himself. His parent stops him, and makes him set up some stumps -and begin to learn to play cricket with another boy. The enjoyment of -the cakes is lost--he has given that up; but after a little while he has -a pleasure which is greater than that of cakes in solitude. He enters -into the life of the game. He has given up, or been forced to give up, -the pleasure he knew, and he has found a greater one. What he thought -about himself before was that he liked cakes, now what he thinks about -himself is that he likes cricket. Which of these is the true thought -about himself? Neither, probably, but at any rate it is more near the -truth to say that he likes the cricket. If now we use the word self to -mean that which a person knows of himself, and it is difficult to see -what other meaning it can have, his self as he knew it at first was -thwarted, was given up, and through that he discovered his true self. -And again with the cricket; he will make the sacrifice of giving that -up, voluntarily or involuntarily, and will find a truer self still. - -In general there is not much difficulty in making a boy find out that he -likes cricket; 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 better. 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 game. 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 hardness, the unpleasantness is of a -double kind, the giving up of a pursuit or indulgence to which he is -accustomed, and the exertion of forming the habits demanded by the new -pursuit. - -Now it is unimportant whether the renunciation is forced or willingly -taken. But as a general rule it may be laid down, that by giving up his -own desires as he feels them at the moment, to the needs and advantage -of those around him, or to the objects which he finds before him -demanding accomplishment, a human being passes to the discovery of his -true self on and on. The process is limited by the responsibilities -which a man finds come upon him. - -The method of moral advance is to acquire a practical knowledge; he must -first see what the advantage of some one other than himself would be, -and then he must act in accordance with that view of things. Then having -acted and formed a habit, he discovers a response in himself. He finds -that he really cares, and that his former limited life was not really -himself. His body and the needs of his body, so far as he can observe -them, externally are the same as before; but he has obtained an inner -and unintellectual, but none the less real, apprehension of what he is. - -Thus altruism, or the sacrifice of egoism to others, is followed by a -truer egoism, or assertion of self, and this process flashed across by -the transcendent lights of religion, wherein, as in the sense of justice -and duty, 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 development, is the normal one by -which from childhood a human being develops into the full -responsibilities of a man. - -Now both in science and in conduct there are self elements. In science, -getting rid of the self elements means a truer apprehension of the facts -about one; in conduct, 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 scale. - -Thus without pretending to any scientific accuracy in the use of terms, -we can assign a certain amount of meaning to the expression--getting rid -of self elements. And all that we can do is to take the rough idea of -this process, and then taking our special subject matter, apply it. In -affairs of life experiments lead to disaster. But happily science is -provided wherein the desire to put theories into practice can be safely -satisfied--and good results sometimes follow. Were it not for this the -human race might before now have been utopiad from off the face of the -earth. - -In experiment, manipulation is everything; we must be certain of all our -conditions, otherwise we fail assuredly and have not even the -satisfaction of knowing that our failure is due to the wrongness of our -conjectures. - -And for our purposes we use a subject matter so simple that the -manipulation is easy. - - -CHAPTER V. - -KNOWLEDGE: SELF-ELEMENTS. - - -I must now go with somewhat of detail into the special subject in which -these general truths will be exhibited. Everything I have to say would -be conceived much more clearly by a very little practical manipulation. - -But here I want to put the subject in as general a light as possible, so -that there may be no hindrance to the judgment of the reader. - -And when I use the word “know,” I assume something else than the -possession of a rule, by which it can be said how facts are. 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 them. -Michael Angelo knew the human frame, he could tell every little fact -about it; if he chose to call up the image, he would see mentally how -each muscle and fold of the skin lay with regard to the surrounding -parts. We want to obtain a knowledge as good as Michael Angelo’s. There -is a great difference between Michael Angelo and us; but let that -difference be expressed, not in our way of knowing, but in the -difference between the things he knew and the things we know. We take a -very simple structure and know it as absolutely as he knew the -complicated structure of the human body. - -And let us take a block of cubes; any number will do, 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 parts. And let each of these -cubes be marked so as to be recognized, and let each have a name so that -it can be referred to. And let us suppose that we have learnt this block -of cubes so that each one is known--that is to say, its position in the -block is known and its relation to the other blocks. - -Now having obtained this knowledge of the block as it stands in front of -us, let us ask ourselves if there is any self element present in our -knowledge of it. - -And there is obviously this self element present. We have learnt the -cubes as they stand in accordance with our own convenience in putting -them up. We put the lowest ones first, and the others on the top of -them, and we distinctly conceive the lower ones as supporting the upper -ones. Now this fact of support has nothing to do with the block of cubes -itself, it depends on the conditions under which we come to apprehend -the block of cubes, it depends on our position on the surface of the -earth, whereby gravity is an all important factor in our experience. In -fact our sight has got so accustomed to take gravity into consideration -in its view of things, that when we look at a landscape or object with -our head upside down we do not see it inverted, but we superinduce on -the direct sense impressions our knowledge of the action of gravity, and -obtain a view differing very little from what we see when in an upright -position. - -It will be found that every fact about the cubes has involved in it a -reference to up and down. It is by being above or below that we chiefly -remember where the cubes are. But above and below is a relation which -depends simply on gravity. If it were not for gravity above and below -would be interchangeable terms, instead of expressing a difference of -marked importance to us under our conditions of existence. Now we put -“being above” or “being below” into the cubes themselves and feel it a -quality in them--it defines their position. But this above or below -really comes from the conditions in which we are. It is a self element, -and as such, to obtain a true knowledge of the cubes we must get rid of -it. - -And now, for the sake of a process which will be explained afterwards, -let us suppose that we cannot move the block of cubes which we have put -up. Let us keep it fixed. - -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 act; at the -centre of the earth, for instance, or in a freely falling shell. - -But this is impossible, so we must choose another way. Let us, then, -since we cannot get rid of gravity, see what we have done already. We -have learnt the cubes, and however they are learnt, it will be found -that there is a certain set of them round which the others are mentally -grouped, as being on the right or left, above or below. Now to get our -knowledge as perfect as we can before getting rid of the self element up -and down, we have to take as central cubes in our mind different sets -again and again, until there are none which are primary to us. - -Then there remains only the distinction of some being above others. Now -this can only be made to sink out of the primary place in our thoughts -by reversing the relation. If we turned the block upside down, and -learnt it in this new position, then we should learn the position of the -cubes with regard to each other with that element in them, which comes -from the action of gravity, reversed. And the true nature of the -arrangement to which we added something in virtue of our sensation of -up and down, would become purer and more isolated in our minds. - -We have, however, supposed that the cubes are fixed. Then, in order to -learn them, we must put up another block showing what they would be like -in the supposed new position. We then take a set of cubes, models of the -original cubes, 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 down. - -And here is the whole point on which the process depends. We can tell -where each cube would come, but we do not _know_ the block in this new -position. I draw a distinction between the two acts, “to tell where it -would be,” and to “know.” Telling where it would be is the preparation -for knowing. The power of assigning the positions may be called the -theory of the block. The actual knowledge is got by carrying out the -theory practically, by putting up the blocks and becoming able to -realize without effort where each one is. - -It is not enough to put up the model blocks in the reverse position. It -is found that this up and down is a very obstinate element indeed, and a -good deal of work is requisite to get rid of it completely. But when it -is got rid of in one set of cubes, the faculty is formed of appreciating -shape independently of the particular parts which are above or below on -first examination. We discover in our own minds the faculty of -appreciating the facts of position independent of gravity and its -influence on us. I have found a very great difference in different minds -in this respect. To some it is easy, to some it is hard. - -And to use our old instance, the discovery of this capacity is like the -discovery of a love of justice in the being who has forced himself to -act justly. It is a capacity for being able to take a view independent -of the conditions under which he is placed, and to feel in accordance -with that view. There is, so far as I know, no means of arriving -immediately at this impartial appreciation of shape. It can only be done -by, as it were, extending our own body so as to include certain cubes, -and appreciating then the relation of the other cubes to those. And -after this, by identifying ourselves with other cubes, and in turn -appreciating the relation of the other cubes to these. And the practical -putting up of the cubes is the way in which this power is gained. It -springs up with a repetition of the mechanical acts. Thus there are -three processes. 1st, An apprehension of what the position of the cubes -would be. 2nd, An actual putting of them up in accordance with that -apprehension, 3rd, The springing up in the mind of a direct feeling of -what the block is, independent of any particular presentation. - -Thus the self element of up and down can be got rid of out of a block of -cubes. - -And when even a little block is known like this, the mind has gained a -great deal. - -Yet in the apprehension and knowledge of the block of cubes with the up -and down relation in them, there is more than in the absolute -apprehension of them. For there is the apprehension of their position -and also of the effect of gravity on them in their position. - -Imagine ourselves to be translated suddenly to another part of the -universe, and to find there intelligent beings, and to hold conversation -with them. If we told them that we came from a world, and were to -describe the sun to them, saying that it was a bright, hot body which -moved round us, they would reply: You have told us something about the -sun, but you have also told us something about yourselves. - -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 round, for we really -in this case apprehend two things--the sun and our own conditions. But -for the purpose of further knowledge it is most important that the more -abstract knowledge should be acquired. 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 out. - -And in our block of cubes, it will be found that feelings about -arrangement, and knowledge of space, which are quite unattainable with -our ordinary view of position, become simple and clear when this -discipline has been gone through. - -And there can be no possible mental harm in going through this bit of -training, 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 view. - - -CHAPTER VI. - -FUNCTION OF MIND. SPACE AGAINST METAPHYSICS. SELF-LIMITATION AND ITS -TEST. A PLANE WORLD. - -We now pass on to the question: Are there any other self elements -present in our knowledge of the block of cubes? - -When we have learnt to free it from up and down, is there anything else -to be got rid of? - -It seems as if, when the cubes were thus learnt, we had got as abstract -and impersonal a bit of knowledge as possible. - -But, in reality, 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 -trifle. If we think we have got absolute knowledge we are indeed walking -on a thin crust in unconsciousness of the depths below. - -We are so certain of that which we are habituated to, we are so sure -that the world is made up of the mechanical forces and principles which -we familiarly deal with, that it is more of a shock than a welcome -surprise to us to find how mistaken we were. - -And after all, 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 parts? If so, where is there room for that -other which we know--more certainly, because inwardly--that reverence -and love which make life worth having? No; these mechanical relations -are our means of knowing about the world; they are not reality itself, -and their primary place in our imaginations is due to the familiarity -which we have with them, and to the peculiar limitations under which we -are. - -But I do not for a moment wish to go in thought beyond physical -nature--I do not suppose that in thought we can. To the mind it is only -the body that appears, and all that I hope to do is to show material -relations, mechanism, arrangements. - -But much depends on what kind of material relations we perceive outside -us. A human being, an animal and a machine are to the mind all merely -portions of matter arranged in certain ways. But the mind can give an -exhaustive account of the machine, account fairly well for the animal, -while the human being it only defines externally, leaving the real -knowledge to be supplied by other faculties. - -But we must not under-estimate the work of the mind, for it is only by -the observation of and thought about the bodies with which we come into -contact that we know human beings. It is the faculty of thought that -puts us in a position to recognize a soul. - -And so, too, about the universe--it is only by correct thought about it -that we can perceive its true moral nature. - -And it will be found that the deadness which we ascribe to the external -world is not really there, but is put in by us because of our own -limitations. It is really the self elements in our knowledge which make -us talk of mechanical necessity, dead matter. When our limitations fall, -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 us. - -Our thought means are sufficient at present to show us human souls; but -all except human beings is, as far as science is concerned, inanimate. -One self element must be got rid of from our perception, and this will -be changed. - -The one thing necessary is, that in matters of thinking we will not -admit anything that is not perfectly clear, palpable and evident. On the -mind the only conceivable demand is to seek for facts. The rock on which -so many systems of philosophy have come to grief is the attempt to put -moral principles into nature. Our only duty is to accept what we find. -Man is no more the centre of the moral world than he is of the physical -world. 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 it. We have to choose between -metaphysics and space thought. In metaphysics we find lofty -ideals--principles enthroned high in our souls, but which reduce the -world to a phantom, and ourselves to the lofty spectators of an arid -solitude. On the other hand, if we follow Kant’s advice, we use our -means and find realities linked together, and in the physical interplay -of forces and connexion of structure we behold the relations between -spirits--those dwelling in man and those above him. - -It is difficult to explain this next self element that has to be removed -from the block of cubes; it requires a little careful preparation, in -fact our language hardly affords us the means. But it is possible to -approach indirectly, and to detect the self-element by means of an -analogy. - -If we suspect there be some condition affecting ourselves which make us -perceive things not as they are, but falsely, then it is possible to -test the matter by making the supposition of other beings subject to -certain conditions, and then examining what the effect on their -experience would be of these conditions. - -Thus if we make up the appearances which would present themselves to a -being subject to a limitation or condition, we shall find that this -limitation or condition, when unrecognized by him, presents itself as a -general law of his outward world, or as properties and qualities of the -objects external to him. He will, moreover, find certain operations -possible, others impossible, 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 operations. - -And if we find that in our experience of the outward world there are -analogous properties and qualities of matter, analogous possibilities -and impossibilities, then it will show to us that we in our turn are -under analogous limitations, and that what we perceive as the external -world is both the external world and our own conditions. And the task -before us will be to separate the two. Now the problem we take up here -is this--to separate the self elements from the true fact. To separate -them not merely as an outward theory and intelligent apprehension, but -to separate them in the consciousness itself, so that our power of -perception is raised to a higher level. We find out that we are under -limitations. Our next step is to so familiarize ourselves with the real -aspect of things, that we perceive like beings not under our -limitations. Or more truly, we find that inward soul which itself not -subject to these limitations, is awakened to its own natural action, -when the verdicts conveyed to it through the senses are purged of the -self elements introduced by the senses. - -Everything depends on this--Is there a native and spontaneous power of -apprehension, which springs into activity when we take the trouble to -present to it a view from which the self elements are eliminated? About -this every one must judge for himself. But the process whereby this -inner vision is called on is a definite one. - -And just as a human being placed in natural human relationships finds in -himself a spontaneous motive towards the fulfilment of them, discovers -in himself a being whose motives transcend the limits of bodily -self-regard, 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 senses. - -I do not mean a theoretical power. A theory is always about it, and -about it only. I mean an inner view, a vision whereby the seeing mind as -it were identifies itself with the thing seen. Not the tree of -knowledge, but of the inner and vital sap which builds up the tree of -knowledge. - -And if this point is settled, it will be of some use in answering the -question: What are we? Are we then bodies only? This question has been -answered in the negative by our instincts. Why should we despair of a -rational answer? Let us adopt our space thought and develop it. - -The supposition which we must make is the following. Let us imagine a -smooth surface--like the surface of a table; but let the solid body at -which we are looking be very thin, so that our surface is more like the -surface of a thin sheet of metal than the top of a table. - -And let us imagine small particles, like particles of dust, to lie on -this surface, and to be attracted downwards so that they keep on the -surface. But let us suppose them to move freely over the surface. Let -them never in their movements rise one over the other; let them all -singly and collectively be close to the surface. And let us suppose all -sorts of attractions and repulsions between these atoms, and let them -have all kinds of movements like the atoms of our matter have. - -Then there may be conceived a whole world, and various kinds of beings -as formed out of this matter. The peculiarity about this world and these -beings would be, that neither the inanimate nor the animate members of -it would move away from the surface. Their movements would all lie in -one plane, a plane parallel to and very near the surface on which they -are. - -And if we suppose a vast mass to be formed out of these atoms, and to -lie like a great round disk on the surface, compact and cohering closely -together, then this great disk would afford a support for the smaller -shapes, which we may suppose to be animate beings. The smaller shapes -would be attracted to the great disk, but would be arrested at its rim. -They would tend to the centre of the disk, but be unable to get nearer -to the centre than its rim. - -Thus, as we are attracted to the centre of the earth, but walk on its -surface, the beings on this disk would be attracted to its centre, but -walk on its rim. The force of attraction which they would feel would be -the attraction of the disk. The other force of attraction, acting -perpendicularly to the plane which keeps them and all the matter of -their world to the surface, they would know nothing about. For they -cannot move either towards this force or away from it; and the surface -is quite smooth, so that they feel no friction in their movement over -it. - -Now let us realize clearly one of these beings as he proceeds along the -rim of his world. Let us imagine him in the form of an outline of a -human being, with no thickness except that of the atoms of his world. As -to the mode in which he walks, we must imagine that he proceeds by -springs or hops, because there would be no room for his limbs to pass -each other. - -Imagine a large disk on the table before you, and a being, such as the -one described, proceeding round it. Let there be small movable particles -surrounding him, which move out of his way as he goes along, and let -these serve him for respiration; let them constitute an atmosphere. - -Forwards and backwards would be to such a being direction along the -rim--the direction in which he was proceeding and its reverse. - -Then up and down would evidently be the direction away from the disk’s -centre and towards it. Thus backwards and forwards, up and down, would -both lie in the plane in which he was. - -And he would have no other liberty of movement except these. Thus the -words right and left would have no meaning to him. All the directions in -which he could move, or could conceive movement possible, would be -exhausted when he had thought of the directions along the rim and at -right angles to it, both in the plane. - -What he would call solid bodies, would be groups of the atoms of -his world cohering together. Such a mass of atoms would, we know, -have a slight thickness; namely, the thickness of a single atom. But -of this he would know nothing. He would say, “A solid body has two -dimensions--height (by how much it goes away from the rim) and thickness -(by how much it lies along the rim).” Thus a solid would be a -two-dimensional body, and a solid would be bounded by lines. Lines would -be all that he could see of a solid body. - -Thus one of the results of the limitations under which he exists would -be, that he would say, “There are only two dimensions in real things.” - -In order for his world to be permanent, we must suppose the surface on -which he is to be very compact, compared to the particles of his matter; -to be very rigid; and, if he is not to observe it by the friction of -matter moving on it, to be very smooth. And if it is very compact with -regard to his matter, the vibrations of the surface must have the effect -of disturbing the portions of his matter, and of separating compound -bodies up into simpler ones. - -[Illustration: Fig. 1.] - -[Illustration: Fig. 2.] - -Another consequence of the limitation under which this being lies, would -be the following:--If we cut out from the corners of a piece of paper -two triangles, A B C and A′ B′ C′, and suppose them to be reduced to -such a thinness that they are capable of being put on to the imaginary -surface, and of being observed by the flat being like other bodies known -to him; he will, after studying the bounding lines, which are all that -he can see or touch, come to the conclusion that they are equal and -similar in every respect; and he can conceive the one occupying the same -space as the other occupies, without its being altered in any way. - -If, however, instead of putting down these triangles into the surface on -which the supposed being lives, as shown in Fig. 1, we first of all turn -one of them over, and then put them down, then the plane-being has -presented to him two triangles, as shown in Fig. 2. - -And if he studies these, he finds that they are equal in size and -similar in every respect. But he cannot make the one occupy the same -space as the other one; this will become evident if the triangles be -moved about on the surface of a table. One will not lie on the same -portion of the table that the other has marked out by lying on it. - -Hence the plane-being by no means could make the one triangle in this -case coincide with the space occupied by the other, nor would he be able -to conceive the one as coincident with the other. - -The reason of this impossibility is, not that the one cannot be made to -coincide, but that before having been put down on his plane it has been -turned round. It has been turned, using a direction of motion which the -plane-being has never had any experience of, and which therefore he -cannot use in his mental work any more than in his practical endeavours. - -Thus, owing to his limitations, there is a certain line of possibility -which he cannot overstep. But this line does not correspond to what is -actually possible and impossible. It corresponds to a certain condition -affecting him, not affecting the triangle. His saying that it is -impossible to make the two triangles coincide, is an assertion, not -about the triangles, but about himself. - -Now, to return to our own world, no doubt there are many assertions -which we make about the external world which are really assertions about -ourselves. And we have a set of statements which are precisely similar -to those which the plane-being would make about his surroundings. - -Thus, he would say, there are only two independent directions; we say -there are only three. - -He would say that solids are bounded by lines; we say that solids are -bounded by planes. - -Moreover, there are figures about which we assert exactly the same kind -of impossibility as his plane-being did about the triangles in Fig. 2. - -We know certain shapes which are equal the one to the other, which are -exactly similar, and yet which we cannot make fit into the same portion -of space, either practically or by imagination. - -If we look at our two hands we see this clearly, though the two hands -are a complicated case of a very common fact of shape. Now, there is one -way in which the right hand and the left hand may practically be brought -into likeness. If we take the right-hand glove and the left-hand glove, -they will not fit any more than the right hand will coincide with the -left hand. But if we turn one glove inside out, then it will fit. Now, -to suppose the same thing done with the solid hand as is done with the -glove when it is turned inside out, we must suppose it, so to speak, -pulled through itself. If the hand were inside the glove all the time -the glove was being turned inside out, then, if such an operation were -possible, the right hand would be turned into an exact model of the left -hand. Such an operation is impossible. But curiously enough there is a -precisely similar operation which, if it were possible, would, in a -plane, turn the one triangle in Fig. 2 into the exact copy of the other. - -[Illustration] - -Look at the triangle in Fig. 2, A B C, and imagine the point A to move -into the interior of the triangle and to pass through it, carrying after -it the parts of the lines A B and A C to which it is attached, we should -have finally a triangle A B C, which was quite like the other of the two -triangles A′ B′ C′ in Fig. 2. - -Thus we know the operation which produces the result of the “pulling -through” is not an impossible one when the plane-being is concerned. -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 performed? The -question is an open one. Our feeling of it being impossible to produce -this result in any way, may be because it really is impossible, or it -may be a useful bit of information about ourselves. - -Now at this point my special work comes in. If there be really a -four-dimensional world, and we are limited to a space or -three-dimensional view, then either we are absolutely three-dimensional -with no experience at all or capacity of apprehending four-dimensional -facts, or we may be, as far as our outward experience goes, so limited; -but we may really be four-dimensional beings whose consciousness is by -certain undetermined conditions limited to a section of the real space. - -Thus we may really be like the plane-beings mentioned above, or we may -be in such a condition that our perceptions, not ourselves, are so -limited. The question is one which calls for experiment. - -We know that if we take an animal, such as a dog or cat, we can by -careful training, and by using rewards and punishment, make them act in -a certain way, in certain defined cases, in accordance with justice; we -can produce the mechanical action. But the feeling of justice will not -be aroused; it will be but a mere outward conformity. But a human being, -if so trained, and seeing others so acting, gets a feeling of justice. - -Now, if we are really four-dimensional, by going through those acts -which correspond to a four-dimensional experience (so far as we can), we -shall obtain an apprehension of four-dimensional existence--not with the -outward eye, but essentially with the mind. - -And after a number of years of experiment which were entirely nugatory, -I can now lay it down as a verifiable fact, that by taking the proper -steps we can feel four-dimensional existence, that the human being -somehow, and in some way, is not simply a three-dimensional being--in -what way it is the province of science to discover. All that I shall do -here is, to put forward certain suppositions which, in an arbitrary and -forced manner, give an outline of the relation of our body to -four-dimensional existence, and show how in our minds we have faculties -by which we recognise it. - - -CHAPTER VII. - -SELF ELEMENTS IN OUR CONSCIOUSNESS. - -It is often taken for granted that our consciousness of ourselves and of -our own feelings has a sort of direct and absolute value. - -It is supposed to afford a testimony which does not require to be sifted -like our consciousness of external events. But in reality it needs far -more criticism to be applied to it than any other mode of apprehension. - -To a certain degree we can sift our experience of the external world, -and divide it into two portions. We can determine the self elements and -the realities. But with regard to our own nature and emotions, the -discovery which makes a science possible has yet to be made. - -There are certain indications, however, springing from our observation -of our own bodies, which have a certain degree of interest. - -It is found that the processes of thought and feeling are connected with -the brain. If the brain is disturbed, thoughts, sights, and sounds come -into the consciousness which have no objective cause in the external -world. Hence we may conclusively say that the human being, whatever he -is, is in contact with the brain, and through the brain with the body, -and through the body with the external world. - -It is the structures and movements in the brain which the human being -perceives. It is by a structure in the brain that he apprehends nature, -not immediately. 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 consciousness. - -Hence, clearly, it is the movements and structure of the minute portions -of matter forming the brain which the consciousness perceives. And it is -only by models and representations made in the stuff of the brain that -the mind knows external changes. - -Now, our brains are well furnished with models and representations of -the facts and events of the external world. - -But a most important fact still requires its due weight to be laid upon -it. - -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 minuteness. Hence the -consciousness primarily apprehends the movements of matter of a degree -of smallness which is beyond the power of observation in any other way. - -Hence we have a means of observing the movements of the minute portions -of matter. 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 molecules. - -Now, these brain molecules are very minute portions of matter indeed; -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 us. - -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 matter. - -It may be that these brain molecules have the power of four-dimensional -movement, and that they can go through four-dimensional movements and -form four-dimensional structures. - -If so, there is a practical way of learning the movements of the very -small particles of matter--by observing, not what we can see, but what -we can think. - -For, 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 world, but in accordance with -their own activity, then they might go through four-dimensional -movements and form four-dimensional structures. - -And these movements and structures would be apprehended by the -consciousness along with the other movements and structures, and would -seem as real as the others--but would have no correspondence in the -external world. - -They would be thoughts and imaginations, not observations of external -facts. - -Now, this field of investigation is one which requires to be worked at. - -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 consistently. 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 existence. It only requires a certain amount of care to -build up mental models of higher space existences. In fact, it is -probably part of the difficulty of forming three-dimensional brain -models, 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 on. - - _Note._--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--as, for instance, the difficulty in remembering which - way to turn a screw, and the numerous cases of images in thought - transference--may be due to a toppling over in the brain, - four-dimensionalwise, of the structures formed--which structures would - be absolutely safe from being turned into image structures if the - brain molecules moved only three-dimensionalwise. - -It is remarkable how in science “explaining” 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 matter. - -Thus, the behaviour of gaseous bodies--the pressure which they exert, -the laws of their cooling and intermixture are explained by tracing the -movements of the very minute particles of which they are composed. - - -CHAPTER VIII. - -RELATION OF LOWER TO HIGHER SPACE. THEORY OF THE ÆTHER. - -At this point of our inquiries the best plan is to turn to the practical -work, and try if the faculty of thinking in higher space can be awakened -in the mind. - -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 cubes. We supposed that the block was fixed; and to get the sense of -what it would be when gravity acted in a different way with regard to -it, we made a model of it as it would be under the new circumstances. We -thought out the relations which would exist; and by practising this new -arrangement we gradually formed the direct apprehension. - -And so with higher-space arrangements. We cannot put them up actually, -but we can say how they would look and be to the touch from various -sides. And we can put up the actual appearances of them, not altogether, -but as models succeeding one another; and by contemplation and active -arrangement of these different views we call upon our inward power to -manifest itself. - -In preparing our general plan of work, it is necessary to make definite -assumptions with regard to our world, our universe, or we may call it -our space, in relation to the wider universe of four-dimensional space. - -What our relation to it may be, is altogether undetermined. The real -relationship will require a great deal of study to apprehend, and when -apprehended will seem as natural to us as the position of the earth -among the other planets does to us now. - -But we have not got to wait for this exploration in order to commence -our work of higher-space thought, for we know definitely that whatever -our real physical relationship to this wider universe may be, 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 space. - -And this relationship of a surface to a solid or of a solid, as we -conjecture, to a higher solid, is one which we often find in nature. A -surface is nothing more nor less than the relation between two things. -Two bodies touch each other. The surface is the relationship of one to -the other. - -Again, we see the surface of water. - -Thus our solid existence may be the contact of two four-dimensional -existences with each other; and just as sensation of touch is limited to -the surface of the body, so sensation on a larger scale may be limited -to this solid surface. - -And it is a fact worthy of notice, that in the surface of a fluid -different laws obtain from those which hold throughout the mass. There -are a whole series of facts which are grouped together under the name of -surface tensions, which are of great importance in physics, and by which -the behaviour of the surfaces of liquids is governed. - -And it may well be that the laws of our universe are the surface -tensions of a higher universe. - -But these expressions, it is evident, afford us no practical basis for -investigation. We must assume something more definite, and because more -definite (in the absence of details drawn from experience), more -arbitrary. - -And we will assume that the conditions under which we human beings are, -exactly resemble those under which the plane-beings are placed, which -have been described. - -This forms the basis of our work; and the practical part of it consists -in doing, with regard to higher space, that which a plane-being would do -with regard to our space in order to enable himself to realize what it -was. - -If we imagine one of these limited creatures whose life is cramped and -confined studying the facts of space existence, we find that he can do -it in two ways. He can assume another direction in addition to those -which he knows; and he can, by means of abstract reasoning, say what -would take place in an ampler kind of space than his own. All this would -be formal work. The conclusions would be abstract possibilities. - -The other mode of study is this. He can take some of these facts of his -higher space and he can ponder over them in his mind, and can make up in -his plane world those different appearances which one and the same solid -body would present to him, and then he may try to realize inwardly what -his higher existence is. - -Now, it is evident that if the creature is absolutely confined to a -two-dimensional existence, then anything more than such existence will -always be a mere abstract and formal consideration to him. - -But if this higher-space thought becomes real to him, if he finds in his -mind a possibility of rising to it, then indeed he knows that somehow he -is not limited to his apparent world. Everything he sees and comes into -contact with may be two-dimensional; but essentially, somehow, himself -he is not two-dimensional merely. - -And a precisely similar piece of work is before us. Assuming as we must -that our outer experience is limited to three-dimensional space, we -shall make up the appearances which the very simplest higher bodies -would present to us, and we shall gradually arrive at a more than merely -formal and abstract appreciation of them. We shall discover in ourselves -a faculty of apprehension of higher space similar to that which we have -of space. And thus we shall discover, each for himself, that, limited as -his senses are, he essentially somehow is not limited. - -The mode and method in which this consciousness will be made general, is -the same in which the spirit of an army is formed. - -The individuals enter into the service from various motives, but each -and all have to go through those movements and actions which correspond -to the unity of a whole formed out of different members. The inner -apprehension which lies in each man of a participation in a life wider -than that of his individual body, is awakened and responds; and the -active spirit of the army is formed. So with regard to higher space, -this faculty of apprehending intuitively four-dimensional relationships -will be taken up because of its practical use. Individuals will be -practically employed to do it by society because of the larger faculty -of thought which it gives. In fact, this higher-space thought means as -an affair of mental training simply the power of apprehending the -results arising from four independent causes. It means the power of -dealing with a greater number of details. - -And when this faculty of higher-space thought has been formed, then the -faculty of apprehending that higher existence in which men have part, -will come into being. - -It is necessary to guard here against there being ascribed to this -higher-space thought any other than an intellectual value. It has no -moral value whatever. 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 now. There is a gradual process -going on which may be described as the getting rid of self elements. -This process is one of knowledge and feeling, and either may be -independent of the other. At present, in respect of feeling, we are much -further on than in respect to understanding, and the reason is very much -this: When a self element has been got rid of in respect of feeling, the -new apprehension is put into practice, and we live it into our -organization. But when a self element has been got rid of -intellectually, it is allowed to remain a matter of theory, not vitally -entering into the mental structure of individuals. - -Thus up and down was discovered to be a self element more than a -thousand years ago; but, except as a matter of theory, we are perfect -barbarians in this respect up to the present day. - -We have supposed a being living in a plane world, that is, a being of a -very small thickness in a direction perpendicular to the surface on -which he is. - -Now, if we are situated analogously with regard to an ampler space, -there must be some element in our experience corresponding to each -element in the plane-being’s experience. - -And it is interesting to ask, in the case of the plane-being, what his -opinion would be with respect to the surface on which he was. - -He would not recognize it as a surface with which he was in contact; he -would have no idea of a motion away from it or towards it. - -But he would discover its existence by the fact that movements were -transmitted along it. By its vibrating and quivering, it would impart -movement to the particles of matter lying on it. - -Hence, he would consider this surface to be a medium lying between -bodies, and penetrating them. It would appear to him to have no weight, -but to be a powerful means of transmitting vibrations. Moreover, it -would be unlike any other substance with which he was acquainted, -inasmuch as he could never get rid of it. However perfect a vacuum be -made, there would be in this vacuum just as much of this unknown medium -as there was before. - -Moreover, this surface would not hinder the movement of the particles of -matter over it. Being smooth, matter would slide freely over it. And -this would seem to him as if matter went freely through the medium. - -Then he would also notice the fact that vibrations of this medium would -tear asunder portions of matter. The plane surface, being very compact, -compared to the masses of matter on it, would, by its vibrations, shake -them into their component parts. - -Hence he would have a series of observations which tended to show that -this medium was unlike any ordinary matter with which he was acquainted. -Although matter passed freely through it, still by its shaking it could -tear matter in pieces. These would be very difficult properties to -reconcile in one and the same substance. Then it is weightless, and it -is everywhere. - -It might well be that he would regard the supposition of there being a -plane surface, on which he was, as a preferable one to the hypothesis of -this curious medium; and thus he might obtain a proof of his limitations -from his observations. - -Now, is there anything in our experience which corresponds to this -medium which the plane-being gets to observe? - -Do we suppose the existence of any medium through which matter freely -moves, which yet by its vibrations destroys the combinations of -matter--some medium which is present in every vacuum, however perfect, -which penetrates all bodies, and yet can never be laid hold of? - -These are precisely observations which have been made. - -The substance which possesses all these qualities is called the æther. -And the properties of the æther are a perpetual object of investigation -in science. - -Now, it is not the place here to go into details, as all we want is a -basis for work; and however arbitrary it may be, it will serve if it -enables us to investigate the properties of higher space. - -We will suppose, then, that we are not in, but on the æther, only not on -it in any known direction, but that the new direction is that which -comes in. The æther is a smooth body, along which we slide, being -distant from it at every point about the thickness of an atom; or, if we -take our mean distance, being distant from it by half the thickness of -an atom measured in this new direction. - -Then, just as in space objects, a cube, for instance, can stand on the -surface of a table, or on the surface over which the plane-being moves, -so on the æther can stand a higher solid. - -All that the plane-being sees or touches of a cube, is the square on -which it rests. - -So all that we could see or touch of a higher solid would be that part -by which it stood on the æther; and this part would be to us exactly -like any ordinary solid body. The base of a cube would be to the -plane-being like a square which is to him an ordinary solid. - -Now, the two ways, in which a plane-being would apprehend a solid body, -would be by the successive appearances to him of it as it passed through -his plane; and also by the different views of one and the same solid -body which he got by turning the body over, so that different parts of -its surface come into contact with his plane. - -And the practical work of learning to think in four-dimensional space, -is to go through the appearances which one and the same higher solid -has. - -Often, in the course of investigation in nature, we come across objects -which have a certain similarity, and yet which are in parts entirely -different. The work of the mind consists in forming an idea of that -whole in which they cohere, and of which they are simple presentations. - -The work of forming an idea of a higher solid is the most simple and -most definite of all such mental operations. - -If we imagine a plane world in which there are objects which correspond -to our sun, to the planets, and, in fact, to all our visible universe, -we must suppose a surface of enormous extent on which great disks slide, -these disks being worlds of various orders of magnitude. - -These disks would some of them be central, and hot, like our sun; round -them would circulate other disks, like our planets. - -And the systems of sun and planets must be conceived as moving with -great velocity over the surface which bears them all. - -And the movements of the atoms of these worlds will be the course of -events in such worlds. As the atoms weave together, and form bodies -altering, becoming, and ceasing, so will bodies be formed and -disappear. - -And the plane which bears them all on its smooth surface will simply be -a support to all these movements, and influence them in no way. - -Is to be conscious of being conscious of being hot, the same thing as to -be conscious of being hot? It is not the same. There is a standing -outside, and objectivation of a state of mind which every one would say -in the first state was very different from the simple consciousness. But -the consciousness must do as much in the first case as in the second. -Hence the feeling hot is very different from the consciousness of -feeling hot. - -A feeling which we always have, we should not be conscious of--a sound -always present ceases to be heard. Hence consciousness is a concomitant -of change, that is, of the contact between one state and another. - -If a being living on such a plane were to investigate the properties, he -would have to suppose the solid to pass through his plane in order to -see the whole of its surface. Thus we may imagine a cube resting on a -table to begin to penetrate through the table. If the cube passes -through the surface, making a clean cut all round it, so that the -plane-being can come up to it and investigate it, then the different -parts of the cube as it passes through the plane will be to him squares, -which he apprehends by the boundary lines. The cut which there is in his -plane must be supposed not to be noticed, he must be able to go right up -to the cube without hindrance, and to touch and see that thin slice of -it which is just above the plane. - -And so, when we study a higher solid, we must suppose that it passes -through the æther, 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 æther. - -When we look on a solid as a section of a higher solid, we have to -suppose the æther broken through, only we must suppose that it runs up -to the edge of the body which is penetrating it, so that we are aware of -no breach of continuity. - -The surface of the æther must then be supposed to have the properties of -the surface of a fluid; only, of course, it is a solid three-dimensional -surface, not a two-dimensional surface. - - -CHAPTER IX. - -ANOTHER VIEW OF THE ÆTHER. MATERIAL AND ÆTHERIAL BODIES. - -We have supposed in the case of a plane world that the surface on which -the movements take place is inactive, except by its vibrations. It is -simply a smooth support. - -For the sake of simplicity let us call this smooth surface “the æther” -in the case of a plane world. - -The æther then we have imagined to be simply a smooth, thin sheet, not -possessed of any definite structure, but excited by real disturbances of -the matter on it into vibrations, which carry the effect of these -disturbances as light and heat to other portions of matter. Now, it is -possible to take an entirely different view of the æther in the case of -a plane world. - -Let us imagine that, instead of the æther being a smooth sheet serving -simply as a support, it is definitely marked and grooved. - -Let us imagine these grooves and channels to be very minute, but to be -definite and permanent. - -Then, let us suppose that, instead of the matter which slides in the -æther having attractions and repulsions of its own, that it is quite -inert, and has only the properties of inertia. - -That is to say, taking a disk or a plane world as a specimen, the whole -disk is sliding on the æther in virtue of a certain momentum which it -has, and certain portions of its matter fit into the grooves in the -æther, and move along those grooves. - -The size of the portions is determined by the size of the grooves. And -let us call those portions of matter which occupy the breadth of a -groove, atoms. Then it is evident that the disk sliding along over the -æther, its atoms will move according to the arrangement of the grooves -over which the disk slides. If the grooves at any one particular place -come close together, there will be a condensation of matter at that -place when the disk passes over it; and if the grooves separate, there -will be a rarefaction of matter. - -If we imagine five particles, each slipping along in its own groove, if -the particles are arranged in the form of a regular pentagon, and the -grooves are parallel, then these five particles, moving evenly on, will -maintain their positions with regard to one another, and a body would -exist like a pentagon, lasting as long as the groves remained parallel. - -But if, after some distance had been traversed by the disk, and these -five particles were brought into a region where one of the grooves -tended away from the others, the shape of the pentagon would be -destroyed, it would become some irregular figure. And it is easy to see -that if the grooves separated, and other grooves came in amongst them, -along which other portions of matter were sliding, that the pentagon -would disappear as an isolated body, that its constituent matter would -be separated, and that its particles would enter into other shapes as -constituents of them, and not of the original pentagon. - -Thus, in cases of greater complication, an elaborate structure may be -supposed to be formed, to alter, and to pass away; its origin, growth, -and decay being due, not to any independent motion of the particles -constituting it, but to the movement of the disk whereby its portions of -matter were brought to regions where there was a particular disposition -of the grooves. - -Then the nature of the shape would really be determined by the grooves, -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 them, -but they would subsist independently of the disk; and if another disk -were to pass over the same grooves, exactly the same material structures -would spring up as came into being before. - -If we make a similar supposition about our æther along which our earth -slides, we may conceive the movements of the particles of matter to be -determined, not by attractions or repulsions exerted on one another, but -to be set in existence by the alterations in the directions of the -grooves of the æther along which they are proceeding. - -If the grooves were all parallel, the earth would proceed without any -other motion than that of its path in the heavens. - -But with an alteration in the direction of the grooves, the particles, -instead of proceeding uniformly with the mass of the earth, would begin -to move amongst each other. 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 -grooves. - -Thus the movements, which any body goes through, would depend on the -arrangement of the æthereal grooves along which it was passing. As long -as the grooves remain grouped together in approximately the same way, it -would maintain its existence as the same body; but when the grooves -separated, and became involved with the grooves of other objects, this -body would cease to exist separately. - -Thus the separate existences of the earth might conceivably be due to -the disposition of those parts of the æther over which the earth -passed. And thus any object would have to be separated into two parts, -one the æthereal form, or modification which lasted, the other the -material particles which, coming on with blind momentum, were directed -into such movements as to produce the actual objects around us. - -In this way there would be two parts in any organism, the material part -and the æthereal part. There would be the material body, which soon -passes and becomes indistinguishable from any other material body, and -the æthereal body which remains. - -Now, if we direct our attention to the material body, we see the -phenomena of growth, decay, and death, the coming and the passing away -of a living being, isolated during his existence, absolutely merged at -his death into the common storehouse of matter. - -But if we regard the æthereal body, we find something different. We find -an organism which is not so absolutely separated from the surrounding -organisms--an organism which is part of the æther, and which is linked -to other æthereal organisms by its very substance--an organism between -which and others there exists a unity incapable of being broken, and a -common life which is rather marked than revealed by the matter which -passes over it. The æthereal body moreover remains permanently when the -material body has passed away. - -The correspondences between the æthereal body and the life of an -organism such as we know, is rather to be found in the emotional region -than in the one of outward observation. To the æthereal form, all parts -of it are equally one; but part of this form corresponds to the future -of the material being, part of it to his past. Thus, care for the future -and regard for the past would be the way in which the material being -would exhibit the unity of the æthereal body, which is both his past, -his present, and his future. That is to say, suppose the æthereal body -capable of receiving an injury, an injury in one part of it would -correspond to an injury in a man’s past; an injury in another -part,--that which the material body was traversing,--would correspond to -an injury to the man at the present moment; injury to the æthereal body -at another part, would correspond to injury coming to the man at some -future time. And the self-preservation of the æthereal body, supposing -it to have such a motive, would in the last case be the motive of -regarding his own future to the man. And inasmuch as the man felt the -real unity of his æthereal body, and did not confine his attention to -his material body, which is absolutely disunited at every moment from -its future and its past--inasmuch as he apprehended his æthereal unity, -insomuch would he care for his future welfare, and consider it as equal -in importance to his present comfort. The correspondence between emotion -and physical fact would be, that the emotion of regard corresponded to -an undiscerned æthereal unity. And then also, just as the two tips of -two fingers put down on a plane, would seem to a plane-being to be two -completely different bodies, not connected together, so one and the same -æthereal body might appear as two distinct material bodies, and any -regard between the two would correspond to an apprehension of their -æthereal unity. In the supposition of an æthereal body, it is not -necessary to keep to the idea of the rigidity and permanence of the -grooves defining the motion of the matter which, passing along, exhibits -the material body. The æthereal body may have a life of its own, -relations with other æthereal bodies, and a life as full of vicissitudes -as that of the material body, which in its total orbit expresses in the -movements of matter one phase in the life of the æthereal body. - -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 apprehensions,--such as that of the -essential unity of the race,--and affords a possible clue to -correspondences between the emotional and the physical life. - -The whole question of our relation to the æther has to be settled. That -which we call the æther is far more probably the surface of a liquid, -and the phenomena we observe due to surface tensions. Indeed, the -physical questions concern us here nothing at all. It is easy enough to -make some supposition which gives us a standing ground to discipline our -higher-space perception; and when that is trained, we shall turn round -and look at the facts. - -The conception which we shall form of the universe will undoubtedly be -as different from our present one, as the Copernican view differs from -the more pleasant view of a wide immovable earth beneath a vast vault. -Indeed, any conception of our place in the universe will be more -agreeable than the thought of being on a spinning ball, kicked into -space without any means of communication with any other inhabitants of -the universe. - - -CHAPTER X. - -HIGHER SPACE AND HIGHER BEING. PERCEPTION AND INSPIRATION. - -In the instinctive and sense perception of man and nature there is all -hidden, which reflection afterwards brings into consciousness. - -We are conscious of somewhat higher than each individual man when we -look at men. In some, this consciousness reaches an extreme pitch, and -becomes a religious apprehension. But in none is it otherwise than -instinctive. The apprehension is sufficiently definite to be certain. -But it is not expressible to us in terms of the reason. - -Now, 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 whole. Our apparent isolation as bodies from each -other is by no means so necessary to assume as it would appear. But, of -course, a supposition of that kind is of no value, except as showing a -possibility. If we came to examine into the matter closely, we should -find a natural relationship which accounted for our consciousness being -limited as at present it is. - -The first thing to be done, is to organize our higher-space perception, -and then look. We cannot tell what external objects will blend together -into the unity of a higher being. But just as the riddle of the two -hands becomes clear to us from our first inspection of higher space, so -will there grow before our eyes greater unities and greater surprises. - -We have been subject to a limitation of the most absurd character. Let -us open our eyes and see the facts. - -Now, it requires some training to open the eyes. For many years I worked -at the subject without the slightest success. All was mere formalism. -But by adopting the simplest means, and by a more thorough knowledge of -space, the whole flashed clear. - -Space shapes can only be symbolical of four-dimensional shapes; and if -we do not deal with space shapes directly, 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 symbols, and the task is -hopeless. But a direct study of space leads us to the knowledge of -higher space. And with the knowledge of higher space there come into our -ken boundless possibilities. All those things may be real, whereof -saints and philosophers have dreamed. - -Looking on the fact of life, it has become clear to the human mind, that -justice, truth, purity, are to be sought--that they are principles which -it is well to serve. And men have invented an abstract devotion to -these, and all comes together in the grand but vague conception of Duty. - -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 -itself. On the piled-up clouds falls the shadow--vast, imposing, but -dark, colourless. If the beholder but turns, he beholds the mountain -itself, towering grandly with verdant pines, the snowline, and the awful -peaks. - -So all these conceptions are the way in which now, with vision confined, -we apprehend the great existences of the universe. Instead of an -abstraction, what we have to serve is a reality, to which even our real -things are but shadows. We are parts of a great being, in whose -service, and with whose love, the utmost demands of duty are satisfied. - -How can it not be a struggle, when the claims of righteousness mean -diminished life,--even death,--to the individual who strives? And yet to -a clear and more rational view it will be seen that in his extinction -and loss, that which he loves,--that real being which is to him shadowed -forth in the present existence of wife and child,--that being lives more -truly, and in its life those he loves are his for ever. - -But, of course, there are mistakes in what we consider to be our duty, -as in everything else; and this is an additional reason for pursuing the -quest of this reality. For by the rational observance of other material -bodies than our own, we come to the conclusion that there are other -beings around us like ourselves, whom we apprehend in virtue of two -processes--the one simply a sense one of observation and reflection--the -other a process of direct apprehension. - -Now, if we did not go through the sense process of observation, we -might, it is true, know that there were other human beings around us in -some subtle way--in some mesmeric feeling; but we should not have that -organized human life which, dealing with the things of the world, grows -into such complicated forms. We should for ever be good-humoured -babies--a sensuous, affectionate kind of jelly-fish. - -And just so now with reference to the high intelligences by whom we are -surrounded. We feel them, but we do not realize them. - -To realize them, it will be necessary to develop our power of -perception. - -The power of seeing with our bodily eye is limited to the -three-dimensional section. - -But I have shown that the inner eye is not thus limited; that we can -organize our power of seeing in higher space, and that we can form -conceptions of realities in this higher space, just as we can in our -ordinary space. - -And this affords the groundwork for the perception and study of these -other beings than man. Just as some mechanical means are necessary for -the apprehension of our fellows in space, so a certain amount of -mechanical education is necessary for the perception of higher beings in -higher space. - -Let us turn the current of our thought right round; instead of seeking -after abstractions, and connecting our observations by ideas, let us -train our sense of higher space and build up conceptions of greater -realities, more absolute existences. - -It is really a waste of time to write or read more generalities. Here is -the grammar of the knowledge of higher being--let us learn it, not spend -time in speculating as to whither it will lead us. - -Yet one thing more. We are, with reference to the higher things of life, -like blind and puzzled children. We know that we are members of one -body, limbs of one vine; but we cannot discern, except by instinct and -feeling, what that body is, what the vine is. If to know it would take -away our feeling, then it were well never to know it. But fuller -knowledge of other human beings does not take away our love for them; -what reason is there then to suppose that a knowledge of the higher -existences would deaden our feelings? - -And then, again, we each of us have a feeling that we ourselves have a -right to exist. We demand our own perpetuation. No man, I believe, is -capable of sacrificing his life to any abstract idea; in all cases it is -the consciousness of contact with some being that enables him to make -the last human sacrifice. And what we can do by this study of higher -space, is to make this consciousness, which has been reserved for a few, -the property of all. Do we not all feel that there is a limit to our -devotion to abstractions, none to beings whom we love. And to love them, -we must know them. - -Then, just as our own individual life is empty and meaningless without -those we love, so the life of the human race is empty and meaningless -without a knowledge of those that surround it. And although to some an -inner knowledge of the oneness of all men is vouchsafed, it remains to -be demonstrated to the many. - -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 each. - -And this search, it seems to me, affords the needful supplement to the -inquiries of one with whose thought I have been very familiar, and to -which I return again, after having abandoned it for the purely -materialistic views which seem forced upon us by the facts of science. - -All that he said seemed to me unsupported by fact, unrelated to what we -know. - -But when I found that my knowledge was merely an empty pretence, 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 nothingness, then I was forced into that simple -quest for fact, which, when persisted in and lived in, opens out to the -thoughts like a flower to the life-giving sun. - -It is indeed a far safer course, to believe that which appeals to us as -noble, than simply to ask what is true; to take that which great minds -have given, than to demand that our puny ones should be satisfied. But -I suppose there is some good to some one in the scepticism and struggle -of those who cannot follow in the safer course. - -The thoughts of the inquirer to whom I allude may roughly be stated -thus:-- - -He saw in human life the working out of a great process, in the toil and -strain of our human history he saw the becoming of man. There is a -defect whereby we fall short of the true measure of our being, and that -defect is made good in the course of history. - -It is owing to that defect that we perceive evil; and in the perception -of evil and suffering lies our healing, for we shall be forced into that -path at last, after trying every other, which is the true one. - -And this, the history of the redemption of man, is what he saw in all -the scenes of life; each most trivial occurrence was great and -significant in relation to this. - -And, further, he put forward a definite statement with regard to this -defect, this lack of true being, for it lay, he said, in the -self-centredness of our emotions, in the limitation of them to our -bodily selves. He looked for a time when, driven from all thoughts of -our own pain or pleasure, good or evil, we should say, in view of the -miseries of our fellow-creatures, Let me be anyhow, use my body and my -mind in any way, so that I serve. - -And this, it seems to me, is the true aspiration; for, just as a note of -music flings itself into the march of the melody, and, losing itself in -it, is used for it and lost as a separate being, so we should throw -these lives of ours as freely into the service of--whom? - -Here comes the difficulty. Let it be granted that we should have no -self-rights, limit our service in no way, still the question comes, What -shall we serve? - -It is far happier to have some concrete object to which we are devoted, -or to be bound up in the ceaseless round of active life, wherein each -day presents so many necessities that we have no room for choice. - -But besides and apart from all these, there comes to some the question, -“What does it all mean?” To others, an unlovable and gloomy aspect is -presented, wherein their life seems to be but used as a material -worthless in itself and ungifted with any dignity or honour; while to -others again, with the love of those they love, comes a cessation of all -personal interest in life, and a disappointment and feeling of -valuelessness. - -And in all these cases some answer is needed. And here human duty -ceases. We cannot make objects to love. We can make machines and works -of art, but nothing which directly excites our love. To give us that -which rouses our love, is the duty of one higher than ourselves. - -And yet in one respect we have a duty--we must look. - -What good would it be, to surround us with objects of loving interest, -if we bury our regards in ourselves and will not see? - -And does it not seem as if with lowered eyelids, till only the thinnest -slit was open, we gazed persistently, not on what is, but on the -thinnest conceivable section of it? - -Let it be granted that our right attitude is, so to devote ourselves -that there is no question as to what we will do or what we will not do, -but we are perfectly obedient servants. The question is, Whom are we to -serve? - -It cannot be each individual, for their claims are conflicting, and as -often as not there is more need of a master than of a servant. Moreover, -the aspect of our fellows does not always excite love, which is the only -possible inducer of the right attitude of service. If we do not love, -we can only serve for a self motive, because it is in some way good for -ourselves. - -Thus it seems to me that we are reduced to this: our only duty is to -look for that which it is given us to love. - -But this looking is not mere gazing. To know, we must act. - -Let any one try it. He will find that unless he goes through a series of -actions corresponding to his knowledge, he gets merely a theoretic and -outside view of any facts. The way to know is this: Get somehow a means -of telling what your perceptions would be if you knew, and act in -accordance with those perceptions. - -Thus, with regard to a fellow-creature, if we knew him we should feel -what his feelings are. Let us then learn his feelings, and act as if we -had them. It is by the practical work of satisfying his needs that we -get to know him. - -Then, may-be, we love him; or perchance it is said we may find that -through him we have been brought into contact with one greater than him. - -This is our duty--to know--to know, not merely theoretically, but -practically; and then, when we know, we have done our part; if there is -nothing, we cannot supply it. All we have to do is to look for -realities. - -We must not take this view of education--that we are horribly pressed -for time, and must learn, somehow, a knack of saying how things must be, -without looking at them. - -But rather, we must say that we have a long time--all our lives, in -which we will press facts closer and closer to our minds; and we begin -by learning the simplest. There is an idea in that home of our -inspiration--the fact that there are certain mechanical processes by -which men can acquire merit. This is perfectly true. It is by mechanical -processes that we become different; and the science of education -consists largely in systematizing these processes. - -Then, just as space perceptions are necessary for the knowledge of our -fellow-men, and enable us to enter into human relationships with them in -all the organized variety of civilized life, so it is necessary to -develop our perceptions of higher space, so that we can apprehend with -our minds the relationship which we have to beings higher than -ourselves, and bring our instinctive knowledge into clearer -consciousness. - -It appears to me self-evident, that in the particular disposition of any -portion of matter, that is, in any physical action, there can be neither -right nor wrong; the thing done is perfectly indifferent. - -At the same time, it is only in things done that we come into -relationship with the beings about us and higher than us. Consequently, -in the things we do lies the whole importance of our lives. - -Now, many of our impulses are directly signs of a relationship in us to -a being of which we are not immediately conscious. The feeling of love, -for instance, is always directed towards a particular individual; but by -love man tends towards the preservation and improvement of his race; -thus in the commonest and most universal impulses lie his relations to -higher beings than the individuals by whom he is surrounded. Now, along -with these impulses are many instincts of a modifying tendency; and, -being altogether in the dark as to the nature of the higher beings to -whom we are related, it is difficult to say in what the service of the -higher beings consists, in what it does not. The only way is, as in -every other pre-rational department of life, to take the verdict of -those with the most insight and inspiration. - -And any striving against such verdicts, and discontent with them, should -be turned into energy towards finding out exactly what relation we have -towards these higher beings by the study of Space. - -Human life at present is an art constructed in its regulations and rules -on the inspirations of those who love the undiscerned higher beings, of -which we are a part. They love these higher beings, and know their -service. - -But our perceptions are coarser; and it is only by labour and toil that -we shall be brought also to see, and then lose the restraints that now -are necessary to us in the fulness of love. - -Exactly what relationship there is towards us on the part of these -higher beings we cannot say in the least. We cannot even say whether -there is more than humanity before the highest; and any conception which -we form now must use the human drama as its only possible mode of -presentation. - -But that there is such a relation seems clear; and the ludicrous manner, -in which our perceptions have been limited, is a sufficient explanation -of why they have not been scientifically apprehended. - -The mode, in which an apprehension of these higher beings or being is at -present secured, is as follows; and it bears a striking analogy to the -mode by which the self is cut out of a block of cubes. - -When we study a block of cubes, we first of all learn it, by starting -from a particular cube, and learning how all the others come with regard -to that. All the others are right or left, up or down, near or far, with -regard to that particular cube. And the line of cubes starting from this -first one, which we take as the direction in which we look, is, as it -were, an axis about which the rest of the cubes are grouped. We learn -the block with regard to this axis, so that we can mentally conceive the -disposition of every cube as it comes regarded from one point of view. -Next we suppose ourselves to be in another cube at the extremity of -another axis; and, looking from this axis, we learn the aspects of all -the cubes, and so on. - -Thus we impress on the feeling what the block of cubes is like from -every axis. In this way we get a knowledge of the block of cubes. - -Now, to get a knowledge of humanity, we must feel with many individuals. -Each individual is an axis as it were, and we must regard human beings -from many different axes. And as, in learning the block of cubes, -muscular action, as used in putting up the block of cubes, is the means -by which we impress on the feeling the different views of the block; so, -with regard to humanity, it is by acting with regard to the view of each -individual that a knowledge is obtained. That is to say, that, besides -sympathizing with each individual, we must act with regard to his view; -and acting so, we shall feel his view, and thus get to know humanity -from more than one axis. Thus there springs up a feeling of humanity, -and of more. - -Those who feel superficially with a great many people, are like those -learners who have a slight acquaintance with a block of cubes from many -points of view. Those who have some deep attachments, are like those who -know them well from one or two points of view. - -Thus there are two definite paths--one by which the instinctive feeling -is called out and developed, the other by which we gain the faculty of -rationally apprehending and learning the higher beings. - -In the one way it is by the exercise of a sympathetic and active life; -in the other, by the study of higher space. - -Both should be followed; but the latter way is more accessible to those -who are not good. For we at any rate have the industry to go through -mechanical operations, and know that we need something. - -And after all, perhaps, the difference between the good and the rest of -us, lies rather in the former being aware. There is something outside -them which draws them to it, which they see while we do not. - -There is no reason, however, why this knowledge should not become -demonstrable fact. Surely, it is only by becoming demonstrable fact that -the errors which have been necessarily introduced into it by human -weakness will fall away from it. - -The rational knowledge will not replace feeling, but will form the -vehicle by which the facts will be presented to our consciousness. Just -as we learn to know our fellows by watching their deeds,--but it is -something beyond the mere power of observing them that makes us regard -them,--so the higher existences need to be known; and, when known, 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 -another. - -And when we reflect on what surrounds us, when we think that the beauty -of fruit and flower, the blue depths of the sky, the majesty of rock and -ocean,--all these are but the chance and arbitrary view which we have of -true being,--then we can imagine somewhat of the glories that await our -coming. How set out in exquisite loveliness are all the budding trees -and hedgerows on a spring day--from here, where they almost sing to us -in their nearness, to where, in the distance, they stand up delicately -distant and distinct in the amethyst ocean of the air! And there, quiet -and stately, revolve the slow moving sun and the stars of the night. All -these are the fragmentary views which we have of great beings to whom we -are related, to whom we are linked, did we but realize it, by a bond of -love and service in close connexions of mutual helpfulness. - -Just as here and there on the face of a woman sits the divine spirit of -beauty, so that all cannot but love who look--so, presenting itself to -us in all this mingled scene of air and ocean, plain and mountain, is a -being of such loveliness that, did we but know with one accord in one -stream, all our hearts would be carried in a perfect and willing -service. It is not that we need to be made different; we have but to -look and gaze, and see that centre whereunto with joyful love all -created beings move. - -But not with effortless wonder will our days be filled, but in toil and -strong exertion; for, just as now we all labour and strive for an -object, our service is bound up with things which we do--so then we find -no rest from labour, but the sense of solitude and isolation is gone. -The bonds of brotherhood with our fellow-men grow strong, for we know -one common purpose. And through the exquisite face of nature shines the -spiritual light that gives us a great and never-failing comrade. - -Our task is a simple one--to lift from our mind that veil which somehow -has fallen on us, to take that curious limitation from our perception, -which at present is only transcended by inspiration. - -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 value. We too often sit as judges -of nature, when all we can be are her humble learners. We have but to -drink in of the inexhaustible fulness of being, pressing it close into -our minds, and letting our pride of being able to foretell vanish into -dust. - -There is a curious passage in the works of Immanuel Kant,[1] in which he -shows that space must be in the mind before we can observe things in -space. “For,” he says, “since everything we conceive is conceived as -being in space, there is nothing which comes before our minds from which -the idea of space can be derived; it is equally present in the most -rudimentary perception and the most complete.” Hence he says that space -belongs to the perceiving soul itself. Without going into this argument -to abstract regions, it has a great amount of practical truth. All our -perceptions are of things in space; we cannot think of any detail, -however limited or isolated, which is not in space. - - [1] The idea of space can “nicht aus den Verhältnissen der äusseren - Erscheinung durch Erfahrung erborgt sein, sondern diese äussere - Erfahrung ist nur durch gedachte Vorstellung allererst möglich.” - -Hence, in order to exercise our perceptive powers, it is well to have -prepared beforehand a strong apprehension of space and space relations. - -And so, as we pass on, is it not easily conceivable that, with our power -of higher space perception so rudimentary and so unorganized, we should -find it impossible to perceive higher existences? That mode of -perception which it belongs to us to exercise is wanting. What wonder, -then, that we cannot see the objects which are ready, were but our own -part done? - -Think how much has come into human life through exercising the power of -the three-dimensional space perception, and we can form some measure, in -a faint way, of what is in store for us. - -There is a certain reluctance in us in bringing anything, which before -has been a matter of feeling, within the domain of conscious reason. We -do not like to explain why the grass is green, flowers bright, and, -above all, why we have the feelings which we pass through. - -But this objection and instinctive reluctance is chiefly derived from -the fact that explaining has got to mean explaining away. We so often -think that a thing is explained, when it can be shown simply to be -another form of something which we know already. And, in fact, the -wearied mind often does long to have a phenomenon shown to be merely a -deduction from certain known laws. - -But explanation proper is not of this kind; it is introducing into the -mind the new conception which is indicated by the phenomenon already -present. Nature consists of many entities towards the apprehension of -which we strive. If for a time we break down the bounds which we have -set up, and unify vast fields of observation under one common law, it is -that the conceptions we formed at first are inadequate, and must be -replaced by greater ones. But it is always the case, that, to understand -nature, a conception must be formed in the mind. This process of growth -in the mental history is hidden; but it is the really important one. The -new conception satisfies more facts than the old ones, is truer -phenomenally; and the arguments for it are its simplicity, its power of -accounting for many facts. But the conception has to be formed first. -And the real history of advance lies in the growth of the new -conceptions which every now and then come to light. - -When the weather-wise savage looked at the sky at night, he saw many -specks of yellow light, like fire-flies, sprinkled amidst whitish -fleece; and sometimes the fleece remained, the fire-spots went, and rain -came; sometimes the fire-spots remained, and the night was fine. He did -not see that the fire-points were ever the same, the clouds different; -but by feeling dimly, he knew enough for his purpose. - -But when the thinking mind turned itself on these appearances, there -sprang up,--not all at once, but gradually,--the knowledge of the -sublime existences of the distant heavens, and all the lore of the -marvellous forms of water, of air, and the movements of the earth. -Surely these realities, in which lies a wealth of embodied poetry, are -well worth the delighted sensuous apprehension of the savage as he -gazed. - -Perhaps something is lost, but in the realities, of which we know, there -is compensation. And so, when we learn to understand the meaning of -these mysterious changes, this course of natural events, we shall find -in the greater realities amongst which we move a fair exchange for the -instinctive reverence, which they now awaken in us. - -In this book the task is taken up of forming the most simple and -elementary of the great conceptions that are about us. In the works of -the poets, and still more in the pages of religious thinkers, lies an -untold wealth of conception, the organization of which in our every-day -intellectual life is the work of the practical educator. - -But none is capable of such simple demonstration and absolute -presentation as this of higher space, and none so immediately opens our -eyes to see the world as a different place. And, indeed, it is very -instructive; for when the new conception is formed, it is found to be -quite simple and natural. We ask ourselves what we have gained; and we -answer: Nothing; we have simply removed an obvious limitation. - -And this is universally true; it is not that we must rise to the higher -by a long and laborious process. We may have a long and laborious -process to go through, but, when we find the higher, it is this: we -discover our true selves, our essential being, the fact of our lives. In -this case, we pass from the ridiculous limitation, to which our eyes -and hands seem to be subject, of acting in a mere section of space, to -the fuller knowledge and feeling of space as it is. How do we pass to -this truer intellectual life? Simply by observing, by laying aside our -intellectual powers, and by looking at what is. - -We take that which is easiest to observe, not that which is easiest to -define; we take that which is the most definitely limited real thing, -and use it as our touchstone whereby to explore nature. - -As it seems to me, Kant made the great and fundamental statement in -philosophy when he exploded all previous systems, and all physics were -reft from off the perceiving soul. But what he did once and for all, was -too great to be a practical means of intellectual work. The dynamic form -of his absolute insight had to be found; and it is in other works that -the practical instances of the Kantian method are to be found. For, -instead of looking at the large foundations of knowledge, the ultimate -principles of experience, late writers turned to the details of -experience, and tested every phenomenon, not with the question, What is -this? but with the question, “What makes me perceive thus?” - -And surely the question, as so put, is more capable of an answer; for it -is only the percipient, as a subject of thought, about which we can -speak. The absolute soul, since it is the thinker, can never be the -subject of thought; but, as physically conditioned, it can be thought -about. Thus we can never, without committing a ludicrous error, think of -the mind of man except as a material organ of some kind; and the path of -discovery lies in investigating what the devious line of his thought -history is due to, which winds between two domains of physics--the -unknown conditions which affect the perceiver, the partially known -physics which constitute what we call the external world. - -It is a pity to spend time over these reflections; if they do not seem -tame and poor compared to the practical apprehension which comes of -working with the models, then there is nothing in the whole subject. 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 -thought, then all these words are not only useless, but false. If, on -the other hand, there is true work to be done with them, then these -suggestions will be felt to be but mean and insufficient apprehensions. - -For, in the simplest apprehension of a higher space lies a knowledge of -a reality which is, to the realities we know, as spirit is to matter; -and yet to this new vision all our solid facts and material conditions -are but as a shadow is to that which casts it. In the awakening light of -this new apprehension, the flimsy world quivers and shakes, rigid solids -flow and mingle, all our material limitations turn into graciousness, -and the new field of possibility waits for us to look and behold. - - -CHAPTER XI. - -SPACE THE SCIENTIFIC BASIS OF ALTRUISM AND RELIGION. - -The reader will doubtless ask for some definite result corresponding to -these words--something not of the nature of an hypothesis or a might-be. -And in that I can only satisfy him after my own powers. My only strength -is in detail and patience; and if he will go through the practical part -of the book, it will assuredly dawn upon him that here is the beginning -of an answer to his request. I only study the blocks and stones of the -higher life. But here they are definite enough. And the more eager he is -for personal and spiritual truth, the more eagerly do I urge him to take -up the practical work, for the true good comes to us through those who, -aspiring greatly, still submit their aspirations to fact, and who, -desiring to apprehend spirit, still are willing to manipulate matter. - -The particular problem at which I have worked for more than ten years, -has been completely solved. It is possible for the mind to acquire a -conception of higher space as adequate as that of our three-dimensional -space, and to use it in the same manner. - -There are two distinct ways of studying space--our familiar space at -present in use. One is that of the analyst, who treats space relations -by his algebra, and discovers marvellous relations. The other is that of -the observer or mechanician, who studies the shapes of things in space -directly. - -A practical designer of machines would not find the knowledge of -geometrical analysis of immediate help to him; and an artist or -draughtsman still less so. - -Now, my inquiry was, whether it was possible to get the same power of -conception of four-dimensional space, as the designer and draughtsman -have of three-dimensional space. It is possible. - -And with this power it is possible for us to design machines in higher -space, and to conceive objects in this space, just as a draughtsman or -artist does. - -Analytical skill is not of much use in designing a statue or inventing a -machine, or in appreciating the detail of either a work of art or a -mechanical contrivance. - -And hitherto the study of four-dimensional space has been conducted by -analysis. Here, for the first time, the fact of the power of conception -of four-dimensional space is demonstrated, and the means of educating it -are given. - -And I propose a complete system of work, of which the volume on four -space[2] is the first instalment. - - [2] “Science Romance,” No. I., by C. H. Hinton. Published by Swan - Sonnenschein & Co. - -I shall bring forward a complete system of four-dimensional -thought--mechanics, science, and art. The necessary condition is, that -the mind acquire the power of using four-dimensional space as it now -does three-dimensional. - -And there is another condition which is no less important. We can never -see, for instance, four-dimensional pictures with our bodily eyes, but -we can with our mental and inner eye. The condition is, that we should -acquire the power of mentally carrying a great number of details. - -If, for instance, we could think of the human body right down to every -minute part in its right position, and conceive its aspect, we should -have a four-dimensional picture which is a solid structure. Now, to do -this, we must form the habit of mental painting, that is, of putting -definite colours in definite positions, not with our hands on paper, but -with our minds in thought, so that we can recall, alter, and view -complicated arrangements of colour existing in thought with the same -ease with which we can paint on canvas. This is simply an affair of -industry; and the mental power latent in us in this direction is simply -marvellous. - -In any picture, a stroke of the brush put on without thought is -valueless. The artist is not conscious of the thought process he goes -through. For our purpose it is necessary that the manipulation of colour -and form which the artist goes through unconsciously, should become a -conscious power, and that, at whatever sacrifice of immediate beauty, -the art of mental painting should exist beside our more unconscious art. -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 heart, so that, -just as an artist thinks over the outlines of a figure he wants to draw, -so we think over each stroke in our pictures. The means by which this -can be done will be given in a future volume. - -We throw ourselves on an enterprise in which we have to leave altogether -the direct presentation to the senses. 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 -see. We have a vast work of organization, but it is merely organization. -The power really exists and shows itself when it is looked for. - -Much fault may be found with the system of organization which I have -adopted, but it is the survivor of many attempts; and although I could -better it in parts, still I think it is best to use it until, the full -importance of the subject being realized, it will be the lifework of men -of science to reorganize the methods. - -The one thing on which I must insist is this--that knowledge is of no -value, it does not exist unless it comes into the mind. To know that a -thing must be is no use at all. It must be clearly realized, and in -detail as it _is_, before it can be used. - -A whole world swims before us, the apprehension of which simply demands -a patient cultivation of our powers; and then, when the faculty is -formed, we shall recognize what the universe in which we are is like. We -shall learn about ourselves and pass into a new domain. - -And I would speak to some minds who, like myself, share to a large -extent the feeling of unsettledness and unfixedness of our present -knowledge. - -Religion has suffered in some respects from the inaccuracy of its -statements; 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 us, and to our own bodies; another, a set of rules as to our -relationship to beings higher than ourselves. - -Now, on the former of these subjects, on physical facts, on the laws of -health, science has a fair standing ground of criticism, and can correct -the religious doctrines in many important respects. - -But on the other part of the subject matter, as to our relationship to -beings higher than ourselves, science has not yet the materials for -judging. The proposition which underlies this book is, that we should -begin to acquire the faculties for judging. - -To judge, we must first appreciate; and how far we are from appreciating -with science the fundamental religious doctrines I leave to any one to -judge. - -There is absolutely no scientific basis for morality, 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 being. Science does not -give us any information which is not equally acceptable to the most -selfish and most generous man; it simply tells him of means by which he -may attain his own ends, it does not show him ends. - -The prosecution of science is an ennobling pursuit; but it is of -scientific knowledge that I am now speaking in itself. We have no -scientific knowledge of any existences higher than ourselves--at least, -not recognized as higher. But we have abundant knowledge of the actions -of beings less developed than ourselves, from the striking unanimity -with which all inorganic beings tend to move towards the earth’s centre, -to the almost equally uniform modes of response in elementary organized -matter to different stimuli. - -The question may be put: In what way do we come into contact with these -higher beings at present? And evidently the answer is, In those ways in -which we tend to form organic unions--unions in which the activities of -individuals coalesce in a living way. - -The coherence of a military empire or of a subjugated population, -presenting no natural nucleus of growth, is not one through which we -should hope to grow into direct contact with our higher destinies. But -in friendship, in voluntary associations, and above all, in the family, -we tend towards our greater life. - -And it seems that the instincts of women are much more relative to this, -the most fundamental and important side of life, than are those of men. -In fact, until we know, the line of advance had better be left to the -feeling of women, as they organize the home and the social life -spreading out therefrom. It is difficult, perhaps, for a man to be -still and perceive; but if he is so, he finds that what, when thwarted, -are meaningless caprices and empty emotionalities, are, on the part of -woman, when allowed to grow freely and unchecked, the first beginnings -of a new life--the shadowy filaments, as it were, by which an organism -begins to coagulate together from the medium in which it makes its -appearance. - -In very many respects men have to make the conditions, and then learn to -recognize. How can we see the higher beings about us, when we cannot -even conceive the simplest higher shapes? We may talk about space, and -use big words, but, after all, the preferable way of putting our efforts -is this: let us look first at the simplest facts of higher existence, -and then, when we have learnt to realize these, We shall be able to see -what the world presents. And then, also, light will be thrown on the -constituent organisms of our own bodies, 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 frame. - -The problem, as it comes to me, is this: 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 done, but things -degrading and vile, so that they serve. - -I am to sign any list of any number of deeds which the most foul -imagination can suggest, as things which I would do did the occasion -come when I could benefit another by doing them; and, in fact, 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 sake. -And I believe that the soul is absolutely unstained by the action, -provided the regard is for another. - -But this is, in truth, a dangerous doctrine; at one Sweep it puts away -all absolute commandments, all absolute verdicts of right about things, -and leaves the agent to his own judgment. - -It is a kind of rule of life which requires most absolute openness, and -demands that society should frame severe and insuperable regulations; -for otherwise, with the motives of the individual thus liberated from -absolute law, endless varieties of conduct would spring forth, and the -wisdom of individual men is hardly enough to justify their irresponsible -action. - -Still, it does seem that, as an ideal, the absolute absence of -self-regard is to be aimed at. - -With a strong religious basis, this would work no harm, for the rules of -life, as laid down by religions, would suffice. But there are many who -do not accept these rules as any absolute indication of the will of God, -but only as the regulations of good men, which have a claim to respect -and nothing more. - -And thus it seems to me that altruism--thoroughgoing altruism--hands -over those who regard it as an ideal, and who are also of a sceptical -turn of mind, to the most absolute unfixedness of theory, and, very -possibly, to the greatest errors in life. - -And here we come to the point where the study of space becomes so -important. - -For if this rule of altruism is the right one, if it appeals with a -great invitation to us, we need not therefore try it with less -precaution than we should use in other affairs of infinitely less -importance. When we want to know if a plank will bear, we entrust it -with a different load from that of a human body. - -And if this law of altruism is the true one, let us try it where failure -will not mean the ruin of human beings. - -Now, in knowledge, pure altruism means so to bury the mind in the thing -known that all particular relations of one’s self pass away. The -altruistic knowledge of the heavens would be, to feel that the stars -were vast bodies, and that I am moving rapidly. It would be, to know -this, not as a matter of theory, but as a matter of habitual feeling. - -Whether this is possible, I do not know; but a somewhat similar attempt -can be made with much simpler means. - -In a different place I have described the process of acquiring an -altruistic knowledge of a block of cubes; and the results of the -laborious processes involved are well worth the trouble. For as a -clearly demonstrable fact this comes before one. To acquire an absolute -knowledge of a block of cubes, so that all self relations are cast out, -means that one has to take the view of a higher being. - -It suddenly comes before one, that the particular relations which are so -fixed and important, and seem so absolutely sure when one begins the -process of learning, are by no means absolute facts, but marks of a -singular limitation, almost a degradation, on one’s own part. In the -determined attempt to know the most insignificant object perfectly and -thoroughly, there flashes before one’s eyes an existence infinitely -higher than one’s own. And with that vision there comes,--I do not speak -from my own experience only,--a conviction that our existence also is -not what we suppose--that this bodily self of ours is but a limit too. -And the question of altruism, as against self-regard, seems almost to -vanish, for by altruism we come to know what we truly are. - -“What we truly are,” I do not mean apart from space and matter, but what -we really are as beings having a space existence; for our way of -thinking about existence is to conceive it as the relations of bodies in -space. To think is to conceive realities in space. - -Just as, to explore the distant stars of the heavens, a particular -material arrangement is necessary which we call a telescope, so to -explore the nature of the beings who are higher than us, a mental -arrangement is necessary. We must prepare our power of thinking as we -prepare a more extended power of looking. We want a structure developed -inside the skull for the one purpose, while an exterior telescope will -do for the other. - -And thus it seems that the difficulties which we first apprehended fall -away. - -To us, looking with half-blinded eyes at merely our own little slice of -existence, our filmy all, it seemed that altruism meant disorder, -vagary, danger. - -But when we put it into practice in knowledge, 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--nay, a consciousness comes that we are -higher than we know. - -And so with our moral life as with our intellectual life. Is it not the -case that those, who truly accept the rule of altruism, learn life in -new dangerous ways? - -It is true that we must give up the precepts of religion as being the -will of God; but then we shall learn that the will of God shows itself -partly in the religious precepts, and comes to be more fully and more -plainly known as an inward spirit. - -And that difficulty, too, about what we may do and what we may not, -vanishes also. For, if it is the same about our fellow-creatures as it -is about the block of cubes, when we have thrown out the self-regard -from our relationship to them, we shall feel towards them as a higher -being than man feels towards them, we shall feel towards them as they -are in their true selves, not in their outward forms, but as eternal -loving spirits. - -And then those instincts which humanity feels with a secret impulse to -be sacred and higher than any temporary good will be justified--or -fulfilled. - -There are two tendencies--one towards the direct cultivation of the -religious perceptions, the other to reducing everything to reason. It -will be but just for the exponents of the latter tendency to look at the -whole universe, not the mere section of it which we know, before they -deal authoritatively with the higher parts of religion. - -And those who feel the immanence of a higher life in us will be needed -in this outlook on the wider field of reality, so that they, being -fitted to recognize, may tell us what lies ready for us to know. - -The true path of wisdom consists in seeing that our intellect is -foolishness--that our conclusions are absurd and mistaken, 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 existences. To think of -ourselves as any other than things in space and subject to material -conditions, is absurd, it is absurd on either of two hypotheses. If we -are really things in space, then of course it is absurd to think of -ourselves as if we were not so. On the other hand, if we are not things -in space, then conceiving in space is the mode in which that unknown -which we are exists as a mind. Its mental action is space-conception, -and then to give up the idea of ourselves as in space, is not to get a -truer idea, but to lose the only power of apprehension of ourselves -which we possess. - -And yet there is, it must be confessed, one way in which it may be -possible for us to think without thinking of things in space. - -That way is, not to abandon the use of space-thought, but to pass -through it. - -When we think of space, we have to think of it as infinity extended, and -we have to think of it as of infinite dimensions. Now, as I have shown -in “The Law of the Valley,”[3] when we come upon infinity in any mode of -our thought, it is a sign that that mode of thought is dealing with a -higher reality than it is adapted for, and in struggling to represent -it, can only do so by an infinite number of terms. Now, space has an -infinite number of positions and turns, and this may be due to the -attempt forced upon us to think of things higher than space as in space. -If so, then the way to get rid of space from our thoughts, is, not to go -away from it, but to pass through it--to think about larger and larger -systems of space, and space of more and more dimensions, till at last we -get to such a representation in space of what is higher than space, 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 is. - - [3] “Science Romances,” No. II. - -Again and again has human nature aspired and fallen. The vision has -presented itself of a law which was love, a duty which carried away the -enthusiasm, and in which the conflict of the higher and lower natures -ceased because all was enlisted in one loving service. But again and -again have such attempts failed. The common-sense view, that man is -subject to law, external law, remains--that there are fates whom he must -propitiate and obey. And there is a strong sharp curb, which, if it be -not brought to bear by the will, is soon pulled tight by the world, and -one more tragedy is enacted, and the over-confident soul is brought low. - -And the rock on which such attempts always split, is in the indulgence -of some limited passion. Some one object fills the soul with its image, -and in devotion to that, other things are sacrificed, until at last all -comes to ruin. - -But what does this mean? Surely it is simply this, that where there -should be knowledge there is ignorance. It is not that there is too much -devotion, too much passion, but that we are ignorant and blind, and -wander in error. We do not know what it is we care for, and waste our -effort on the appearance. There is no such thing as wrong love; there is -good love and bad knowledge, and men who err, clasp phantoms to -themselves. Religion is but the search for realities; and thought, -conscious of its own limitations, is its best aid. - -Let a man care for any one object--let his regard for it be as -concentrated and exclusive as you will, there will be no danger if he -truly apprehends that which he cares for. Its true being is bound up -with all the rest of existence, and, if his regard is true to one, then, -if that one is really known, his regard is true to all. - -There is a question sometimes asked, which shows the mere formalism into -which we have fallen. - -We ask: What is the end of existence? A mere play on words! For to -conceive existence is to feel ends. The knowledge of existence is the -caring for objects, the fear of dangers, the anxieties of love. Immersed -in these, the triviality of the question, what is the end of existence? -becomes obvious. If, however, letting reality fade away, we play with -words, some questions of this kind are possible; but they are mere -questions of words, and all content and meaning has passed out of them. - -The task before us is this: we strive to find out that physical unity, -that body which men are parts of, and in the life of which their true -unity lies. The existence of this one body we know from the utterances -of those whom we cannot but feel to be inspired; we feel certain -tendencies in ourselves which cannot be explained except by a -supposition of this kind. - -And, now, we set to work deliberately to form in our minds the means of -investigation, the faculty of higher-space conception. To our ordinary -space-thought, men are isolated, distinct, in great measure -antagonistic. But with the first use of the weapon of higher thought, it -is easily seen that all men may really be members of one body, their -isolation may be but an affair of limited consciousness. There is, of -course, no value as science in such a supposition. But it suggests to us -many possibilities; it reveals to us the confined nature of our present -physical views, and stimulates us to undertake the work necessary to -enable us to deal adequately with the subject. - -The work is entirely practical and detailed; it is the elaboration, -beginning from the simplest objects of an experience in thought, of a -higher-space world. - -To begin it, we take up those details of position and relation which are -generally relegated to symbolism or unconscious apprehension, and bring -these waste products of thought into the central position of the -laboratory of the mind. We turn all our attention on the most simple and -obvious details of our every-day experience, and thence we build up a -conception of the fundamental facts of position and arrangement in a -higher world. We next study more complicated higher shapes, and get our -space perception drilled and disciplined. Then we proceed to put a -content into our framework. - -The means of doing this are twofold--observation and inspiration. - -As to observation, it is hardly possible to describe the feelings of -that investigator who shall distinctly trace in the physical world, and -experimentally demonstrate the existence of the higher-space facts which -are so curiously hidden from us. He will lay the first stone for the -observation and knowledge of the higher beings to whom we are related. - -As to the other means, it is obvious, surely, that if there has ever -been inspiration, there is inspiration now. Inspiration is not a unique -phenomenon. It has existed in absolutely marvellous degree in some of -the teachers of the ancient world; but that, whatever it was, which they -possessed, must be present now, and, if we could isolate it, be a -demonstrable fact. - -And I would propose to define inspiration as the faculty, which, to take -a particular instance, does the following:-- - -If a square penetrates a line cornerwise, it marks out on the line a -segment bounded by two points--that is, we suppose a line drawn on a -piece of paper, and a square lying on the paper to be pushed so that its -corner passes over the line. Then, supposing the paper and the line to -be in the same plane, the line is interrupted by the square; and, of the -square, all that is observable in the line, is a segment bounded by two -points. - -Next, suppose a cube to be pushed cornerwise through a plane, and let -the plane make a section of the cube. The section will be a plane -figure, and it will be a triangle. - -Now, first, the section of a square by a line is a segment bounded by -two points; second, the section of a cube by a plane is a triangle -bounded by three lines. - -Hence, we infer that the section of a figure in four dimensions -analogous to a cube, by three-dimensional space, will be a -tetrahedron--a figure bounded by four planes. - -This is found to be true; with a little familiarity with -four-dimensional movements this is seen to be obvious. But I would -define inspiration as the faculty by which without actual experience -this conclusion is formed. - -How it is we come to this conclusion I am perfectly unable to say. -Somehow, looking at mere formal considerations, there comes into the -mind a conclusion about something beyond the range of actual experience. - -We may call this reasoning from analogy; but using this phrase does not -explain the process. 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 consciousness, as Plato shows with the boy’s knowledge of the cube. -We must be really four-dimensional creatures, or we could not think -about four dimensions. - -But whatever name we give to this peculiar and inexplicable faculty, -that we do possess it is certain; and in our investigations it will be -of service to us. We must carefully investigate existence in a plane -world, and then, making sure, and impressing on our inward sense, as we -go, every step we take with regard to a higher world, we shall be -reminded continually of fresh possibilities of our higher existence. - - - - -PART II. - - -CHAPTER I. - -THREE-SPACE. GENESIS OF A CUBE. APPEARANCES OF A CUBE TO A PLANE-BEING. - -The models consist of a set of eight and a set of four cubes. They are -marked with different colours, so as to show the properties of the -figure in Higher Space, to which they belong. - -The simplest figure in one-dimensional space, that is, in a straight -line, is a straight line bounded at the two extremities. The figure in -this case consists of a length bounded by two points. - -Looking at Cube 1, and placing it so that the figure 1 is uppermost, we -notice a straight line in contact with the table, which is coloured -Orange. It begins in a Gold point and ends in a Fawn point. The Orange -extends to some distance on two faces of the Cube; but for our present -purpose we suppose it to be simply a thin line. - -This line we conceive to be generated in the following way. Let a point -move and trace out a line. Let the point be the Gold point, and let it, -moving, trace out the Orange line and terminate in the Fawn point. Thus -the figure consists of the point at which it begins, the point at which -it ends, and the portion between. We may suppose the point to start as a -Gold point, to change its colour to Orange during the motion, and when -it stops to become Fawn. The motion we suppose from left to right, and -its direction we call X. - -If, now, this Orange line move away from us at right angles, it will -trace out a square. Let this be the Black square, which is seen -underneath Model 1. The points, which bound the line, will during this -motion trace out lines, and to these lines there will be terminal -points. Also, the Square will be terminated by a line on the opposite -side. Let the Gold point in moving away trace out a Blue line and end in -a Buff point; the Fawn point a Crimson line ending in a Terracotta -point. The Orange line, having traced a Black square, ends in a -Green-grey line. This direction, away from the observer, we call Y. - -Now, let the whole Black square traced out by the Orange line move -upwards at right angles. It will trace out a new figure, a Cube. And the -edges of the square, while moving upwards, will trace out squares. -Bounding the cube, and opposite to the Black square, will be another -square. Let the Orange line moving upwards trace a Dark Blue square and -end in a Reddish line. The Gold point traces a Brown line; the Fawn -point traces a French-grey line, and these lines end in a Light-blue and -a Dull-purple point. Let the Blue line trace a Vermilion square and end -in a Deep-yellow line. Let the Buff point trace a Green line, and end in -a Red point. The Green-grey line traces a Light-yellow square and ends -in a Leaden line; the Terracotta point traces a Dark-slate line and ends -in a Deep-blue point. The Crimson line traces a Blue-green square and -ends in a Bright-blue line. - -Finally, the Black square traces a Cube, the colour of which is -invisible, and ends in a white square. We suppose the colour of the cube -to be a Light-buff. The upward direction we call Z. Thus we say: The -Gold point moved Z, traces a Brown line, and ends in a Light-blue point. - -We can now clearly realize and refer to each region of the cube by a -colour. - -At the Gold point, lines from three directions meet, the X line Orange, -the Y line Blue, the Z line Brown. - -Thus we began with a figure of one dimension, a line, we passed on to a -figure of two dimensions, a square, and ended with a figure of three -dimensions, a cube. - - * * * * * - -The square represents a figure in two dimensions; but if we want to -realize what it is to a being in two dimensions, we must not look down -on it. Such a view could not be taken by a plane-being. - -Let us suppose a being moving on the surface of the table and unable to -rise from it. Let it not know that it is upon anything, but let it -believe that the two directions and compounds of those two directions -are all possible directions. Moreover, let it not ask the question: “On -what am I supported?” Let it see no reason for any such question, but -simply call the smooth surface, along which it moves, Space. - -Such a being could not tell the colour of the square traced by the -Orange line. The square would be bounded by the lines which surround it, -and only by breaking through one of those lines could the plane-being -discover the colour of the square. - -In trying to realize the experience of a plane-being it is best to -suppose that its two dimensions are upwards and sideways, _i.e._, Z and -X, because, if there be any matter in the plane-world, it will, like -matter in the solid world, exert attractions and repulsions. The matter, -like the beings, must be supposed very thin, that is, of so slight -thickness that it is quite unnoticed by the being. Now, if there be a -very large mass of such matter lying on the table, and a plane-being be -free to move about it, he will be attracted to it in every direction. -“Towards this huge mass” would be “Down,” and “Away from it” would be -“Up,” just as “Towards the earth” is to solid beings “Down,” and “Away -from it” is “Up,” at whatever part of the globe they may be. Hence, if -we want to realize a plane-being’s feelings, we must keep the sense of -up and down. Therefore we must use the Z direction, and it is more -convenient to take Z and X than Z and Y. - -Any direction lying between these is said to be compounded of the two; -for, if we move slantwise for some distance, the point reached might -have been also reached by going a certain distance X, and then a certain -distance Z, or _vice versâ_. - -Let us suppose the Orange line has moved Z, and traced the Dark-blue -square ending in the Reddish line. If we now place a piece of stiff -paper against the Dark-blue square, and suppose the plane-beings to move -to and fro on that surface of the paper, which touches the square, we -shall have means of representing their experience. - -To obtain a more consistent view of their existence, let us suppose the -piece of paper extended, so that it cuts through our earth and comes out -at the antipodes, thus cutting the earth in two. Then suppose all the -earth removed away, both hemispheres vanishing, and only a very thin -layer of matter left upon the paper on that side which touches the -Dark-blue square. This represents what the world would be to a -plane-being. - -It is of some importance to get the notion of the directions in a -plane-world, as great difficulty arises from our notions of up and down. -We miss the right analogy if we conceive of a plane-world without the -conception of up and down. - -A good plan is, to use a slanting surface, a stiff card or book cover, -so placed that it slopes upwards to the eye. Then gravity acts as two -forces. It acts (1) as a force pressing all particles upon the slanting -surface into it, and (2) as a force of gravity along the plane, making -particles tend to slip down its incline. We may suppose that in a -plane-world there are two such forces, one keeping the beings thereon to -the plane, the other acting between bodies in it, 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 it. In both cases the larger draws the smaller to itself, and -creates the sensations of up and down. - -If we hold the cube so that its Dark-blue side touches a sheet of paper -held upwards to the eye, and if we then look straight down along the -paper, confining our view to that which is in actual contact with the -paper, we see the same view of the cube as a plane-being would get. We -see a Light-blue point, a Reddish line, and a Dull-purple point. The -plane-being only sees a line, just as we only see a square of the cube. - -The line where the paper rests on the table may be taken as -representative of the surface of the plane-being’s earth. It would be -merely a line to him, but it would have the same property in relation to -the plane-world, as a square has in relation to a solid world; in -neither case can the notion of what in the latter is termed solidity be -quite excluded. If the plane-being broke through the line bounding his -earth, he would find more matter beyond it. - -Let us now leave out of consideration the question of “up and down” in -a plane-world. Let us no longer consider it in the vertical, or ZX, -position, but simply take the surface (XY) of the table as that which -supports a plane-world. Let us represent its inhabitants by thin pieces -of paper, which are free to move over the surface of the table, but -cannot rise from it. Also, let the thickness (_i.e._, height above the -surface) of these beings be so small that they cannot discern it. Lastly -let us premise there is no attraction in their world, so that they have -not any up and down. - -Placing Cube 1 in front of us, let us now ask how a plane-being could -apprehend such a cube. The Black face he could easily study. He would -find it bounded by Gold point, Orange line, Fawn point, Crimson line, -and so on. And he would discover it was Black by cutting through any of -these lines and entering it. (This operation would be equivalent to the -mining of a solid being). - -But of what came above the Black square he would be completely ignorant. -Let us now suppose a square hole to be made in the table, so that the -cube could pass through, and let the cube fit the opening so exactly -that no trace of the cutting of the table be visible to the plane-being. -If the cube began to pass through, it would seem to him simply to -change, for of its motion he could not be aware, as he would not know -the direction in which it moved. Let it pass down till the White square -be just on a level with the surface of the table. The plane-being would -then perceive a Light-blue point, a Reddish line, a Dull-purple point, a -Bright-blue line, and so on. These would surround a White square, which -belonged to the same body as that to which the Black square belonged. -But in this body there would be a dimension, which was not in the -square. Our upward direction would not be apprehended by him directly. -Motion from above downwards would only be apprehended as a change in the -figure before him. He would not say that he had before him different -sections of a cube, but only a changing square. If he wanted to look at -the upper square, he could only do so when the Black square had gone an -inch below his plane. To study the upper square simultaneously with the -lower, he would have to make a model of it, and then he could place it -beside the lower one. - -Looking at the cube, we see that the Reddish line corresponds precisely -to the Orange line, and the Deep-yellow to the Blue line. But if the -plane-being had a model of the upper square, and placed it on the -right-hand side of the Black square, the Deep-yellow line would come -next to the Crimson line of the Black square. There would be a -discontinuity about it. All that he could do would be to observe which -part in the one square corresponded to which part in the other. -Obviously too there lies something between the Black square and the -White. - -The plane-being would notice that when a line moves in a direction not -its own, it traces out a square. When the Orange line is moved away, it -traces out the Black square. The conception of a new direction thus -obtained, he would understand that the Orange line moving so would trace -out a square, and the Blue line moving so would do the same. To us these -squares are visible as wholes, the Dark-blue, and the Vermilion. To him -they would be matters of verbal definition rather than ascertained -facts. However, given that he had the experience of a cube being pushed -through his plane, he would know there was some figure, whereof his -square was part, which was bounded by his square on one side, and by a -White square on another side. We have supposed him to make models of -these boundaries, a Black square and a White square. The Black square, -which is his solid matter, is only one boundary of a figure in Higher -Space. - -But we can suppose the cube to be presented to him otherwise than by -passing through his plane. It can be turned round the Orange line, in -which case the Blue line goes out, and, after a time, the Brown line -comes in. It must be noticed that the Brown line comes into a direction -opposite to that in which the Blue line ran. These two lines are at -right angles to each other, and, if one be moved upwards till it is at -right angles to the surface of the table, the other comes on to the -surface, but runs in a direction opposite to that in which the first -ran. Thus, by turning the cube about the Orange line and the Blue line, -different sides of it can be shown to a plane-being. By combining the -two processes of turning and pushing through the plane, all the sides -can be shown to the plane-being. For instance, if the cube be turned so -that the Dark-blue square be on the plane, and it be then passed -through, the Light-yellow square will come in. - -Now, if the plane-being made a set of models of these different -appearances and studied them, he could form some rational idea of the -Higher Solid which produced them. He would become able to give some -consistent account of the properties of this new kind of existence; he -could say what came into his plane space, if the other space penetrated -the plane edge-wise or corner-wise, and could describe all that would -come in as it turned about in any way. - -He would have six models. Let us consider two of them--the Black and the -White squares. We can observe them on the cube. Every colour on the one -is different from every colour on the other. If we now ask what lies -between the Orange line and the Reddish line, we know it is a square, -for the Orange line moving in any direction gives a square. And, if the -six models were before the plane-being, he could easily select that -which showed what he wanted. For that which lies between Orange line and -Reddish line must be bounded by Orange and Reddish lines. He would -search among the six models lying beside each other on his plane, till -he found the Dark-blue square. It is evident that only one other square -differs in all its colours from the Black square, viz., the White -square. For it is entirely separate. The others meet it in one of their -lines. This total difference exists in all the pairs of opposite -surfaces on the cube. - -Now, suppose the plane-being asked himself what would appear if the cube -turned round the Blue line. The cube would begin to pass through his -space. The Crimson line would disappear beneath the plane and the -Blue-green square would cut it, so that opposite to the Blue line in the -plane there would be a Blue-green line. The French-grey line and the -Dark-slate line would be cut in points, and from the Gold point to the -French-grey point would be a Dark-blue line; and opposite to it would be -a Light-yellow line, from the Buff point to the Dark-slate point. Thus -the figure in the plane world would be an oblong instead of a square, -and the interior of it would be of the same Light-buff colour as the -interior of the cube. It is assumed that the plane closes up round the -passing cube, as the surface of a liquid does round any object immersed. - -[Illustration: Fig. 1.] - -[Illustration: Fig. 2.] - -[Illustration: Fig. 3.] - -[Illustration: Fig. 4.] - -[Illustration: Fig. 5.] - -But, in order to apprehend what would take place when this twisting -round the Blue line began, the plane-being would have to set to work by -parts. He has no conception of what a solid would do in twisting, but he -knows what a plane does. Let him, then, instead of thinking of the -whole Black square, think only of the Orange line. The Dark-blue square -stands on it. As far as this square is concerned, twisting round the -Blue line is the same as twisting round the Gold point. Let him imagine -himself in that plane at right angles to his plane-world, which contains -the Dark-blue square. Let him keep his attention fixed on the line where -the two planes meet, viz., that which is at first marked by the Orange -line. We will call this line the line of his plane, for all that he -knows of his own plane is this line. Now, let the Dark-blue square turn -round the Gold point. The Orange line at once dips below the line of his -plane, and the Dark-blue square passes through it. Therefore, in his -plane he will see a Dark-blue line in place of the Orange one. And in -place of the Fawn point, only further off from the Gold point, will be a -French-grey point. The Diagrams (1), (2) show how the cube appears as it -is before and after the turning. G is the Gold, F the Fawn point. In (2) -G is unmoved, and the plane is cut by the French-grey line, Gr. - -Instead of imagining a direction he did not know, the plane-being could -think of the Dark-blue square as lying in his plane. But in this case -the Black square would be out off his plane, and only the Orange line -would remain in it. Diagram (3) shows the Dark-blue square lying in his -plane, and Diagram (4) shows it turning round the Gold point. Here, -instead of thinking about his plane and also that at right angles to it, -he has only to think how the square turning round the Gold point will -cut the line, which runs left to right from G, viz., the dotted line. -The French-grey line is cut by the dotted line in a point. To find out -what would come in at other parts, 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 square. Every such section -would turn round a point, as the whole cube turned round the Blue line. -Thus he would treat the cube as a number of squares by taking parallel -sections from the Dark-blue to the Light-yellow square, and he would -turn each of these round a corner of the same colour as the Blue line. -Combining these series of appearances, he would discover what came into -his plane as the cube turned round the Blue line. Thus, the problem of -the turning of the cube could be settled by the consideration of the -turnings of a number of squares. - -As the cube turned, a number of different appearances would be presented -to the plane-being. The Black square would change into a Light-buff -oblong, with Dark-blue, Blue-green, Light-yellow, and Blue sides, and -would gradually elongate itself until it became as long as the diagonal -of the square side of the cube; and then the bounding line opposite to -the Blue line would change from Blue-green to Bright-blue, the other -lines remaining the same colour. If the cube then turned still further, -the Bright-blue line would become White, and the oblong would diminish -in length. It would in time become a Vermilion square, with a -Deep-yellow line opposite to the Blue line. It would then pass wholly -below the plane, and only the Blue line would remain. - -If the turning were continued till half a revolution had been -accomplished, the Black square would come in again. But now it would -come up into the plane from underneath. It would appear as a Black -square exactly similar to the first; but the Orange line, instead of -running left to right from Gold point, would run right to left. The -square would be the same, only differently disposed with regard to the -Blue line. It would be the looking-glass image of the first square. -There would be a difference in respect of the lie of the particles of -which it was composed. If the plane-being could examine its thickness, -he would find that particles which, in the first case, lay above others, -now lay below them. But, if he were really a plane-being, he would have -no idea of thickness in his squares, and he would find them both quite -identical. Only the one would be to the other as if it had been pulled -through itself. In this phenomenon of symmetry he would apprehend the -difference of the lie of the line, which went in the, to him, unknown -direction of up-and-down. - - -CHAPTER II. - -FURTHER APPEARANCES OF A CUBE TO A PLANE-BEING. - -Before leaving the observation of the cube, it is well to look at it for -a moment as it would appear to a plane-being, in whose world there was -such a fact as attraction. To do this, let the cube rest on the table, -so that its Dark-blue face is perpendicular in front of us. Now, let a -sheet of paper be placed in contact with the Dark-blue square. Let up -and sideways be the two dimensions of the plane-being, and away the -unknown direction. Let the line where the paper meets the table, -represent the surface of his earth. Then, there is to him, as all that -he can apprehend of the cube, a Dark-blue square standing upright; and, -when we look over the edge of the paper, and regard merely the part in -contact with the paper, we see what the plane-being would see. - -If the cube be turned round the up line, the Brown line, the Orange line -will pass to the near side of the paper, and the section made by the -cube in the paper will be an oblong. Such an oblong can be cut out; and -when the cube is fitted into it, it can be seen that it is bounded by a -Brown line and a Blue-green line opposite thereto, while the other -boundaries are Black and White lines. Next, if we take a section -half-way between the Black and White squares, we shall have a square -cutting the plane of the aforesaid paper in a single line. With regard -to this section, all we have to inquire is, What will take the place of -this line as the cube turns? Obviously, the line will elongate. From a -Dark-blue line it will change to a Light-buff line, the colour of the -inside of the section, and will terminate in a Blue-green point instead -of a French-grey. Again, it is obvious that, if the cube turns round the -Orange line, it will give rise to a series of oblongs, stretching -upwards. This turning can be continued till the cube is wholly on the -near side of the paper, and only the Orange line remains. And, when the -cube has made half a revolution, the Dark-blue square will return into -the plane; but it will run downwards instead of upwards as at first. -Thereafter, if the cube turn further, a series of oblongs will appear, -all running downwards from the Orange line. Hence, if all the -appearances produced by the revolution of the cube have to be shown, it -must be supposed to be raised some distance above the plane-being’s -earth, so that those appearances may be shown which occur when it is -turned round the Orange line downwards, as well as when it is turned -upwards. The unknown direction comes into the plane either upwards or -downwards, but there is no special connection between it and either of -these directions. If it come in upwards, the Brown line goes nearwards -or -Y; if it come in downwards, or -Z, the Brown line goes away, or Y. - -Let us consider more closely the directions which the plane-being would -have. Firstly, he would have up-and-down, that is, away from his earth -and towards it on the plane of the paper, the surface of his earth being -the line where the paper meets the table. Then, if he moved along the -surface of his earth, there would only be a line for him to move in, the -line running right and left. But, being the direction of his movement, -he would say it ran forwards and backwards. Thus he would simply have -the words up and down, forwards and backwards, and the expressions right -and left would have no meaning for him. If he were to frame a notion of -a world in higher dimensions, he must invent new words for distinctions -not within his experience. - -To repeat the observations already made, let the cube be held in front -of the observer, and suppose the Dark-blue square extended on every side -so as to form a plane. Then let this plane be considered as independent -of the Dark-blue square. Now, holding the Brown line between finger and -thumb, and touching its extremities, the Gold and Light-blue points, -turn the cube round the Brown line. The Dark-blue square will leave the -plane, the Orange line will tend towards the -Y direction, and the Blue -line will finally come into the plane pointing in the +X direction. If -we move the cube so that the line which leaves the plane runs +Y, 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 plane. The Blue line, -which runs in the unknown direction can come into either of the two -known directions of the plane. It can take the place of the Orange line -by turning the cube round the Brown line, or the place of the Brown line -by turning it round the Orange line. If the plane-being made models to -represent these two appearances of the cube, he would have identically -the same line, the Blue line, running in one of his known directions in -the first model, and in the other of his known directions in the second. -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 sense. In -that case, it would come into the plane in the negative sense of the -known directions. - -Starting with the cube in front of the observer, there are two ways in -which the Vermilion square can be brought into the imaginary plane, that -is the extension of the Dark-blue square. If the cube turn round the -Brown line so that the Orange line goes away, (_i.e._ +Y), the Vermilion -square comes in on the left of the Brown line. If it turn in the -opposite direction, the Vermilion square comes in on the right of the -Brown line. Thus, if we identify the plane-being with the Brown line, -the Vermilion square would appear either behind or before him. These two -appearances of the Vermilion square would seem identical, but they could -not be made to coincide by any movement in the plane. The diagram (Fig. -5.) shows the difference in them. It is obvious that no turn in the -plane could put one in the place of the other, part for part. Thus the -plane-being apprehends the reversal of the unknown direction by the -disposition of his figures. If a figure, which lay on one side of a -line, changed into an identical figure on the other side of it, he could -be sure that a line of the figure, which at first ran in the positive -unknown direction, now ran in the negative unknown direction. - -We have dwelt at great length on the appearances, which a cube would -present to a plane-being, and it will be found that all the points which -would be likely to cause difficulty hereafter, have been explained in -this obvious case. - -There is, however, one other way, open to a plane-being of studying a -cube, to which we must attend. This is, by steady motion. Let the cube -come into the imaginary plane, which is the extension of the Dark-blue -square, _i.e._ let it touch the piece of paper which is standing -vertical on the table. Then let it travel through this plane at right -angles to it at the rate of an inch a minute. The plane-being would -first perceive a Dark-blue square, that is, he would see the coloured -lines bounding that square, and enclosed therein would be what he would -call a Dark-blue solid. In the movement of the cube, however, this -Dark-blue square would not last for more than a flash of time. (The -edges and points on the models are made very large; in reality they must -be supposed very minute.) This Dark-blue square would be succeeded by -one of the colour of the cube’s interior, _i.e._ by a Light-buff square. -But this colour of the interior would not be visible to the plane-being. -He would go round the square on his plane, and would see the bounding -lines, _viz._ Vermilion, White, Blue-green, Black. And at the corners he -would see Deep-yellow, Bright-blue, Crimson, and Blue points. These -lines and points would really be those parts of the faces and lines of -the cube, which were on the point of passing through his plane. Now, -there would be one difference between the Dark-blue square and the -Light-buff with their respective boundaries. The first only lasted for a -flash; the second would last for a minute or all but a minute. Consider -the Vermilion square. It appears to the plane-being as a line. The Brown -line also appears to him as a line. But there is a difference between -them. The Brown line only lasts for a flash, whereas the Vermilion line -lasts for a minute. Hence, in this mode of presentation, we may say that -for a plane-being a lasting line is the mode of apprehending a plane, -and a lasting plane (which is a plane-being’s solid) is the mode of -apprehending our solids. In the same way, the Blue line, as it passes -through his plane, gives rise to a point. This point lasts for a minute, -whereas the Gold point only lasted for a flash. - - -CHAPTER III. - -FOUR-SPACE. GENESIS OF A TESSARACT. ITS REPRESENTATION IN THREE-SPACE. - -Hitherto we have only looked at Model 1. This, with the next seven, -represent what we can observe of the simplest body in Higher Space. A -few words will explain their construction. A point by its motion traces -a line. A line by its motion traces either a longer line or an area; if -it moves at right angles to its own direction, it traces a rectangle. -For the sake of simplicity, we will suppose all movements to be an inch -in length and at right angles to each other. Thus, a point moving traces -a line an inch long; a line moving traces a square inch; a square moving -traces a cubic inch. In these cases each of these movements produces -something intrinsically different from what we had before. A square is -not a longer line, nor a cube a larger square. When the cube moves, we -are unable to see any new direction in which it can move, and are -compelled to make it move in a direction which has previously been used. -Let us suppose there is an unknown direction at right angles to all our -known directions, just as a third direction would be unknown to a being -confined to the surface of the table. And let the cube move in this -unknown direction for an inch. We call the figure it traces a Tessaract. -The models are representations of the appearances a Tessaract would -present to us if shown in various ways. Consider for a moment what -happens to a square when moved to form a cube. Each of its lines, moved -in the new direction, traces a square; the square itself traces a new -figure, a cube, which ends in another square. Now, our cube, moved in a -new direction, will trace a tessaract, whereof the cube itself is the -beginning, and another cube the end. These two cubes are to the -tessaract as the Black square and White square are to the cube. A -plane-being could not see both those squares at once, but he could make -models of them and let them both rest in his plane at once. So also we -can make models of the beginning and end of the tessaract. Model 1 is -the cube, which is its beginning; Model 2 is the cube which is its end. -It will be noticed that there are no two colours alike in the two -models. The Silver point corresponds to the Gold point, that is, the -Silver point is the termination of the line traced by the Gold point -moving in the new direction. The sides correspond in the following -manner:-- - -SIDES. - - _Model 1._ _Model 2._ - 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 quickly. Then there are the -following correspondencies in points and lines. - -POINTS. - - _Model 1._ _Model 2._ - 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 - -LINES. - - _Model 1._ _Model 2._ - 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 invisible, as it is covered by its -boundaries. We suppose it to be Sage-green. - -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 plane. He cannot see the squares in their right position with -regard to each other, nor can we see the cubes in theirs. - -Let us now consider the vermilion side of Model 1. If it move in the X -direction, it traces the cube of Model 1. Its Gold point travels along -the Orange line, and itself, after tracing the cube, ends in the -Blue-green square. But if it moves in the new direction, it will also -trace a cube, for the new direction is at right angles to the up and -away directions, in which the Brown and Blue lines run. Let this square, -then, move in the unknown direction, and trace a cube. This cube we -cannot see, because the unknown direction runs out of our space at once, -just as the up direction runs out of the plane of the table. But a -plane-being could see the square, which the Blue line traces when moved -upwards, by the cube being turned round the Blue line, the Orange line -going upwards; then the Brown line comes into the plane of the table in -the -X direction. So also with our cube. As treated above, it runs from -the Vermilion square out of our space. But if the tessaract were turned -so that the line which runs from the Gold point in the unknown direction -lay in our space, and the Orange line lay in the unknown direction, we -could then see the cube which is formed by the movement of the Vermilion -square in the new direction. - -Take Model 5. There is on it a Vermilion square. Place this so that it -touches the Vermilion square on Model 1. All the marks of the two -squares are identical. This Cube 5, is the one traced by the Vermilion -square moving in the unknown direction. In Model 5, the whole figure, -the tessaract, produced by the movement of the cube in the unknown -direction, is supposed to be so turned that the Orange line passes into -the unknown direction, and that the line which went in the unknown -direction, runs opposite to the old direction of the Orange line. -Looking at this new cube, we see that there is a Stone line running to -the left from the Gold point. This line is that which the Gold point -traces when moving in the unknown direction. - -It is obvious that, if the Tessaract turns so as to show us the side, of -which Cube 5 is a model, then Cube 1 will no longer be visible. The -Orange line will run in the unknown or fourth direction, and be out of -our sight, together with the whole cube which the Vermilion square -generates, when the Gold point moves along the Orange line. Hence, if we -consider these models as real portions of the tessaract, we must not -have more than one before us at once. When we look at one, the others -must necessarily be beyond our sight and touch. But we may consider them -simply as models, and, as such, we may let them lie alongside of each -other. In this case, we must remember that their real relationships are -not those in which we see them. - -We now enumerate the sides of the new Cube 5, so that, when we refer to -it, any colour may be recognised by name. - -The square Vermilion traces a Pale-green cube, and ends in an Indian-red -square. - -(The colour Pale-green of this cube is not seen, as it is entirely -surrounded by squares and lines of colour.) - -Each Line traces a Square and ends in a Line. - - 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 „. - -Each Point traces a Line and ends in a Point. - - 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 „. - -It will be noticed that besides the Vermilion square of this cube -another square of it has been seen before. A moment’s comparison with -the experience of a plane-being will make this more clear. If a -plane-being has before him models of the Black and White squares of the -Cube, he sees that all the colours of the one are different from all the -colours of the other. Next, if he looks at a model of the Vermilion -square, he sees that it starts from the Blue line and ends in a line of -the White square, the Deep-yellow line. In this square he has two lines -which he had before, the Blue line with Gold and Buff points, the -Deep-yellow line with Light-blue and Red points. To him the Black and -White squares are his Models 1 and 2, and the Vermilion square is to him -as our Model 5 is to us. The left-hand square of Model 5 is Indian-red, -and is identical with that of the same colour on the left-hand side of -Model 2. In fact, Model 5 shows us what lies between the Vermilion face -of 1, and the Indian-red face of 2. - -From the Gold point we suppose four perfectly independent lines to -spring forth, each of them at right angles to all the others. In our -space there is only room for three lines mutually at right angles. It -will be found, if we try to introduce a fourth at right angles to each -of three, that we fail; hence, of these four lines one must go out of -the space we know. The colours of these four lines are Brown, Orange, -Blue, Stone. In Model 1 are shown the Brown, Orange, and Blue. In Model -5 are shown the Brown, Blue, and Stone. These lines might have had any -directions at first, but we chose to begin with the Brown line going up, -or Z, the Orange going X, the Blue going Y, and the Stone line going in -the unknown direction, which we will call W. - -Consider for a moment the Stone and the Orange lines. They can be seen -together on Model 7 by looking at the lower face of it. They are at -right angles to each other, and if the Orange line be turned to take the -place of the Stone line, the latter will run into the negative part of -the direction previously occupied by the former. This is the reason that -the Models 3, 5, and 7 are made with the Stone line always running in -the reverse direction of that line of Model 1, which is wanting in each -respectively. It will now be easy to find out Models 3 and 7. All that -has to be done is, to discover what faces they have in common with 1 and -2, and these faces will show from which planes of 1 they are generated -by motion in the unknown direction. - -Take Model 7. On one side of it there is a Dark-blue square, which is -identical with the Dark-blue square of Model 1. Placing it so that it -coincides with 1 by this square line for line, we see that the square -nearest to us is Burnt-sienna, the same as the near square on Model 2. -Hence this cube is a model of what the Dark-blue square traces on moving -in the unknown direction. Here the unknown direction coincides with the -negative away direction. In fact, to see this cube, we have been obliged -to suppose the Blue line turned into the unknown direction, for we -cannot look at more than three of these rectangular lines at once in our -space, and in this Model 7 we have the Brown, Orange, and Stone lines. -The faces, lines, and points of Cube 7 can be identified by the -following list. - -The Dark-blue square traces a Dark-stone cube (whose interior is -rendered invisible by the bounding squares), and ends in a Burnt-sienna -square. - -Each Line traces a Square and ends in a Line. - - 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 „. - -Each Point traces a Line and ends in a Point. - - 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 „. - -If we now take Model 3, we see that it has a Black square uppermost, and -has Blue and Orange lines. Hence, it evidently proceeds from the Black -square in Model 1; and it has in it Blue and Orange lines, which proceed -from the Gold point. But besides these, it has running downwards a Stone -line. The line wanting is the Brown line, and, as in the other cases, -when one of the three lines of Model 1 turns out into the unknown -direction, the Stone line turns into the direction opposite to that from -which the line has turned. Take this Model 3 and place it underneath -Model 1, raising the latter so that the Black squares on the two -coincide line for line. Then we see what would come into our view if the -Brown line were to turn into the unknown direction, and the Stone line -come into our space downwards. Looking at this cube, we see that the -following parts of the tessaract have been generated. - -The Black square traces a Brick-red cube (invisible because covered by -its own sides and edges), and ends in a Bright-green square. - -Each Line traces a Square and ends in a Line. - - 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 „. - -Each Point traces a Line and ends in a Point. - - 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 „. - -This completes the enumeration of the regions of Cube 3. It may seem a -little unnatural that it should come in downwards; 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-far. - -And if, instead of thinking of a plane-being as living on the surface of -a table, we suppose his world to be the surface of the sheet of paper -touching the Dark-blue square of Cube 1, then we see that a turn round -the Orange line, which makes the Brown line go into the plane-being’s -unknown direction, brings the Blue line into his downwards direction. - -There still remain to be described Models 4, 6, and 8. It will be shown -that Model 4 is to Model 3 what Model 2 is to Model 1. That is, if, when -3 is in our space, it be moved so as to trace a tessaract, 4 will be -the opposite cube in which the tessaract ends. There is no colour common -to 3 and 4. Similarly, 6 is the opposite boundary of the tessaract -generated by 5, and 8 of that by 7. - -A little closer consideration will reveal several points. Looking at -Cube 5, we see proceeding from the Gold point a Brown, a Blue, and a -Stone line. The Orange line is wanting; therefore, it goes in the -unknown direction. If we want to discover what exists in the unknown -direction from Cube 5, we can get help from Cube 1. For, since the -Orange line lies in the unknown direction from Cube 5, the Gold point -will, if moved along the Orange line, pass in the unknown direction. So -also, the Vermilion square, if moved along in the direction of the -Orange line, will proceed in the unknown direction. Looking at Cube 1 we -see that the Vermilion square thus moved ends in a Blue-green square. -Then, looking at Model 6, on it, corresponding to the Vermilion square -on Cube 5, is a Blue-green square. - -Cube 6 thus shows what exists an inch beyond 5 in the unknown direction. -Between the right-hand face on 5 and the right-hand face on 6 lies a -cube, the one which is shown in Model 1. Model 1 is traced by the -Vermilion square moving an inch along the direction of the Orange line. -In Model 5, the Orange line goes in the unknown direction; and looking -at Model 6 we see what we should get at the end of a movement of one -inch in that direction. We have still to enumerate the colours of Cubes -4, 6, and 8, and we do so in the following list. In the first column is -designated the part of the cube; in the columns under 4, 6, 8, come the -colours which 4, 6, 8, respectively have in the parts designated in the -corresponding line in the first column. - -Cube itself:-- - - 4 6 8 - Chocolate Oak-yellow Salmon - -Squares:-- - - 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 - -Lines:-- - -On ground, going round the square from left to right:-- - - 4 6 8 - 1. Brown-green Smoke Dark-purple - 2. Dark-green Crimson Magenta - 3. Pale-yellow Magenta Green-grey - 4. Dark Dull-green Light-green - -Vertical, going round the sides from left to right:-- - - 1. Rich-red Dark-pink Indigo - 2. Green-blue French-grey Pale-pink - 3. Sea-green Dark-slate Dark-slate - 4. Emerald Pale-pink Green - -Round upper face in same order:-- - - 1. Reddish Green-blue Pale-yellow - 2. Bright-blue Bright-blue Sea-green - 3. Leaden Sea-green Leaden - 4. Deep-yellow Dark-green Emerald - -Points:-- - -On lower face, going from left to right:-- - - 1. Quaker-green Turquoise Blue-tint - 2. Peacock-blue Fawn Earthen - 3. Orange-vermilion Terra-cotta Terra-cotta - 4. Purple Earthen Buff - -On upper face:-- - - 1. Light-blue Peacock-blue Purple - 2. Dull-purple Dull-purple Orange-vermilion - 3. Deep-blue Deep-blue Deep-blue - 4. Red Orange-vermilion Red - -If any one of these cubes be taken at random, it is easy enough to find -out to what part of the Tessaract it belongs. In all of them, except 2, -there will be one face, which is a copy of a face on 1; this face is, in -fact, identical with the face on 1 which it resembles. And the model -shows what lies in the unknown direction from that face. This unknown -direction is turned into our space, so that we can see and touch the -result of moving a square in it. And we have sacrificed one of the three -original directions in order to do this. It will be found that the line, -which in 1 goes in the 4th direction, in the other models always runs in -a negative direction. - -Let us take Model 8, for instance. Searching it for a face we know, we -come to a Light-yellow face away from us. We place this face parallel -with the Light-yellow face on Cube 1, and we see that it has a Green -line going up, and a Green-grey line going to the right from the Buff -point. In these respects it is identical with the Light-yellow face on -Cube 1. But instead of a Blue line coming towards us from the Buff -point, there is a Light-green line. This Light-green line, then, is that -which proceeds in the unknown direction from the Buff point. The line is -turned towards us in this Model 8 in the negative Y direction; and -looking at the model, we see exactly what is formed when in the motion -of the whole cube in the unknown direction, the Light-yellow face is -moved an inch in that direction. It traces out a Salmon cube (_v._ Table -on p. 127), and it has Sea-blue and Deep-green sides below and above, -and Deep-crimson and Dark-grey sides left and right, and Dun and -Light-yellow sides near and far. If we want to verify the correctness of -any of these details, we must turn to Models 1 and 2. What lies an inch -from the Light-yellow square in the unknown direction? Model 2 tells -us, a Dun square. Now, looking at 8, we see that towards us lies a Dun -square. This is what lies an inch in the unknown direction from the -Light-yellow square. It is here turned to face us, and we can see what -lies between it and the Light-yellow square. - - -CHAPTER IV. - -TESSARACT MOVING THROUGH THREE-SPACE. MODELS OF THE SECTIONS. - -In order to obtain a clear conception of the higher solid, a certain -amount of familiarity with the facts shown in these models is necessary. -But the best way of obtaining a systematic knowledge is shown hereafter. -What these models enable us to do, is to take a general review of the -subject. In all of them we see simply the boundaries of the tessaract in -our space; we can no more see or touch the tessaract’s solidity than a -plane-being can touch the cube’s solidity. - -There remain the four models 9, 10, 11, 12. Model 9 represents what lies -between 1 and 2. If 1 be moved an inch in the unknown direction, it -traces out the tessaract and ends in 2. But, obviously, between 1 and 2 -there must be an infinite number of exactly similar solid sections; -these are all like Model 9. - -Take the case of a plane-being on the table. He sees the Black -square,--that is, he sees the lines round it,--and he knows that, if it -moves an inch in some mysterious direction, it traces a new kind of -figure, the opposite boundary whereof is the White square. If, then, he -has models of the White and Black squares, he has before him the end and -beginning of our cube. But between these squares are any number of -others, the plane sections of the cube. We can see what they are. The -interior of each is a Light-buff (the colour of the substance of the -cube), the sides are of the colours of the vertical faces of the cube, -and the points of the colours of the vertical lines of the cube, viz., -Dark-blue, Blue-green, Light-yellow, Vermilion lines, and Brown, -French-grey, Dark-slate, Green points. Thus, the square, in moving in -the unknown direction, traces out a succession of squares, the -assemblage of which makes the cube in layers. So also the cube, moving -in the unknown direction, will at any point of its motion, still be a -cube; and the assemblage of cubes thus placed constitutes the tessaract -in layers. We suppose the cube to change its colour directly it begins -to move. Its colour between 1 and 2 we can easily determine by finding -what colours its different parts assume, as they move in the unknown -direction. The Gold point immediately begins to trace a Stone-line. We -will look at Cube 5 to see what the Vermilion face becomes; we know the -interior of that cube is Pale-green (_v._ Table, p. 122). Hence, as it -moves in the unknown direction, the Vermilion square forms in its course -a series of Pale-green squares. The Brown line gives rise to a Yellow -square; hence, at every point of its course in the fourth direction, it -is a Yellow line, until, on taking its final position, it becomes a -Dull-blue line. Looking at Cube 5, we see that the Deep yellow line -becomes a Light-red line, the Green line a Deep Crimson one, the Gold -point a Stone one, the Light-blue point a Rich-red one, the Red point an -Emerald one, and the Buff point a Light-green one. Now, take the Model -9. Looking at the left side of it, we see exactly that into which the -Vermilion square is transformed, as it moves in the unknown direction. -The left side is an exact copy of a section of Cube 5, parallel to the -Vermilion face. - -But we have only accounted for one side of our Model 9. There are five -other sides. Take the near side corresponding to the Dark-blue square on -Cube 1. When the Dark-blue square moves, it traces a Dark-stone cube, of -which we have a copy in Cube 7. Looking at 7 (_v._ Table, p. 124), we -see that, as soon as the Dark-blue square begins to move, it becomes of -a Dark-stone colour, and has Yellow, Ochre, Yellow-green, and Azure -sides, and Stone, Rich-red, Green-blue, Smoke lines running in the -unknown direction from it. Now, the side of Model 9, which faces us, has -these colours the squares being seen as lines, and the lines as points. -Hence Model 9 is a copy of what the cube becomes, so far as the -Vermilion and Dark-blue sides are concerned, when, moving in the unknown -direction, it traces the tessaract. - -We will now look at the lower square of our model. It is a Brick-red -square, with Azure, Rose, Sea-blue, and Light-brown lines, and with -Stone, Smoke, Magenta, and Light-green points. This, then, is what the -Black square should change into, as it moves in the unknown direction. -Let us look at Model 3. Here the Stone line, which is the line in the -unknown direction, runs downwards. It is turned into the downwards -direction, so that the cube traced by the Black square may be in our -space. The colour of this cube is Brick-red; the Orange line has traced -an Azure, the Blue line a Light-brown, the Crimson line a Rose, and the -Green-grey line a Sea-blue square. Hence, the lower square of Model 9 -shows what the Black square becomes, as it traces the tessaract; or, in -other words, the section of Model 3 between the Black and Bright-green -squares exactly corresponds to the lower face of Model 9. - -Therefore, it appears that Model 9 is a model of a section of the -tessaract, that it is to the tessaract what a square between the Black -and White squares is to the cube. - -To prove the other sides correct, we have to see what the White, -Blue-green, and Light-yellow squares of Cube 1 become, as the cube moves -in the unknown direction. This can be effected by means of the Models 4, -6, 8. Each cube can be used as an index for showing the changes through -which any side of the first model passes, as it moves in the unknown -direction till it becomes Cube 2. Thus, what becomes of the White -square? Look at Cube 4. From the Light-blue corner of its White square -runs downwards the Rich-red line in the unknown direction. If we take a -parallel section below the White square, we have a square bounded by -Ochre, Deep-brown, Deep-green, and Light-red lines; and by Rich-red, -Green-blue, Sea-green, and Emerald points. The colour of the cube is -Chocolate, and therefore its section is Chocolate. This description is -exactly true of the upper surface of Model 9. - -There still remain two sides, those corresponding to the Light-yellow -and Blue-green of Cube 1. What the Blue-green square becomes midway -between Cubes 1 and 2 can be seen on Model 6. The colour of the -last-named is Oak-yellow, and a section parallel to its Blue-green side -is surrounded by Yellow-green, Deep-brown, Dark-grey and Rose lines and -by Green-blue, Smoke, Magenta, and Sea-green points. This is exactly -similar to the right side of Model 9. Lastly, that which becomes of the -Light-yellow side can be seen on Model 8. The section of the cube is a -Salmon square bounded by Deep-crimson, Deep-green, Dark-grey and -Sea-blue lines and by Emerald, Sea-green, Magenta, and Light-green -points. - -Thus the models can be used to answer any question about sections. For -we have simply to take, instead of the whole cube, a plane, and the -relation of the whole tessaract to that plane can be told by looking at -the model, which, starting with that plane, stretches from it in the -unknown direction. - -We have not as yet settled the colour of the interior of Model 9. It is -that part of the tessaract which is traced out by the interior of Cube -1. The unknown direction starts equally and simultaneously from every -point of every part of Cube 1, just as the up direction starts equally -and simultaneously from every point of a square. Let us suppose that the -cube, which is Light-buff, changes to a Wood-colour directly it begins -to trace the tessaract. Then the internal part of the section between 1 -and 2 will be a Wood-colour. The sides of the Model 9 are of the -greatest importance. They are the colour of the six cubes, 3, 4, 5, 6, -7, and 8. The colours of 1 and 2 are wanting, viz. Light-buff and -Sage-green. Thus the section between 1 and 2 can be found by its wanting -the colours of the Cubes 1 and 2. - -Looking at Models 10, 11, and 12 in a similar manner, the reader will -find they represent the sections between Cubes 3 and 4, Cubes 5 and 6, -and Cubes 7 and 8 respectively. - - -CHAPTER V. - -REPRESENTATION OF THREE-SPACE BY NAMES, AND IN A PLANE. - -We may now ask ourselves the best way of passing on to a clear -comprehension of the facts of higher space. Something can be effected by -looking at these models; but it is improbable that more than a slight -sense of analogy will be obtained thus. Indeed, we have been trusting -hitherto to a method which has something vicious about it--we have been -trusting to our sense of what _must_ be. The plan adopted, as the -serious effort towards the comprehension of this subject, is to learn a -small portion of higher space. If any reader feel a difficulty in the -foregoing chapters, or if the subject is to be taught to young minds, it -is far better to abandon all attempt to see what higher space _must_ be, -and to learn what it _is_ from the following chapters. - - -NAMING A PIECE OF SPACE. - -The diagram (Fig. 6) represents a block of 27 cubes, which form Set 1 of -the 81 cubes. The cubes are coloured, and it will be seen that the -colours are arranged after the pattern of Model 1 of previous chapters, -which will serve as a key to the block. In the diagram, G. denotes Gold, -O. Orange, F. Fawn, Br. Brown, and so on. We will give names to the -cubes of this block. They should not be learnt, but kept for reference. -We will write these names in three sets, the lowest consisting of the -cubes which touch the table, the next of those immediately above them, -and the third of those at the top. Thus the Gold cube is called Corvus, -the Orange, Cuspis, the Fawn, Nugæ, and the central one below, Syce. The -corresponding colours of the following set can easily be traced. - - Olus Semita Lama - Via Mel Iter - Ilex Callis Sors - - Bucina Murex Daps - Alvus Mala Proes - Arctos Mœna Far - - Cista Cadus Crus - Dos Syce Bolus - Corvus Cuspis Nugæ - -Thus the central or Light-buff cube is called Mala; the middle one of -the lower face is Syce; of the upper face Mel; of the right face, Proes; -of the left, Alvus; of the front, Mœna (the Dark-blue square of Model -1); and of the back, Murex (the Light-yellow square). - -Now, if Model 1 be taken, and considered as representing a block of 64 -cubes, the Gold corner as one cube, the Orange line as two cubes, the -Fawn point as one cube, the Dark-blue square as four cubes, the -Light-buff interior as eight cubes, and so on, it will correspond to the -diagram (Fig. 7). This block differs from the last in the number of -cubes, but the arrangement of the colours is the same. The following -table gives the names which we will use for these cubes. There are no -new names; they are only applied more than once to all cubes of the same -colour. - - {Olus Semita Semita Lama - Fourth{Via Mel Mel Iter - Floor.{Via Mel Mel Iter - {Ilex Callis Callis Sors - - {Bucina Murex Murex Daps - Third {Alvus Mala Mala Proes - Floor.{Alvus Mala Mala Proes - {Arctos Mœna Mœna Far - - {Bucina Murex Murex Daps - Second{Alvus Mala Mala Proes - Floor.{Alvus Mala Mala Proes - {Arctos Mœna Mœna Far - - {Cista Cadus Cadus Crus - First {Dos Syce Syce Bolus - Floor.{Dos Syce Syce Bolus - {Corvus Cuspis Cuspis Nugæ - -[Illustration: Fig. 6.] - -[Illustration: Fig. 7.] - -[Illustration: Fig. 8.] - -If we now consider Model 1 to represent a block, five cubes each way, -built up of inch cubes, and colour it in the same way, that is, with -similar colours for the corner-cubes, edge-cubes, face-cubes, and -interior-cubes, we obtain what is represented in the diagram (Fig. 8). -Here we have nine Dark-blue cubes called Mœna; that is, Mœna denotes the -nine Dark-blue cubes, forming a layer on the front of the cube, and -filling up the whole front except the edges and points. Cuspis denotes -three Orange, Dos three Blue, and Arctos three Brown cubes. - -Now, the block of cubes can be similarly increased to any size we -please. The corners will always consist of single cubes; that is, Corvus -will remain a single cubic inch, even though the block be a hundred -inches each way. Cuspis, in that case, will be 98 inches long, and -consist of a row of 98 cubes; Arctos, also, will be a long thin line of -cubes standing up. Mœna will be a thin layer of cubes almost covering -the whole front of the block; the number of them will be 98 times 98. -Syce will be a similar square layer of cubes on the ground, so also -Mel, Alvus, Proes, and Murex in their respective places. Mala, the -interior of the cube, will consist of 98 times 98 times 98 inch cubes. - -[Illustration: Fig. 9] - -Now, if we continued in this manner till we had a very large block of -thousands of cubes in each side Corvus would, in comparison to the whole -block, be a minute point of a cubic shape, and Cuspis would be a mere -line of minute cubes, which would have length, but very small depth or -height. Next, if we suppose this much sub-divided block to be reduced in -size till it becomes one measuring an inch each way, the cubes of which -it consists must each of them become extremely minute, and the corner -cubes and line cubes would be scarcely discernible. But the cubes on the -faces would be just as visible as before. For instance, the cubes -composing Mœna would stretch out on the face of the cube so as to fill -it up. They would form a layer of extreme thinness, but would cover the -face of the cube (all of it except the minute lines and points). Thus we -may use the words Corvus and Nugæ, etc., to denote the corner-points of -the cube, the words Mœna, Syce, Mel, Alvus, Proes, Murex, to denote the -faces. It must be remembered that these faces have a thickness, but it -is extremely minute compared with the cube. Mala would denote all the -cubes of the interior except those, which compose the faces, edges, and -points. Thus, Mala would practically mean the whole cube except the -colouring on it. And it is in this sense that these words will be used. -In the models, the Gold point is intended to be a Corvus, only it is -made large to be visible; so too the Orange line is meant for Cuspis, -but magnified for the same reason. Finally, the 27 names of cubes, with -which we began, come to be the names of the points, lines, and faces of -a cube, as shown in the diagram (Fig. 9). With these names it is easy -to express what a plane-being would see of any cube. Let us suppose that -Mœna is only of the thickness of his matter. We suppose his matter to be -composed of particles, which slip about on his plane, and are so thin -that he cannot by any means discern any thickness in them. So he has no -idea of thickness. But we know that his matter must have some thickness, -and we suppose Mœna to be of that degree of thickness. If the cube be -placed so that Mœna is in his plane, Corvus, Cuspis, Nugæ, Far, Sors, -Callis, Ilex and Arctos will just come into his apprehension; they will -be like bits of his matter, while all that is beyond them in the -direction he does not know, will be hidden from him. Thus a plane-being -can only perceive the Mœna or Syce or some one other face of a cube; -that is, he would take the Mœna of a cube to be a solid in his -plane-space, and he would see the lines Cuspis, Far, Callis, Arctos. To -him they would bound it. The points Corvus, Nugæ, Sors, and Ilex, he -would not see, for they are only as long as the thickness of his matter, -and that is so slight as to be indiscernible to him. - -We must now go with great care through the exact processes by which a -plane-being would study a cube. For this purpose we use square slabs -which have a certain thickness, but are supposed to be as thin as a -plane-being’s matter. Now, let us take the first set of 81 cubes again, -and build them from 1 to 27. We must realize clearly that two kinds of -blocks can be built. It may be built of 27 cubes, each similar to Model -1, in which case each cube has its regions coloured, but all the cubes -are alike. Or it may be built of 27 differently coloured cubes like Set -1, in which case each cube is coloured wholly with one colour in all its -regions. If the latter set be used, we can still use the names Mœna, -Alvus, etc. to denote the front, side, etc., of any one of the cubes, -whatever be its colour. When they are built up, place a piece of card -against the front to represent the plane on which the plane-being lives. -The front of each of the cubes in the front of the block touches the -plane. In previous chapters we have supposed Mœna to be a Blue square. -But we can apply the name to the front of a cube of any colour. Let us -say the Mœna of each front cube is in the plane; the Mœna of the Gold -cube is Gold, and so on. To represent this, take nine slabs of the same -colours as the cubes. Place a stiff piece of cardboard (or a book-cover) -slanting from you, and put the slabs on it. They can be supported on the -incline so as to prevent their slipping down away from you by a thin -book, or another sheet of cardboard, which stands for the surface of the -plane-being’s earth. - -We will now give names to the cubes of Block 1 of the 81 Set. We call -each one Mala, to denote that it is a cube. They are written in the -following list in floors or layers, and are supposed to run backwards or -away from the reader. Thus, in the first layer, Frenum Mala is behind or -farther away than Urna Mala; in the second layer, Ostrum is in front, -Uncus behind it, and Ala behind Uncus. - - Third, or {Mars Mala Merces Mala Tyro Mala - Top {Spicula Mala Mora Mala Oliva Mala - Floor. {Comes Mala Tibicen Mala Vestis Mala - - Second, or{Ala Mala Cortis Mala Aer Mala - Middle {Uncus Mala Pallor Mala Tergum Mala - Floor. {Ostrum Mala Bidens Mala Scena Mala - - First, or {Sector Mala Hama Mala Remus Mala - Bottom {Frenum Mala Plebs Mala Sypho Mala - Floor. {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 name. They must be -learnt in every order, that is, in each of the three directions -backwards and forwards, _e.g._ Urna to Saltus, Urna to Sector, Urna to -Comes; and the same reversed, viz., Comes to Urna, Sector to Urna, etc. -Only by so learning them can the mind identify any one individually -without even a momentary reference to the others around it. It is well -to make it a rule not to proceed from one cube to a distant one without -naming the intermediate cubes. For, in Space we cannot pass from one -part to another without going through the intermediate portions. And, in -thinking of Space, it is well to accustom our minds to the same -limitations. - -Urna Mala is supposed to be solid Gold an inch each way; so too all the -cubes are supposed to be entirely of the colour which they show on their -faces. Thus any section of Moles Mala will be Orange, of Plebs Mala -Black, and so on. - -[Illustration: Fig. 10.] - -Let us now draw a pair of lines on a piece of paper or cardboard like -those in the diagram (Fig. 10). In this diagram the top of the page is -supposed to rest on the table, and the bottom of the page to be raised -and brought near the eye, so that the plane of the diagram slopes -upwards to the reader. Let Z denote the upward direction, and X the -direction from left to right. Let us turn the Block of cubes with its -front upon this slope _i.e._ so that Urna fits upon the square marked -Urna. Moles will be to the right and Ostrum above Urna, _i.e._ nearer -the eye. We might leave the block as it stands and put the piece of -cardboard against it; in this case our plane-world would be vertical. It -is difficult to fix the cubes in this position on the plane, and -therefore more convenient if the cardboard be so inclined that they will -not slip off. But the upward direction must be identified with Z. Now, -taking the slabs, let us compose what a plane-being would see of the -Block. He would perceive just the front faces of the cubes of the Block, -as it comes into his plane; these front faces we may call the Moenas of -the cubes. Let each of the slabs represent the Moena of its -corresponding cube, the Gold slab of the Gold cube and so on. They are -thicker than they should be; but we must overlook this and suppose we -simply see the thickness as a line. We thus build a square of nine slabs -to represent the appearance to a plane-being of the front face of the -Block. The middle one, Bidens Moena, would be completely hidden from him -by the others on all its sides, and he would see the edges of the eight -outer squares. If the Block now begin to move through the plane, that -is, to cut through the piece of paper at right angles to it, it will not -for some time appear any different. For the sections of Urna are all -Gold like the front face Moena, so that the appearance of Urna at any -point in its passage will be a Gold square exactly like Urna Moena, seen -by the plane-being as a line. Thus, if the speed of the Block’s passage -be one inch a minute, the plane-being will see no change for a minute. -In other words, this set of slabs lasting one minute will represent what -he sees. - -When the Block has passed one inch, a different set of cubes appears. -Remove the front layer of cubes. There will now be in contact with the -paper nine new cubes, whose names we write in the order in which we -should see them through a piece of glass standing upright in front of -the Block: - - 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 cubes, and -placed in order they show what the plane-being sees of the second set -of cubes as they pass through. Similarly the third wall of the Block -will come into the plane, and looking at them similarly, as it were -through an upright piece of glass, we write their names: - - Mars Mala Merces Mala Tyro Mala - Ala Mala Cortis Mala Aer Mala - Sector Mala Hama Mala Remus Mala - -Now, it is evident that these slabs stand at different times for -different parts of the cubes. We can imagine them to stand for the Moena -of each cube as it passes through. In that case, the first set of slabs, -which we put up, represents the Moenas of the front wall of cubes; the -next set, the Moenas of the second wall. Thus, if all the three sets of -slabs be together on the table, we have a representation of the sections -of the cube. For some purposes it would be better to have four sets of -slabs, the fourth set representing the Murex of the third wall; for the -three sets only show the front faces of the cubes, and therefore would -not indicate anything about the back faces of the Block. For instance, -if a line passed through the Block diagonally from the point Corvus -(Gold) to the point Lama (Deep-blue), it would be represented on the -slabs by a point at the bottom left-hand corner of the Gold slab, a -second point at the same corner of the Light-buff slab, and a third at -the same corner of the Deep-blue slab. Thus, we should have the points -mapped at which the line entered the fronts of the walls of cubes, but -not the point in Lama at which it would leave the Block. - -Let the Diagrams 1, 2, 3 (Fig. 11), be the three sets of slabs. To see -the diagrams properly, the reader must set the top of the page on the -table, and look along the page from the bottom of it. The line in -question, 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 A, B, C. To complete the diagram of its course, -we need a fourth set of slabs for the Murex of the third wall; the same -object might be attained, if we had another Block of 27 cubes behind the -first Block and represented its front or Moenas by a set of slabs. For -the point, at which the line leaves the first Block is identical with -that at which it enters the second Block. - -[Illustration: Fig. 11.] - -If we suppose a sheet of glass to be the plane-world, the Diagrams 1, 2, -3 (Fig. 11), may be drawn more naturally to us as Diagrams α, β, γ (Fig. -12). Here α represents the Moenas of the first wall, β those of the -second, γ those of the third. But to get the plane-being’s view we must -look over the edge of the glass down the Z axis. - -[Illustration: Fig. 12.] - -Set 2 of slabs represent the Moenas of Wall 2. These Moenas are in -contact with the Murex of Wall 1. Thus Set 2 will show where the line -issues from Wall 1 as well as where it enters Wall 2. - -The plane-being, therefore, could get an idea of the Block of cubes by -learning these slabs. He ought not to call the Gold slab Urna Mala, but -Urna Moena, and so on, because all that he learns are Moenas, merely the -thin faces of the cubes. By introducing the course of time, he can -represent the Block more nearly. For, if he supposes it to be passing an -inch a minute, he may give the name Urna Mala to the Gold slab enduring -for a minute. - -But, when he has learnt the slabs in this position and sequence, he has -only a very partial view of the Block. Let the Block turn round the Z -axis, as Model 1 turns round the Brown line. A different set of cubes -comes into his plane, and now they come in on the Alvus faces. -(Alvus is here used to denote the left-hand faces of the cubes, and is -not supposed to be Vermilion; it is simply the thinnest slice on the -left hand of the cube and of the same colour as the cube.) To represent -this, the plane-being should employ a fresh set of slabs, for there is -nothing common to the Moena and Alvus faces except an edge. But, since -each cube is of the same colour throughout, the same slab may be used -for its different faces. Thus the Alvus of Urna Mala can be represented -by a Gold slab. Only it must never be forgotten that it is meant to be a -new slab, and is not identical with the same slab used for Moena. - -[Illustration: Fig. 13.] - -[Illustration: Fig. 14.] - -Now, when the Block of cubes has turned round the Brown line into the -plane, it is clear that they will be on the side of the Z axis opposite -to that on which were the Moena slabs. The line, which ran Y, now runs --X. Thus the slabs will occupy the second quadrant marked by the axes, -as shown in the diagram (Fig. 13). Each of these slabs we will name -Alvus. In this view, as before, the book is supposed to be tilted up -towards the reader, so that the Z axis runs from O to his eye. Then, if -the Block be passed at right angles through the plane, there will come -into view the two sets of slabs represented in the Diagrams (Fig. 13). -In copying this arrangement with the slabs, the cardboard on which they -are arranged must slant upwards to the eye, _i.e._, OZ must run up to -the eye, and the sides of the slabs seen are in Diagram 2 (Fig. 13), the -upper edges of Tibicen, Mora, Merces; in Diagram 3, the upper edges of -Vestis, Oliva, Tyro. - -There is another view of the Block possible to a plane-being. If the -Block be turned round the X axis, the lower face comes into the vertical -plane. This corresponds to turning Model 1 round the Orange line. On -referring to the diagram (Fig. 14), we now see that the name of the -faces of the cubes coming into the plane is Syce. Here the plane-being -looks from the extremity of the Z axis and the squares, which he sees -run from him in the -Z direction. (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 Moena, 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 plane.) Next, if the Block be -passed through the plane, the sections shown in the Diagrams 2 and 3 -(Fig. 14) are brought into view. - -Thus, there are three distinct ways of regarding the cubic Block, each -of them equally primary; and if the plane-being is to have a correct -idea of the Block, he must be equally familiar with each view. By means -of the slabs each aspect can be represented; but we must remember in -each of the three cases, that the slabs represent different parts of the -cube. - -When we look at the cube Pallor Mala in space, we see that it touches -six other cubes by its six faces. But the plane-being could only arrive -at this fact by comparing different views. Taking the three Moena -sections of the Block, he can see that Pallor Mala Moena touches Plebs -Moena, Mora Moena, Uncus Moena, and Tergum Moena by lines. And it takes -the place of Bidens Moena, and is itself displaced by Cortis Moena as -the Block passes through the plane. Next, this same Pallor Mala can -appear to him as an Alvus. In this case, it touches Plebs Alvus, Mora -Alvus, Bidens Alvus, and Cortis Alvus by lines, takes the place of Uncus -Alvus, and is itself displaced by Tergum Alvus as the Block moves. -Similarly he can observe the relations, if the Syce of the Block be in -his plane. - -Hence, this unknown body Pallor Mala appears to him now as one -plane-figure now as another, and comes before him in different -connections. Pallor Mala is that which satisfies all these relations. -Each of them he can in turn present to sense; but the total conception -of Pallor Mala itself can only, if at all, grow up in his mind. The way -for him to form this mental conception, is to go through all the -practical possibilities which Pallor Mala would afford him by its -various movements and turns. In our world these various relations are -found by the most simple observations; but a plane-being could only -acquire them by considerable labour. And if he determined to obtain a -knowledge of the physical existence of a higher world than his own, he -must pass through such discipline. - - * * * * * - -[Illustration: Fig. 15.] - -[Illustration: Fig. 16.] - -We will see what change could be introduced into the shapes he builds by -the movements, which he does not know in his world. Let us build up this -shape with the cubes of the Block: Urna Mala, Moles Mala, Bidens Mala, -Tibicen Mala. To the plane-being this shape would be the slabs, Urna -Moena, Moles Moena, Bidens Moena, Tibicen Moena (Fig. 15). Now let the -Block be turned round the Z axis, so that it goes past the position, in -which the Alvus sides enter the vertical plane. Let it move until, -passing through the plane, the same Moena sides come in again. The mass -of the Block will now have cut through the plane and be on the near side -of it towards us; but the Moena faces only will be on the plane-being’s -side of it. The diagram (Fig. 16) shows what he will see, and it will -seem to him similar to the first shape (Fig. 15) in every respect except -its disposition with regard to the Z axis. It lies in the direction -X, -opposite to that of the first figure. However much he turn these two -figures about in the plane, he cannot make one occupy the place of the -other, part for part. Hence it appears that, if we turn the -plane-being’s figure about a line, it undergoes an operation which is to -him quite mysterious. He cannot by any turn in his plane produce the -change in the figure produced by us. A little observation will show that -a plane-being can only turn round a point. Turning round a line is a -process unknown to him. Therefore one of the elements in a plane-being’s -knowledge of a space higher than his own, will be the conception of a -kind of turning which will change his solid bodies into their own -images. - - -CHAPTER VI. - -THE MEANS BY WHICH A PLANE-BEING WOULD ACQUIRE A CONCEPTION OF OUR -FIGURES. - -Take the Block of twenty-seven Mala cubes, and build up the following -shape (Fig. 18):-- - -Urna Mala, Moles Mala, Plebs Mala, Pallor Mala, Mora Mala. - -If this shape, passed through the vertical plane, the plane-being would -perceive:-- - -(1) The squares Urna Moena and Moles Moena. - -(2) The three squares Plebs Moena, Pallor Moena, Mora Moena, - -and then the whole figure would have passed through his plane. - -If the whole Block were turned round the Z axis till the Alvus sides -entered, and the figure built up as it would be in that disposition of -the cubes, the plane-being would perceive during its passage through the -plane:-- - -(1) Urna Alvus; - -(2) Moles Alvus, Plebs Alvus, Pallor Alvus, Mora Alvus, which would all -enter on the left side of the Z axis. - -Again, if the Block were turned round the X axis, the Syce side would -enter, and the cubes appear in the following order:-- - -(1) Urna Syce, Moles Syce, Plebs Syce; - -(2) Pallor Syce; - -(3) Mora Syce. - -[Illustration: Fig. 17.] - -[Illustration: Fig. 18.] - -A comparison of these three sets of appearances would give the -plane-being a full account of the figure. It is that which can produce -these various appearances. - -Let us now suppose a glass plate placed in front of the Block in its -first position. On this plate let the axes X and Z be drawn. They divide -the surface into four parts, to which we give the following names (Fig. -17):-- - -Z X = that quarter defined by the positive Z and positive X axis. - -Z [=X] = that quarter defined by the positive Z and negative X axis -(which is called “Z negative X”). - -[=Z] [=X] = that quarter defined by the negative Z and negative X axis. - -[=Z] X = that quarter defined by the negative Z and positive X axis. - -The Block appears in these different quarters or quadrants, as it is -turned round the different axes. In Z X the Moenas appear, in Z [=X] the -Alvus faces, in [=Z] X the Syces. In each quadrant are drawn nine -squares, to receive the faces of the cubes when they enter. For -instance, in Z X we have the Moenas of:-- - - Z - | Comes Tibicen Vestis - | Ostrum Bidens Scena - | Urna Moles Saltus - +--------------------------------------X - - And in Z [=X] we have the Alvus of:-- - - Z - Mars Spicula Comes | - Ala Uncus Ostrum | - Sector Frenum Urna | - -X-------------------------------------+ - -And in the [=Z] X we have the Syces of:-- - - +-----------------------------------X - | Urna Moles Saltus - | Frenum Plebs Sypho - | Sector Hama Remus - -Z - -Now, if the shape taken at the beginning of this chapter be looked at -through the glass, and the distance of the second and third walls of the -shape behind the glass be considered of no account--that is, if they be -treated as close up to the glass--we get a plane outline, which occupies -the squares Urna Moena, Moles Moena, Bidens Moena, Tibicen Moena. This -outline is called a projection of the figure. To see it like a -plane-being, we should have to look down on it along the Z axis. - -It is obvious that one projection does not give the shape. For instance, -the square Bidens Moena might be filled by either Pallor or Cortis. All -that a square in the room of Bidens Moena denotes, is that there is a -cube somewhere in the Y, or unknown, direction from Bidens Moena. This -view, just taken, we should call the front view in our space; we are -then looking at it along the negative Y axis. - -When the same shape is turned round on the Z axis, so as to be projected -on the Z [=X] quadrant, we have the squares--Urna Alvus, Frenum Alvus, -Uncus Alvus, Spicula Alvus. When it is turned round the X axis, and -projected on [=Z] X, we have the squares, Urna Syce, Moles Syce, Plebs -Syce, and no more. This is what is ordinarily called the ground plan; -but we have set it in a different position from that in which it is -usually drawn. - -[Illustration: Fig. 19.] - -Now, the best method for a plane-being of familiarizing himself with -shapes in our space, would be to practise the realization of them from -their different projections in his plane. Thus, given the three -projections just mentioned, he should be able to construct the figure -from which they are derived. The projections (Fig. 19) are drawn below -the perspective pictures of the shape (Fig. 18). From the front, or -Moena view, he would conclude that the shape was Urna Mala, Moles Mala, -Bidens Mala, Tibicen Mala; or instead of these, or also in addition to -them, any of the cubes running in the Y direction from the plane. That -is, from the Moena projection he might infer the presence of all the -following cubes (the word Mala is omitted for brevity): Urna, Frenum, -Sector, Moles, Plebs, Hama, Bidens, Pallor, Cortis, Tibicen, Mora, -Merces. - -Next, the Alvus view or projection might be given by the cubes (the word -Mala being again omitted): Urna, Moles, Saltus, Frenum, Plebs, Sypho, -Uncus, Pallor, Tergum, Spicula, Mora, Oliva. Lastly, looking at the -ground plan or Syce view, he would infer the possible presence of Urna, -Ostrum, Comes, Moles, Bidens, Tibicen, Plebs, Pallor, Mora. - -Now, the shape in higher space, which is usually there, is that which is -common to all these three appearances. It can be determined, therefore, -by rejecting those cubes which are not present in all three lists of -cubes possible from the projections. 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 projections. After a time, when he had experience of -the cubes (which, though invisible to him as wholes, he could see part -by part in turn entering his space), the projections would have more -meaning to him, and he might comprehend the shape they expressed -fragmentarily in his space. To practise the realization from -projections, we should proceed in this way. First, we should think of -the possibilities involved in the Moena view, and build them up in cubes -before us. Secondly, we should build up the cubes possible from the -Alvus view. Again, taking the shape at the beginning of the chapter, we -should find that the shape of the Alvus possibilities intersected that -of the Moena possibilities in Urna, Moles, Frenum, Plebs, Pallor, Mora; -or, in other words, these cubes are common to both. Thirdly, we should -build up the Syce possibilities, and, comparing their shape with those -of the Moena and Alvus projections, we should find Urna, Moles, Plebs, -Pallor, Mora, of the Syce view coinciding with the same cubes of the -other views, the only cube present in the intersection of the Moena and -Alvus possibilities, and not present in the Syce view, being Frenum. - -The determination of the figure denoted by the three projections, may be -more easily effected by treating each projection as an indication of -what cubes are to be cut away. Taking the same shape as before, we have -in the Moena projection Urna, Moles, Bidens, Tibicen; and the -possibilities from them are Urna, Frenum, Sector, Moles, Plebs, Hama, -Bidens, Pallor, Cortis, Tibicen, Mora, Merces. This may aptly be called -the Moena solution. Now, from the Syce projection, we learn at once that -those cubes, which in it would produce Frenum, Sector, Hama, Remus, -Sypho, Saltus, are not in the shape. This absence of Frenum and Sector -in the Syce view proves that their presence in the Moena solution is -superfluous. The absence of Hama removes the possibility of Hama, -Cortis, Merces. The absence of Remus, Sypho, Saltus, makes no -difference, as neither they nor any of their Syce possibilities are -present in the Moena solution. Hence, the result of comparison of the -Moena and Syce projections and possibilities is the shape: Urna, Moles, -Plebs, Bidens, Pallor, Tibicen, Mora. This may be aptly called the -Moena-Syce solution. Now, in the Alvus projection we see that Ostrum, -Comes, Sector, Ala, and Mars are absent. The absence of Sector, Ala, and -Mars has no effect on our Moena-Syce solution; as it does not contain -any of their Alvus possibilities. But the absence of Ostrum and Comes -proves that in the Moena-Syce solution Bidens and Tibicen are -superfluous, since their presence in the original shape would give -Ostrum and Comes in the Alvus projection. Thus we arrive at the -Moena-Alvus-Syce solution, which gives us the shape: Urna, Moles, Plebs, -Pallor, Mora. - -It will be obvious on trial that a shape can be instantly recognised -from its three projections, if the Block be thoroughly well known in all -three positions. 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 us. If we remember -Ostrum as above Urna, we are not remembering the Block, but only one -particular relation of the Block to us. That position of Ostrum is a -fact as much related to ourselves as to the Block. There is, of course, -some information about the Block implied in that position; but it is so -mixed with information about ourselves as to be ineffectual knowledge of -the Block. It is of the highest importance to enter minutely into all -the details of solution written above. For, corresponding to every -operation necessary to a plane-being for the comprehension of our world, -there is an operation, with which we have to become familiar, if in our -turn we would enter into some comprehension of a world higher than our -own. Every cube of the Block ought to be thoroughly known in all its -relations. And the Block must be regarded, not as a formless mass out of -which shapes can be made, but as the sum of all possible shapes, from -which any one we may choose is a selection. In fact, to be familiar with -the Block, we ought to know every shape that could be made by any -selection of its cubes; or, in other words, we ought to make an -exhaustive study of it. In the Appendix is given a set of exercises in -the use of these names (which form a language of shape), and in various -kinds of projections. The projections studied in this chapter are not -the only, nor the most natural, projections by which a plane-being would -study higher space. But they suffice as an illustration of our present -purpose. If the reader will go through the exercises in the Appendix, -and form others for himself, he will find every bit of manipulation done -will be of service to him in the comprehension of higher space. - -There is one point of view in the study of the Block, by means of slabs, -which is of some interest. The cubes of the Block, and therefore also -the representative slabs of their faces, can be regarded as forming rows -and columns. There are three sets of them. If we take the Moena view, -they represent the views of the three walls of the Block, as they pass -through the plane. To form the Alvus view, we only have to rearrange the -slabs, and form new sets. The first new set is formed by taking the -first, or left-hand, column of each of the Moena sets. The second Alvus -set is formed by taking the second or middle columns of the three Moena -sets. The third will consist of the remaining or right-hand columns of -the Moenas. - -Similarly, the three Syce sets may be formed from the three horizontal -rows or floors of the Moena sets. - -Hence, it appears that the plane-being would study our space by taking -all the possible combinations of the corresponding rows and columns. He -would break up the first three sets into other sets, and the study of -the Block would practically become to him the study of these various -arrangements. - - -CHAPTER VII. - -FOUR-SPACE: ITS REPRESENTATION IN THREE-SPACE. - -We now come to the essential difficulty of our task. All that has gone -before is preliminary. We have now to frame the method by which we shall -introduce through our space-figures the figures of a higher space. When -a plane-being studies our shapes of cubes, he has to use squares. He is -limited at the outset. A cube appears to him as a square. On Model 1 we -see the various squares as which the cube can appear to him. We suppose -the plane-being to look from the extremity of the Z axis down a vertical -plane. First, there is the Moena square. Then there is the square given -by a section parallel to Moena, which he recognises by the variation of -the bounding lines as soon as the cube begins to pass through his plane. -Then comes the Murex square. Next, if the cube be turned round the Z -axis and passed through, he sees the Alvus and Proes squares and the -intermediate section. So too with the Syce and Mel squares and the -section between them. - -Now, dealing with figures in higher space, we are in an analogous -position. We cannot grasp the element of which they are composed. We can -conceive a cube; but that which corresponds to a cube in higher space is -beyond our grasp. But the plane-being was obliged to use two-dimensional -figures, squares, in arriving at a notion of a three-dimensional figure; -so also must we use three-dimensional figures to arrive at the notion -of a four-dimensional. Let us call the figure which corresponds to a -square in a plane and a cube in our space, a tessaract. Model 1 is a -cube. Let us assume a tessaract generated from it. Let us call the -tessaract Urna. The generating cube may then be aptly called Urna Mala. -We may use cubes to represent parts of four-space, but we must always -remember that they are to us, in our study, only what squares are to a -plane-being with respect to a cube. - -Let us again examine the mode in which a plane-being represents a Block -of cubes with slabs. Take Block 1 of the 81 Set of cubes. The -plane-being represents this by nine slabs, which represent the Moena -face of the block. Then, omitting the solidity of these first nine -cubes, he takes another set of nine slabs to represent the next wall of -cubes. Lastly, he represents the third wall by a third set, omitting the -solidity of both second and third walls. In this manner, he evidently -represents the extension of the Block upwards and sideways, in the Z and -X directions; but in the Y direction he is powerless, and is compelled -to represent extension in that direction by setting the three sets of -slabs alongside in his plane. The second and third sets denote the -height and breadth of the respective walls, but not their depth or -thickness. Now, note that the Block extends three inches in each of the -three directions. The plane-being can represent two of these dimensions -on his plane; but the unknown direction he has to represent by a -repetition of his plane figures. The cube extends three inches in the Y -direction. He has to use 3 sets of slabs. - -The Block is built up arbitrarily in this manner: Starting from Urna -Mala and going up, we come to a Brown cube, and then to a Light-blue -cube. Starting from Urna Mala and going right, we come to an Orange and -a Fawn cube. Starting from Urna Mala and going away from us, we come to -a Blue and a Buff cube. Now, the plane-being represents the Brown and -Orange cubes by squares lying next to the square which represents Urna -Mala. The Blue cube is as close as the Brown cube to Urna Mala, 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 first. So he is forced -to put a Blue square anywhere in his plane and say of it: “This Blue -square represents what I should arrive at, if I started from Urna Mala -and moved away, that is in the Y or unknown direction.” Now, just as -there are three cubes going up, so there are three going away. Hence, -besides the Blue square placed anywhere on the plane, he must also place -a Buff square beyond it, to show that the Block extends as far away as -it does upwards and sideways. (Each cube being a different colour, there -will be as many different colours of squares as of cubes.) It will -easily be seen that not only the Gold square, 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 away, or in the unknown Y direction. - -Coming now to the representation of a four-dimensional block, we see -that we can show only three dimensions by cubic blocks, and that the -fourth can only be represented by repetitions of such blocks. There must -be a certain amount of arbitrary naming and colouring. The colours have -been chosen as now stated. Take the first Block of the 81 Set. We are -familiar with its colours, and they can be found at any time by -reference to Model 1. Now, suppose the Gold cube to represent what we -can see in our space of a Gold tessaract; the other cubes of Block 1 -give the colours of the tessaracts which lie in the three directions X, -Y, and Z from the Gold one. But what is the colour of the tessaract -which lies next to the Gold in the unknown direction, W? Let us suppose -it to be Stone in colour. Taking out Block 2 of the 81 Set and arranging -it on the pattern of Model 9, we find in it a Stone cube. But, just as -there are three tessaracts in the X, Y, and Z directions, as shown by -the cubes in Block 1, so also must there be three tessaracts in the -unknown direction, W. Take Block 3 of the 81 Set. This Block can be -arranged on the pattern of Model 2. In it there is a Silver cube where -the Gold cube lies in Block 1. Hence, we may say, the tessaract which -comes next to the Stone one in the unknown direction from the Gold, is -of a Silver colour. Now, a cube in all these cases represents a -tessaract. Between the Gold and Stone cubes there is an inch in the -unknown direction. The Gold tessaract is supposed to be Gold throughout -in all four directions, and so also is the Stone. We may imagine it in -this way. Suppose the set of three tessaracts, the Gold, the Stone, and -the Silver to move through our space at the rate of an inch a minute. We -should first see the Gold cube which would last a minute, then the Stone -cube for a minute, and lastly the Silver cube a minute. (This is -precisely analogous to the appearance of passing cubes to the -plane-being as successive squares lasting a minute.) After that, nothing -would be visible. - -Now, just as we must suppose that there are three tessaracts proceeding -from the Gold cube in the unknown direction, so there must be three -tessaracts extending in the unknown direction from every one of the 27 -cubes of the first Block. The Block of 27 cubes is not a Block of 27 -tessaracts, but it represents as much of them as we can see at once in -our space; and they form the first portion or layer (like the first -wall of cubes to the plane-being) of a set of eighty-one tessaracts, -extending to equal distances in all four directions. Thus, to represent -the whole Block of tessaracts there are 81 cubes, or three Blocks of 27 -each. - -Now, it is obvious that, just as a cube has various plane boundaries, so -a tessaract has various cube boundaries. The cubes of the tessaract, -which we have been regarding, have been those containing the X, Y, and Z -directions, just as the plane-being regarded the Moena faces containing -the X and Z directions. And, as long as the tessaract is unchanged in -its position with regard to our space, we can never see any portion of -it which is in the unknown direction. Similarly, we saw that a -plane-being could not see the parts of a cube which went in the third -direction, until the cube was turned round one of its edges. In order to -make it quite clear what parts of a cube came into the plane, we gave -distinct names to them. Thus, the squares containing the Z and X -directions were called Moena and Murex; those containing the Z and Y, -Alvus and Proes; and those the X and Y, Syce and Mel. Now, similarly -with our four axes, any three will determine a cube. Let the tessaract -in its normal position have the cube Mala determined by the axes Z, X, -Y. Let the cube Lar be that which is determined by X, Y, W, that is, the -cube which, starting from the X Y plane, stretches one inch in the -unknown or W direction. Let Vesper be the cube determined by Z, Y, W, -and Pluvium by Z, X, W. And let these cubes have opposite cubes of the -following names: - - Mala has Margo - Lar „ Velum - Vesper „ Idus - Pluvium „ Tela - -Another way of looking at the matter is this. When a cube is generated -from a square, each of the lines bounding the square becomes a square, -and the square itself becomes a cube, giving two squares in its initial -and final positions. When a cube moves in the new and unknown direction, -each of its planes traces a cube and it generates a tessaract, giving in -its initial and final positions two cubes. Thus there are eight cubes -bounding the tessaract, six of them from the six plane sides and two -from the cube itself. These latter two are Mala and Margo. The cubes -from the six sides are: Lar from Syce, Velum from Mel, Vesper from -Alvus, Idus from Proes, Pluvium from Moena, Tela from Murex. And just as -a plane-being can only see the squares of a cube, so we can only see the -cubes of a tessaract. It may be said that the cube can be pushed partly -through the plane, so that the plane-being sees a section between Moena -and Murex. Similarly, the tessaract can be pushed through our space so -that we can see a section between Mala and Margo. - -There is a method of approaching the matter, which settles all -difficulties, and provides us with a nomenclature for every part of the -tessaract. We have seen how by writing down the names of the cubes of a -block, and then supposing that their number increases, certain sets of -the names come to denote points, lines, planes, and solid. Similarly, if -we write down a set of names of tessaracts in a block, it will be found -that, when their number is increased, certain sets of the names come to -denote the various parts of a tessaract. - -For this purpose, let us take the 81 Set, and use the cubes to represent -tessaracts. The whole of the 81 cubes make one single tessaractic set -extending three inches in each of the four directions. The names must be -remembered to denote tessaracts. Thus, Corvus is a tessaract which has -the tessaracts Cuspis and Nugæ to the right, Arctos and Ilex above it, -Dos and Cista away from it, and Ops and Spira in the fourth or unknown -direction from it. It will be evident at once, that to write these names -in any representative order we must adopt an arbitrary system. We -require them running in four directions; we have only two on paper. The -X direction (from left to right) and the Y (from the bottom towards the -top of the page) will be assumed to be truly represented. The Z -direction will be symbolized by writing the names in floors, the upper -floors always preceding the lower. Lastly, the fourth, or W, direction -(which has to be symbolized in three-dimensional space by setting the -solids in an arbitrary position) will be signified by writing the names -in blocks, 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 it. Thus, Ops is written in the same place in the -Second Block, Spira in the Third Block, as Corvus occupies in the First -Block. - -Since there are an equal number of tessaracts in each of the four -directions, there will be three floors Z when there are three X and Y. -Also, there will be three Blocks W. If there be four tessaracts in each -direction, there will be four floors Z, and four blocks W. Thus, when -the number in each direction is enlarged, the number of blocks W is -equal to the number of tessaracts in each known direction. - -On pp. 136, 137 were given the names as used for a cubic block of 27 or -64. Using the same and more names for a tessaractic Set, and remembering -that each name now represents, not a cube, but a tessaract, we obtain -the following nomenclature, the order in which the names are written -being that stated above: - -THIRD BLOCK. - - Upper { Solia Livor Talus - Floor. { Lensa Lares Calor - { Felis Tholus Passer - ----------------- - Middle { Lixa Portica Vena - Floor. { Crux Margo Sal - { Pagus Silex Onager - ----------------- - Lower { Panax Mensura Mugil - Floor. { Opex Lappa Mappa - { Spira Luca Ancilla - -SECOND BLOCK. - - Upper { Orsa Mango Libera - Floor. { Creta Velum Meatus - { Lucta Limbus Pator - ----------------- - Middle { Camoena Tela Orca - Floor. { Vesper Tessaract Idus - { Pagina Pluvium Pactum - ----------------- - Lower { Lis Lorica Offex - Floor. { Lua Lar Olla - { Ops Lotus Limus - -FIRST BLOCK. - - Upper { Olus Semita Lama - Floor. { Via Mel Iter - { Ilex Callis Sors - ----------------- - Middle { Bucina Murex Daps - Floor. { Alvus Mala Proes - { Arctos Moena Far - ----------------- - Lower { Cista Cadus Crus - Floor. { Dos Syce Bolus - { Corvus Cuspis Nugæ - -It is evident that this set of tessaracts could be increased to the -number of four in each direction, the names being used as before for the -cubic blocks on pp. 136, 137, and in that case the Second Block would be -duplicated to make the four blocks required in the unknown direction. -Comparing such an 81 Set and 256 Set, we should find that Cuspis, which -was a single tessaract in the 81 Set became two tessaracts in the 256 -Set. And, if we introduced a larger number, it would simply become -longer, and not increase in any other dimension. Thus, Cuspis would -become the name of an edge. Similarly, Dos would become the name of an -edge, and also Arctos. Ops, which is found in the Middle Block of the 81 -Set, occurs both in the Second and Third Blocks of the 256 Set; that is, -it also tends to elongate and not extend in any other direction, and may -therefore be used as the name of an edge of a tessaract. - -Looking at the cubes which represent the Syce tessaracts, we find that, -though they increase in number, they increase only in two directions; -therefore, Syce may be taken to signify a square. But, looking at what -comes from Syce in the W direction, we find in the Middle Block of the -81 Set one Lar, and in the Second and Third Blocks of the 256 Set four -Lars each. Hence, Lar extends in three directions, X, Y, W, and becomes -a cube. Similarly, Moena is a plane; but Pluvium, which proceeds from -it, extends not only sideways and upwards like Moena, but in the unknown -direction also. It occurs in both Middle Blocks of the 256 Set. Hence, -it also is a cube. We have now considered such parts of the Sets as -contain one, two, and three dimensions. But there is one part which -contains four. It is that named Tessaract. In the 256 Set there are -eight such cubes in the Second, and eight in the Third Block; that is, -they extend Z, X, Y, and also W. They may, therefore, be considered to -represent that part of a tessaract or tessaractic Set, which is -analogous to the interior of a cube. - -The arrangement of colours corresponding to these names is seen on Model -1 corresponding to Mala, Model 2 to Margo, and Model 9 to the -intermediate block. - -When we take the view of the tessaract with which we commenced, and in -which Arctos goes Z, Cuspis X, Dos Y, and Ops W, we see Mala in our -space. But when the tessaract is turned so that the Ops line goes -X, -while Cuspis is turned W, the other two remaining as they were, then we -do not see Mala, but that cube which, in the original position of the -tessaract, contains the Z, Y, W, directions, that is, the Vesper cube. - -A plane-being may begin to study a block of cubes by their Syce squares; -but if the block be turned round Dos, he will have Alvus squares in his -space, and he must then use them to represent the cubic Block. So, when -the tessaractic Set is turned round, Mala cubes leave our space, and -Vespers enter. - -There are two ways which can be followed in studying the Set of -tessaracts. - -I. Each tessaract of one inch every way can be supposed to be of the -same colour throughout, so that, whichever way it be turned, whichever -of its edges coincide with our known axes, it appears to us as a cube of -one uniform colour. Thus, if Urna be the tessaract, Urna Mala would be a -Gold cube, Urna Vesper a Gold cube, and so on. This method is, for the -most part, adopted in the following pages. In this case, a whole Set of -4 × 4 × 4 × 4 tessaracts would in colours resemble a set composed of -four cubes like Models 1, 9, 9, and 2. But, when any question about a -particular tessaract has to be settled, it is advantageous, for the sake -of distinctness, to suppose it coloured in its different regions as the -whole set is coloured. - -II. The other plan is, to start with the cubic sides of the inch -tessaract, each coloured according to the scheme of the Models 1 to 8. -In this case, the lines, if shown at all, should be very thin. For, in -fact, only the faces would be seen, as the width of the lines would only -be equal to the thickness of our matter in the fourth dimension, which -is indistinguishable to the senses. If such completely coloured cubes be -used, less error is likely to creep in; but it is a disadvantage that -each cube in the several blocks is exactly like the others in that -block. If the reader make such a set to work with for a time, he will -gain greatly, for the real way of acquiring a sense of higher space is -to obtain those experiences of the senses exactly, which the observation -of a four-dimensional body would give. These Models 1-8 are called sides -of the tessaract. - -To make the matter perfectly clear, it is best to suppose that any -tessaract or set of tessaracts which we examine, has a duplicate exactly -similar in shape and arrangement of parts, but different in their -colouring. In the first, let each tessaract have one colour throughout, -so that all its cubes, apprehended in turn in our space, will be of one -and the same colour. In the duplicate, let each tessaract be so coloured -as to show its different cubic sides by their different colours. Then, -when we have it turned to us in different aspects, we shall see -different cubes, and when we try to trace the contacts of the tessaracts -with each other, we shall be helped by realizing each part of every -tessaract in its own colour. - - -CHAPTER VIII. - -REPRESENTATION OF FOUR-SPACE BY NAME. STUDY OF TESSARACTS. - -We have now surveyed all the preliminary ground, and can study the -masses of tessaracts without obscurity. - -We require a scaffold or framework for this purpose, which in three -dimensions will consist of eight cubic spaces or octants assembled round -one point, as in two dimensions it consisted of four squares or -quadrants round a point. - -These eight octants lie between the three axes Z, X, Y, which intersect -at the given point, and can be named according to their positions -between the positive and negative directions of those axes. Thus the -octant Z, X, Y, is that which is contained by the positive portions of -all three axes; the octant Z, [=X], Y, that which is to the left of Z, -X, Y, and between the positive parts of Z and Y and the negative of X. -To illustrate this quite clearly, let us take the eight cubes--Urna, -Moles, Plebs, Frenum, Uncus, Pallor, Bidens, Ostrum--and place them in -the eight octants. Let them be placed round the point of intersection of -the axes; Pallor Corvus, Plebs Ilex, etc., will be at that point. Their -positions will then be:-- - - 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 cubes, as they are before us, are as follows:-- - -THIRD BLOCK. - - Third { Arcus Mala Ovis Mala Portio Mala - Floor. { Laurus Mala Tigris Mala Segmen Mala - { Axis Mala Troja Mala Aries Mala - - Second { Postis Mala Clipeus Mala Tabula Mala - Floor. { Orcus Mala Lacerta Mala Testudo Mala - { Verbum Mala Luctus Mala Anguis Mala - - First { Telum Mala Nepos Mala Angusta Mala - Floor. { Polus Mala Penates Mala Vulcan Mala - { Cervix Mala Securis Mala Vinculum Mala - -SECOND BLOCK. - - Third { Ara Mala Vomer Mala Pluma Mala - Floor. { Praeda Mala Sacerdos Mala Hydra Mala - { Cortex Mala Mica Mala Flagellum Mala - - Second { Pilum Mala Glans Mala Colus Mala - Floor. { Ocrea Mala Tessera Mala Domitor Mala - { Cardo Mala Cudo Mala Malleus Mala - - First { Agmen Mala Lacus Mala Arvus Mala - Floor. { Crates Mala Cura Mala Limen Mala - { Thyrsus Mala Vitta Mala Sceptrum Mala - -FIRST BLOCK. - - Third { Mars Mala Merces Mala Tyro Mala - Floor. { Spicula Mala Mora Mala Oliva Mala - { Comes Mala Tibicen Mala Vestis Mala - - Second { Ala Mala Cortis Mala Aer Mala - Floor. { Uncus Mala Pallor Mala Tergum Mala - { Ostrum Mala Bidens Mala Scena Mala - - First { Sector Mala Hama Mala Remus Mala - Floor. { Frenum Mala Plebs Mala Sypho Mala - { Urna Mala Moles Mala Saltus Mala - -Their colours can be found by reference to the Models 1, 9, 2, which -correspond respectively to the First, Second, and Third Blocks. Thus, -Urna Mala is Gold; Moles, Orange; Saltus, Fawn; Thyrsus, Stone; Cervix, -Silver. The cubes whose colours are not shown in the Models, are Pallor -Mala, Tessera Mala, and Lacerta Mala, which are equivalent to the -interiors of the Model cubes, and are respectively Light-buff, Wooden, -and Sage-green. These 81 cubes are the cubic sides and sections of the -tessaracts of an 81 tessaractic Set, which measures three inches in -every direction. We suppose it to pass through our space. Let us call -the positive unknown direction Ana (_i.e._, +W) and the negative unknown -direction Kata (-W). Then, as the whole tessaract moves Kata at the rate -of an inch a minute, we see first the First Block of 27 cubes for one -minute, then the Second, and lastly the Third, each lasting one minute. - -Now, when the First Block stands in the normal position, the edges of -the tessaract that run from the Corvus corner of Urna Mala, are: Arctos -in Z, Cuspis in X, Dos in Y, Ops in W. Hence, we denote this position by -the following symbol:-- - - Z X Y W - _a_ _c_ _d_ _o_ - -where _a_ stands for Arctos, _c_ for Cuspis, _d_ for Dos, and _o_ for -Ops, and the other letters for the four axes in space. _a_, _c_, _d_, -_o_ are the axes of the tessaract, and can take up different directions -in space with regard to us. - - * * * * * - -Let us now take a smaller four-dimensional set. Of the 81 Set let us -take the following:-- - - Z X Y W - _a_ _c_ _d_ _o_ - -SECOND BLOCK. - - Second Floor. { Ocrea Mala Tessera Mala - { Cardo Mala Cudo Mala - - First Floor. { Crates Mala Cura Mala - { Thyrsus Mala Vitta Mala - -FIRST BLOCK. - - Second Floor. { Uncus Mala Pallor Mala - { Ostrum Mala Bidens Mala - - First Floor. { Frenum Mala Plebs Mala - { Urna Mala Moles Mala - -Let the First Block be put up before us in Z X Y, (Urna Corvus is at the -junction of our axes Z X Y). The Second Block is now one inch distant in -the unknown direction; and, if we suppose the tessaractic Set to move -through our space at the rate of one inch a minute, the Second will -enter in one minute, and replace the first. But, instead of this, let us -suppose the tessaracts to turn so that Ops, which now goes W, shall go --X. Then we can see in our space that cubic side of each tessaract which -is contained by the lines Arctos, Dos, and Ops, the cube Vesper; and we -shall no longer have the Mala sides but the Vesper sides of the -tessaractic Set in our space. We will now build it up in its Vesper view -(as we built up the cubic Block in its Alvus view). Take the Gold cube, -which now means Urna Vesper, and place it on the left hand of its former -position as Urna Mala, that is, in the octant Z [=X] Y. Thyrsus Vesper, -which previously lay just beyond Urna Vesper in the unknown direction, -will now lie just beyond it in the -X direction, that is, to the left of -it. The tessaractic Set is now in the position - - Z X Y W - _a_ _ō_ _d_ _c_ - -(the minus sign over the _o_ meaning that Ops runs in the negative -direction), and its Vespers lie in the following order:-- - -SECOND BLOCK. - - Second Floor. { Tessara Pallor - { Cudo Bidens - - First Floor. { Cura Plebs - { Vitta Moles - -FIRST BLOCK. - - Second Floor. { Ocrea Uncus - { Cardo Ostrum - - First Floor. { Crates Frenum - { Thyrsus Urna - -The name Vesper is left out in the above list for the sake of brevity, -but should be used in studying the positions. - -[Illustration: Fig. 20.] - -On comparing the two lists of the Mala view and Vesper view, it will be -seen that the cubes presented in the Vesper view are new sides of the -tessaract, and that the arrangement of them is different from that in -the Mala view. (This is analogous to the changes in the slabs from the -Moena to Alvus view of the cubic Block.) Of course, the Vespers of all -these tessaracts are not visible at once in our space, any more than are -the Moenas of all three walls of a cubic Block to a plane-being. But if -the tessaractic Set be supposed to move through space in the unknown -direction at the rate of an inch a minute, the Second Block will present -its Vespers after the First Block has lasted a minute. The relative -position of the Mala Block and the Vesper Block may be represented in -our space as in the diagram, Fig. 20. But it must be distinctly -remembered that this arrangement is quite conventional, 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 faces, -with the solidity omitted, along one of his known directions. - -The Vespers of the First and Second Blocks cannot be in our space -simultaneously, any more than the Moenas of all three walls in plane -space. To render their simultaneous presence possible, the cubic or -tessaractic Block or Set must be broken up, and its parts no longer -retain their relations. This fact is of supreme importance in -considering higher space. Endless fallacies creep in as soon as it is -forgotten that the cubes are merely representative as the slabs were, -and the positions in our space merely conventional and symbolical, like -those of the slabs along the plane. And these fallacies are so much -fostered by again symbolizing the cubic symbols and their symbolical -positions in perspective drawings or diagrams, that the reader should -surrender all hope of learning space from this book or the drawings -alone, and work every thought out with the cubes themselves. - -If we want to see what each individual cube of the tessaractic faces -presented to us in the last example is like, we have only to consider -each of the Malas similar in its parts to Model 1, and each of the -Vespers to Model 5. And it must always be remembered that the cubes, -though used to represent both Mala and Vesper faces of the tessaract, -mean as great a difference as the slabs used for the Moena and Alvus -faces of the cube. - -If the tessaractic Set move Kata through our space, when the Vesper -faces are presented to us, we see the following parts of the tessaract -Urna (and, therefore, also the same parts of the other tessaracts): - -(1) Urna Vesper, which is Model 5. - -(2) A parallel section between Urna Vesper and Urna Idus, which is Model -11. - -(3) Urna Idus, which is Model 6. - -When Urna Idus has passed Kata our space, Moles Vesper enters it; then a -section between Moles Vesper and Moles Idus, and then Moles Idus. Here -we have evidently observed the tessaract more minutely; as it passes -Kata through our space, starting on its Vesper side, we have seen the -parts which would be generated by Vesper moving along Cuspis--that is -Ana. - -Two other arrangements of the tessaracts have to be learnt besides those -from the Mala and Vesper aspect. One of them is the Pluvium aspect. -Build up the Set in Z X [=Y], letting Arctos run Z, Cuspis X, and Ops -[=Y]. In the common plane Moena, Urna Pluvium coincides with Urna Mala, -though they cannot be in our space together; so too Moles Pluvium with -Moles Mala, Ostrum Pluvium with Ostrum Mala. And lying towards us, or -[=Y], is now that tessaract which before lay in the W direction from -Urna, viz., Thyrsus. The order will therefore be the following (a star -denotes the cube whose corner is at point of intersection of the axes, -and the name Pluvium must be understood to follow each of the names): - - Z X Y W - _a_ _c_ _ō_ _d_ - - SECOND BLOCK. - - Second Floor. { Uncus Pallor - { Ocrea Tessera - - First Floor. { Frenum Plebs - { Crates Cura - - FIRST BLOCK. - - Second Floor. { Ostrum Bidens - { Cardo Cudo - - First Floor. {*Urna Moles - { Thyrsus Vitta - -Thus the wall of cubes in contact with that wall of the Mala position -which contains the Urna, Moles, Ostrum, and Bidens Malas, is a wall -composed of the Pluviums of Urna, Moles, Ostrum, and Bidens. The wall -next to this, and nearer to us, is of Thyrsus, Vitta, Cardo, Cudo, -Pluviums. The Second Block is one inch out of our Space, and only enters -it if the Block moves Kata. Model 7 shows the Pluvium cube; and each of -the cubes of the tessaracts seen in the Pluvium position is a Pluvium. -If the tessaractic Set moved Kata, we would see the Section between -Pluvium and Tela for all but a minute; and then Tela would enter our -space, and the Tela of each tessaract would be seen. Model 12 shows the -Section from Pluvium to Tela. Model 8 is Tela. Tela only lasts for a -flash, as it has only the minutest magnitude in the unknown or Ana -direction. Then, Frenum Pluvium takes the place of Urna Tela; and, when -it passes through, we see a similar section between Frenum Pluvium and -Frenum Tela, and lastly Frenum Tela. Then the tessaractic Set passes -out, or Kata, our space. A similar process takes place with every other -tessaract, when the Set of tessaracts moves through our space. - -There is still one more arrangement to be learnt. If the line of the -tessaract, which in the Mala position goes Ana, or W, be changed into -the [=Z] or downwards direction, the tessaract will then appear in our -space below the Mala position; and the side presented to us will not be -Mala, but that which contains the lines Dos, Cuspis, and Ops. This side -is Model 3, and is called Lar. Underneath the place which was occupied -by Urna Mala, will come Urna Lar; under the place of Moles Mala, Moles -Lar; under the place of Frenum Mala, Frenum Lar. The tessaract, which in -the Mala position was an inch out of our space Ana, or W, from Urna -Mala, will now come into it an inch downwards, or [=Z], below Urna -Mala, with its Lar presented to us; that is, Thyrsus Lar will be below -Urna Lar. In the whole arrangement of them written below, the highest -floors are written first, for now they stretch downwards instead of -upwards. The name Lar is understood after each. - - Z X Y W - _ō_ _c_ _d_ _a_ - - SECOND BLOCK. - - Second Floor. { Uncus Pallor - { Ostrum Bidens - - First Floor. { Ocrea Tessera - { Cardo Cudo - - FIRST BLOCK. - - Second Floor. { Frenum Plebs - { *Urna Moles - - First Floor. { Crates Cura - { Thyrsus Vitta - -Here it is evident that what was the lower floor of Malas, Urna, Moles, -Plebs, Frenum, now appears as if carried downwards instead of upwards, -Lars being presented in our space instead of Malas. This Block of Lars -is what we see of the tessaract Set when the Arctos line, which in the -Mala position goes up, is turned into the Ana, or W, direction, and the -Ops line comes in downwards. - -The rest of the tessaracts, which consists of the cubes opposite to the -four treated above, and of the tessaractic space between them, is all -Ana in our space. If the tessaract be moved through our space--for -instance, when the Lars are present in it--we see, taking Urna alone, -first the section between Urna Lar and Urna Velum (Model 10), and then -Urna Velum (Model 4), and similarly the sections and Velums of each -tessaract in the Set. When the First Block has passed Kata our space, -Ostrum Lar enters; and the Lars of the Second Block of tessaracts occupy -the places just vacated by the Velums of the First Block. Then, as the -tessaractic Set moves on Kata, the sections between Velums and Lars of -the Second Block of tessaracts enter our space, and finally their -Velums. Then the whole tessaractic Set disappears from our space. - -When we have learnt all these aspects and passages, we have experienced -some of the principal features of this small Set of tessaracts. - - -CHAPTER IX. - -FURTHER STUDY OF TESSARACTS. - -When the arrangement of a small set has been mastered, the different -views of the whole 81 Set should be learnt. It is now clear to us that, -in the list of the names of the eighty-one tessaracts given above, those -which lie in the W direction appear in different blocks, while those -that lie in the Z, X, or Y directions can be found in the same block. -Therefore, from the arrangement given, which is denoted by - - Z X Y W , - _a_ _c_ _d_ _o_ - -or more briefly by _a c d o_, we can form any other arrangement. - -To confirm the meaning of the symbol _a c d o_ for position, let us -remember that the order of the axes known in our space will invariably -be Z X Y, and the unknown direction will be stated last, thus: Z X Y W. -Hence, if we write _a ō d c_, we know that the position or aspect -intended is that in which Arctos (_a_) goes Z, Ops (_ō_) negative X, Dos -(_d_) Y, and Cuspis (_c_) W. And such an arrangement can be made by -shifting the nine cubes on the left side of the First Block of the -eighty-one tessaracts, and putting them into the Z [=X] Y octant, so -that they just touch their former position. Next to them, to their left, -we set the nine of the left side of the Second Block of the 81 Set; and -next to these again, on their left, the nine of the left side of the -Third Block. This Block of twenty-seven now represents Vesper Cubes, -which have only one square identical with the Mala cubes of the -previous blocks, from which they were taken. - -Similarly the Block which is one inch Ana, can be made by taking the -nine cubes which come vertically in the middle of each of the Blocks in -the first position, and arranging them in a similar manner. Lastly, the -Block which lies two inches Ana, can be made by taking the right sides -of nine cubes each from each of the three original Blocks, and arranging -them so that those in the Second original Block go to the left of those -in the First, and those in the Third to their left. In this manner we -should obtain three new Blocks, which represent what we can see of the -tessaracts, when the direction in which Urna, Moles, Saltus lie in the -original Set, is turned W. - -The Pluvium Block we can make by taking the front wall of each original -Block, and setting each an inch nearer to us, that is -Y. The far sides -of these cubes are Moenas of Pluviums. By continuing this treatment of -the other walls of the three original Blocks parallel to the front wall, -we obtain two other Blocks of tessaracts. The three together are the -tessaractic position _a c ō d_, for in all of them Ops lies in the -Y -direction, and Dos has been turned W. - -The Lar position is more difficult to construct. To put the Lars of the -Blocks in their natural position in our space, we must start with the -original Mala Blocks, at least three inches above the table. The First -Lar Block is made by taking the lowest floors of the three Mala Blocks, -and placing them so that that of the Second is below that of the First, -and that of the Third below that of the Second. The floor of cubes whose -diagonal runs from Urna Lar to Remus Lar, will be at the top of the -Block of Lars; and that whose diagonal goes from Cervix Lar to Angusta -Lar, will be at the bottom. The next Block of Lars will be made by -taking the middle horizontal floors of the three original Blocks, and -placing them in a similar succession--the floor from Ostrum Lar to Aer -Lar being at the top, that from Cardo Lar to Colus Lar in the middle, -and Verbum Lar to Tabula Lar at the bottom. The Third Lar Block is -composed of the top floor of the First Block on the top--that is, of -Comes Lar to Tyro Lar, of Cortex Lar to Pluma Lar in the middle, and -Axis Lar to Portio Lar at the bottom. - - -CHAPTER X. - -CYCLICAL PROJECTIONS. - -Let us denote the original position of the cube, that wherein Arctos -goes Z, Cuspis X, and Dos Y, by the expression, - - Z X Y (1) - _a_ _c_ _d_ - -If the cube be turned round Cuspis, Dos goes [=Z], Cuspis remains -unchanged, and Arctos goes Y, and we have the position, - - 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 intersect. We might write - - [=Z] - _d_ - -but it is preferable to write - - Z - _[=d]_. - -If we next turn the cube round the line, which runs Y, that is, round -Arctos, we obtain the position, - - Z X Y (2) - _c_ _d_ _a_ - -and by means of this double turn we have put _c_ and _d_ in the places -of _a_ and _c_. Moreover, we have no negative directions. This position -we call simply _c d a_. If from it we turn the cube round _a_, which -runs Y, we get - - Z X Y - _d_ _[=c]_ _a_, - -and if, then, we turn it round Dos we get - - Z X Y - _d_ _a_ _c_ - -or simply _d a c_. This last is another position in which all the lines -are positive, and the projections, instead of lying in different -quadrants, will be contained in one. - -The arrangement of cubes in _a c d_ we know. That in _c d a_ is: - - { Vestis Oliva Tyro - Third { Scena Tergum Aer - Floor. { Saltus Sypho Remus - - { Tibicen Mora Merces - Second { Bidens Pallor Cortis - Floor. { Moles Plebs Hama - - { Comes Spicula Mars - First { Ostrum Uncus Ala - Floor. { Urna Frenum Sector - -It will be found that learning the cubes in this position gives a great -advantage, for thereby the axes of the cube become dissociated with -particular directions in space. - -The _d a c_ position gives the following arrangement: - - 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 sides, which touch the vertical plane in the first position, are -respectively, in _a c d_ Moena, in _c d a_ Syce, in _d a c_ Alvus. - -Take the shape Urna, Ostrum, Moles, Saltus, Scena, Sypho, Remus, Aer, -Tyro. This gives in _a c d_ the projection: Urna Moena, Ostrum Moena, -Moles Moena, Saltus Moena, Scena Moena, Vestis Moena. (If the different -positions of the cube are not well known, it is best to have a list of -the names before one, but in every case the block should also be built, -as well as the names used.) The same shape in the position _c d a_ is, -of course, expressed in the same words, but it has a different -appearance. 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 Urna, and we know that Ostrum lies -behind Urna, and does not come in; next we have Moles, Saltus, and we -know that Scena lies behind Saltus and does not come in; lastly, we have -Sypho and Remus, and we know that Aer and Tyro are in the Y direction -from Remus, and so do not come in. Hence, altogether the projection will -consist only of the Syces of Urna, Moles, Saltus, Sypho, and Remus. - -Next, taking the position _d a c_, the cubes in the front face have -their Alvus sides against the plane, and are: - - Sector Ala Mars - Frenum Uncus Spicula - Urna Ostrum Comes - -And, taking the shape, we find Urna, Ostrum; Moles and Saltus are hidden -by Urna, Scena is behind Ostrum, Sypho gives Frenum, Remus gives Sector, -Aer gives Ala, and Tyro gives Mars. All these are Alvus sides. - -Let us now take the reverse problem, and, given the three cyclical -projections, determine the shape. Let the _a c d_ projection be the -Moenas of Urna, Ostrum, Bidens, Scena, Vestis. Let the _c d a_ be the -Syces of Urna, Frenum, Plebs, Sypho, and the _d a c_ be the Alvus of -Urna, Frenum, Uncus, Spicula. Now, from _a c d_ we have Urna, Frenum, -Sector, Ostrum, Uncus, Ala, Bidens, Pallor, Cortis, Scena, Tergum, Aer, -Vestis, Oliva, Tyro. From _c d a_ we have Urna, Ostrum, Comes, Frenum, -Uncus, Spicula, Plebs, Pallor, Mora, Sypho, Tergum, Oliva. In order to -see how these will modify each other, let us consider the _a c d_ -solution as if it were a set of cubes in the _c d a_ arrangement. Here, -those that go in the Arctos direction, go away from the plane of -projection, and must be represented by the Syce of the cube in contact -with the plane. Looking at the _a c d_ solution we write down (keeping -those together which go away from the plane of projection): Urna and -Ostrum, Frenum and Uncus, Sector and Ala, Bidens, Pallor, Cortis, Scena -and Vestis, Tergum and Oliva, Aer and Tyro. Here we see that the whole -_c d a_ face is filled up in the projection, as far as this solution is -concerned. But in the _c d a_ solution we have only Syces of Urna, -Frenum, Plebs, Sypho. These Syces only indicate the presence of a -certain number of the cubes stated above as possible from the Moena -projection, and those are Urna, Ostrum, Frenum, Uncus, Pallor, Tergum, -Oliva. This is the result of a comparison of the Moena projection with -the Syce projection. Now, writing these last named as they come in the -_d a c_ projection, we have Urna, Ostrum, Frenum, Uncus and Pallor and -Tergum, Oliva. And of these Ostrum Alvus is wanting in the _d a c_ -projection as given above. Hence Ostrum will be wanting in the final -shape, and we have as the final solution: Urna, Frenum, Uncus, Pallor, -Tergum, Oliva. - - -CHAPTER XI. - -A TESSARACTIC FIGURE AND ITS PROJECTIONS. - -We will now consider a fourth-dimensional shape composed of tessaracts, -and the manner in which we can obtain a conception of it. The operation -is precisely analogous to that described in chapter VI., by which a -plane being could obtain a conception of solid shapes. It is only a -little more difficult in that we have to deal with one dimension or -direction more, and can only do so symbolically. - -We will assume the shape to consist of a certain number of the 81 -tessaracts, whose names we have given on p. 168. Let it consist of the -thirteen tessaracts: Urna, Moles, Plebs, Frenum, Pallor, Tessera, Cudo, -Vitta, Cura, Penates, Polus, Orcus, Lacerta. - -Firstly, we will consider what appearances or projections these -tessaracts will present to us according as the tessaractic set touches -our space with its (_a_) Mala cubes, (_b_) Vesper cubes, (_c_) Pluvium -cubes, or (_d_) Lar cubes. Secondly, we will treat the converse -question, how the shape can be determined when the projections in each -of those views are given. - -Let us build up in cubes the four different arrangements of the -tessaracts according as they enter our space on their Mala, Vesper, -Pluvium or Lar sides. They can only be printed by symbolizing two of the -directions. In the following tabulations the directions Y, X will at -once be understood. The direction Z (expressed by the wavy line) -indicates that the floors of nine, each printed nearer the top of the -page, lie above those printed nearer the bottom of it. The direction W -is indicated by the dotted line, which shows that the floors of nine -lying to the left or right are in the W direction (Ana) or the -W -direction (Kata) from those which lie to the right or left. For -instance, in the arrangement of the tessaracts, as Malas (Table A) the -tessaract Tessara, which is exactly in the middle of the eighty-one -tessaracts has - - Domitor on its right side or in the X direction. - 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 Urna, with Thyrsus -between them. And to take one more instance, a journey from Saltus to -Arcus would be made by travelling Y to Remus, thence -X to Sector, -thence Z to Mars, and finally W to Arcus. A line from Saltus to Arcus is -therefore a diagonal of the set of 81 tessaracts, because the full -length of its side has been traversed in each of the four directions to -reach one from the other, _i.e._ Saltus to Remus, Remus to Sector, -Sector to Mars, Mars to Arcus. - -TABLE A. - -Mala presentation of 81 Tessaracts. - - 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 A: - Arcus Ovis Portio - Laurus Tigris Segmen - Axis Troja Aries - - Block B: - Ara Vomer Pluma - Praeda Sacerdos Hydra - Cortex Mica Flagellum - - Block C: - Mars Merces Tyro - Spicula Mora Oliva - Comes Tibicen Vestis - - Block D: - Postis Clipeus Tabula - _Orcus_ _Lacerta_ Testudo - Verbum Luctus Anguis - - Block E: - Pilum Glans Coins - Ocrea _Tessera_ Domitor - Cardo _Cudo_ Malleus - - Block F: - Ala Cortis Aer - Uncus‡ _Pallor_‡ Tergum - Ostrum Bidens‡ Scena - - Block G: - Telum Nepos Angusta - _Polus_ _Penates_ Vulcan - Cervix Securis Vinculum - - Block H: - Agmen Lacus Arvus - Crates _Cura_ Limen - Thyrsus _Vitta_ Sceptrum - - Block I: - Sector Hama Remus - _Frenum_‡ _Plebs_‡ Sypho - _Urna_‡ _Moles_‡ Saltus - -TABLE B. - -Vesper presentation of 81 Tessaracts. - - 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 A: - Portio Pluma Tyro - Segmen Hydra Oliva - Aries Flagellum Vestis - - Block B: - Ovis Vomer Merces - Tigris Sacerdos Mora - Troja Mica Tibicen - - Block C: - Arcus Ara Mars - Laurus Praeda Spicula - Axis Cortex Comes - - Block D: - Tabula Colus Aer - Testudo Domitor Tergum - Anguis Malleus Scena - - Block E: - Clipeus Glans Cortis - _Lacerta_* _Tessera_* _Pallor_* - Luctus* _Cudo_* Bidens* - - Block F: - Postis Pilum Ala - _Orcus_* Ocrea* Uncus* - Verbum† Cardo† Ostrum† - - Block G: - Angusta Arvus Remus - Vulcan Limen Sypho - Vinculum Sceptrum Saltus - - Block H: - Nepos Lacus Hama - _Penates_* _Cura_* _Plebs_* - Securis* _Vitta_* _Moles_* - - Block I: - Telum Agmen Sector - _Polus_* Crates* _Frenum_* - Cervix* Thyrsus* _Urna_* - -TABLE C. - -Pluvium presentation of 81 Tessaracts. - - 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 A: - Mars Merces Tyro - Ara Vomer Pluma - Arcus Ovis Portio - - Block B: - Spicula Mora Oliva - Praeda Sacerdos Hydra - Laurus Tigris Segmen - - Block C: - Comes Tibicen Vestis - Cortex Mica Flagellum - Axis Troja Aries - - Block D: - Ala Cortis Aer - Pilum Glans Colus - Postis Clipeus Tabula - - Block E: - Uncus* _Pallor_* Tergum - Ocrea* _Tessera_* Domitor - _Orcus_* _Lacerta_* Testudo - - Block F: - Ostrum† Bidens† Scena - Cardo† _Cudo_* Malleus - Verbum† Luctus† Anguis - - Block G: - Sector Hama Remus - Agmen Lacus Arvus - Telum Nepos Angusta - - Block H: - _Frenum_* _Plebs_* Sypho - Crates* _Cura_* Limen - _Polus_* _Penates_* Vulcan - - Block I: - _Urna_* _Moles_* Saltus - Thyrsus* _Vitta_* Sceptrum - Cervix† Securis† Vinculum - -TABLE D. - -Lar presentation of 81 Tessaracts. - - 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 A: - Mars Merces Tyro - Spicula Mora Oliva - Comes Tibicen Vestis - - Block B: - Ala Cortis Aer - Uncus _Pallor_* Tergum - Ostrum Bidens Scena - - Block C: - Sector Hama Remus - _Frenum_* _Plebs_* Sypho - _Urna_* _Moles_* Saltus - - Block D: - Ara Vomer Pluma - Proeda Sacerdos Hydra - Cortex Mica Flagellum - - Block E: - Pilum Glans Colus - Ocrea _Tessera_* Domitor - Cardo _Cudo_* Malleus - - Block F: - Agmen Laurus Arvus - Crates _Cura_* Limen - Thyrsus _Vitta_* Sceptrum - - Block G: - Arcus Ovis Portio - Laurus Tigris Segmen - Axis Troja Aries - - Block H: - Postis Clipeus Tabula - _Orcus_* _Lacerta_* Testudo - Verbum Luctus Anguis - - Block I: - Telum Nepos Angusta - _Polus_* _Penates_* Vulcan - Cervix Securis Vinculum - -The relation between the four different arrangements shown in the tables -A, B, C, and D, will be understood from what has been said in chapter -VIII. about a small set of sixteen tessaracts. A glance at the lines, -which indicate the directions in each, will show the changes effected by -turning the tessaracts from the Mala presentation. - - In the Vesper presentation: - - The tessaracts-- - (_e.g._ Urna, Ostrum, Comes), which ran Z still run Z. - (_e.g._ Urna, Frenum, Sector), „ Y „ Y. - (_e.g._ Urna, Moles, Saltus), „ X now run W. - (_e.g._ Urna, Thyrsus, Cervix), „ W „ -X. - - In the Pluvium presentation: - - The tessaracts-- - (_e.g._ Urna, Ostrum, Comes), which ran Z still run Z. - (_e.g._ Urna, Moles, Saltus), „ X „ X. - (_e.g._ Urna, Frenum, Sector), „ Y now run W. - (_e.g._ Urna, Thyrsus, Cervix), „ W „ -Y. - - In the Lar presentation: - - The tessaracts-- - (_e.g._ Urna, Moles, Saltus), which ran X still run X. - (_e.g._ Urna, Frenum, Sector), „ Y „ Y. - (_e.g._ Urna, Ostrum, Comes), „ Z now run W. - (_e.g._ Urna, Thyrsus, Cervix), „ W „ -Z. - -This relation was already treated in chapter IX., but it is well to have -it very clear for our present purpose. For it is the apparent change of -the relative positions of the tessaracts in each presentation, which -enables us to determine any body of them. - -In considering the projections, we always suppose ourselves to be -situated Ana or W towards the tessaracts, and any movement to be Kata or --W through our space. For instance, in the Mala presentation we have -first in our space the Malas of that block of tessaracts, which is the -last in the -W direction. Thus, the Mala projection of any given -tessaract of the set is that Mala in the extreme -W block, whose place -its (the given tessaract’s) Mala would occupy, if the tessaractic set -moved Kata until the given tessaract reached our space. Or, in other -words, if all the tessaracts were transparent except those which -constitute the body under consideration, and if a light shone through -Four-space from the Ana (W) side to the Kata (-W) side, there would be -darkness in each of those Malas, which would be occupied by the Mala of -any opaque tessaract, if the tessaractic set moved Kata. - -Let us look at the set of 81 tessaracts we have built up in the Mala -arrangements, and trace the projections in the extreme -W block of the -thirteen of our shape. The latter are printed in italics in Table A, and -their projections are marked ‡. - -Thus the cube Uncus Mala is the projection of the tessaract Orcus, -Pallor Mala of Pallor and Tessera and Tacerta, Bidens Mala of Cudo, -Frenum Mala of Frenum and Polus, Plebs Mala of Plebs and Cura and -Penates, Moles Mala of Moles and Vitta, Urna Mala of Urna. - -Similarly, we can trace the Vesper projections (Table B). Orcus Vesper -is the projection of the tessaracts Orcus and Lacerta, Ocrea Vesper of -Tessera, Uncus Vesper of Pallor, Cardo Vesper of Cudo, Polus Vesper of -Polus and Penates, Crates Vesper of Cura, Frenum Vesper of Frenum and -Plebs, Urna Vesper of Urna and Moles, Thyrsus Vesper of Vitta. Next in -the Pluvium presentation (Table C) we find that Bidens Pluvium is the -projection of the tessaract Pallor, Cudo Pluvium of Cudo and Tessera, -Luctus Pluvium of Lacerta, Verbum Pluvium of Orcus, Urna Pluvium of Urna -and Frenum, Moles Pluvium of Moles and Plebs, Vitta Pluvium of Vitta and -Cura, Securis Pluvium of Penates, Cervix Pluvium of Polus. Lastly, in -the Lar presentation (Table D) we observe that Frenum Lar is the -projection of Frenum, Plebs Lar of Plebs and Pallor, Moles Lar of -Moles, Urna Lar of Urna, Cura Lar of Cura and Tessara, Vitta Lar of -Vitta and Cudo, Penates Lar of Penates and Lacerta, Polur Lar of Polus -and Orcus. - -Secondly, we will treat the converse problem, how to determine the shape -when the projections in each presentation are given. Looking back at the -list just given above, let us write down in each presentation the -projections only. - - Mala projections: - - Uncus, Pallor, Bidens, Frenum, Plebs, Moles, Urna. - - Vesper projections: - - Orcus, Ocrea, Uncus, Cardo, Polus, Crates, Frenum, Urna, Thyrsus. - - Pluvium projections: - - Bidens, Cudo, Luctus, Verbum, Urna, Moles, Vitta, Securis, Cervix. - - Lar projections: - - Frenum, Plebs, Moles, Urna, Cura, Vitta, Polus, Penates. - -Now let us determine the shape indicated by these projections. In now -using the same tables we must not notice the italics, as the shape is -supposed to be unknown. It is assumed that the reader is building the -problem in cubes. 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 presentation. - - (Uncus, Ocrea, Orcus); (Pallor, Tessera, Lacerta); - - (Bidens, Cudo, Luctus); (Frenum, Crates, Polus); - - (Plebs, Cura, Penates); (Moles, Vitta, Securis); - - (Urna, Thyrsus, Cervix). - -Let us suppose them all to be present in our shape, and observe what -their appearance would be in the Vesper presentation. We mark them all -with an asterisk in Table B. In addition to those already marked we must -mark (†) Verbum, Cardo, Ostrum, and then we see all the Vesper -projections, which would be formed by all the tessaracts possible from -the Mala projections. Let us compare these Vesper projections, viz. -Orcus, Ocrea, Uncus, Verbum, Cardo, Ostrum, Polus, Crates, Frenum, -Cervix, Thyrsus, Urna, with the given Vesper projections. We see at once -that Verbum, Ostrum, and Cervix are absent. Therefore, we may conclude -that all the tessaracts, which would be implied as possible by their -presence, are absent, and of the Mala possibilities may exclude the -tessaracts Bidens, Luctus, Securis, and Cervix itself. Thus, of the 21 -tessaracts possible in the Mala view, there remain only 17 possible, -both in the Mala and Vesper views, viz. Uncus, Ocrea, Orcus, Pallor, -Tessera, Lacerta, Cudo, Frenum, Crates, Polus, Plebs, Cura, Penates, -Moles, Vitta, Urna, Thyrsus. This we call the Mala-Vesper solution. - -Next let us take the Pluvium presentation. We again mark with an -asterisk in Table C the possibilities inferred from the Mala-Vesper -solution, and take the projections those possibilities would produce. -The additional projections are again marked (†). There are twelve -Pluvium projections altogether, viz. Bidens, Ostrum, Cudo, Cardo, -Luctus, Verbum, Urna, Moles, Vitta, Thyrsus, Securis, Cervix. Again we -compare these with the given Pluvium projections, and find three are -absent, viz. Ostrum, Cardo, Thyrsus. Hence the tessaracts implied by -Ostrum and Cardo and Thyrsus cannot be in our shape, viz. Uncus, Ocrea, -Crates, nor Thyrsus itself. Excluding these four from the seventeen -possibilities of the Mala-Vesper solution we have left the thirteen -tessaracts: Orcus, Pallor, Tessera, Lacerta, Cudo, Frenum, Polus, -Plebs, Cura, Penates, Moles, Vitta, Urna. This we call the -Mala-Vesper-Pluvium solution. - -Lastly, we have to consider whether these thirteen tessaracts are -consistent with the given Lar projections. We mark them again on Table D -with an asterisk, and we find that the projections are exactly those -given, viz. Frenum, Plebs, Moles, Urna, Cura, Vitta, Polus, Penates. -Therefore, we have not to exclude any of the thirteen, and can infer -that they constitute the shape, which produces the four different given -views or projections. - -In fine, any shape in space consists of the possibilities common to the -projections of its parts upon the boundaries of that space, whatever be -the number of its dimensions. Hence the simple rule for the -determination of the shape would be to write down all the possibilities -of the sets of projections, and then cancel all those possibilities -which are not common to all. But the process adopted above is much -preferable, as through it we may realize the gradual delimitation of the -shape view by view. For once more we must remind ourselves that our -great object is, not to arrive at results by symbolical operations, but -to realize those results piece by piece through realized processes. - - - - -APPENDICES. - - -APPENDIX A. - -This set of 100 names is useful for studying Plane Space, and forms a -square 10 × 10. - - Aiōn Bios Hupar Neas Kairos Enos Thlipsis Cheimas Theion Epei - - Itea Hagios Phaino Geras Tholos Ergon Pachūs Kiōn Eris Cleos - - Loma Etēs Trochos Klazo Lutron Hēdūs Ischūs Paigma Hedna Demas - - Numphe Bathus Pauo Euthu Holos Para Thuos Karē Pylē Spareis - - Ania Eōn Seranx Mesoi Dramo Thallos Aktē Ozo Onos Magos - - Notos Mēnis Lampas Ornis Thama Eni Pholis Mala Strizo Rudon - - Labo Helor Rupa Rabdos Doru Epos Theos Idris Ēdē Hepo - - Sophos Ichor Kaneōn Ephthra Oxis Lukē Blue Helos Peri Thelus - - Eunis Limos Keedo Igde Matē Lukos Pteris Holmos Oulo Dokos - - Aeido Ias Assa Muzo Hippeus Eōs Atē Akme Ōrē Gua - - -APPENDIX B. - -The following list of names is used to denote cubic spaces. It makes a -cubic block of six floors, the highest being the sixth. - - _ 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 r. 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 r. 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 r. 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 r. 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 r. 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 r. Turba Ager Trabs Myrtus Fibra Nauta - t _ Decus Pulvis Meta Rota Palma Terra - - -APPENDIX C. - -The following names are used for a set of 256 Tessaracts. - - FOURTH BLOCK. THIRD BLOCK. - - _Fourth Floor._ _Fourth Floor._ - Dolium Caballus Python Circaea Charta Cures Quaestor Cliens - Cussis Pulsus Drachma Cordax Frux Pyra Lena Procella - Porrum Consul Diota Dyka Hera Esca Secta Rugæ - Columen Ravis Corbis Rapina Eurus Gloria Socer Sequela - - _Third Floor._ _Third Floor._ - 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 Floor._ _Second Floor._ - 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 Floor._ _First Floor._ - 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 BLOCK. FIRST BLOCK. - - _Fourth Floor._ _Fourth Floor._ - 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 Floor._ _Third Floor._ - 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 Floor._ _Second Floor._ - 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 Floor._ _First Floor._ - 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 D. - -The following list gives the colours, and the various uses for them. -They have already been used in the foregoing pages to distinguish the -various regions of the Tessaract, and the different individual cubes or -Tessaracts in a block. The other use suggested in the last column of the -list has not been discussed; but it is believed that it may afford great -aid to the mind in amassing, handling, and retaining the quantities of -formulae requisite in scientific training and work. - - _Region of _Tessaract - _Colour._ Tessaract._ in 81 Set._ _Symbol._ - Black Syce Plebs 0 - White Mel Mora 1 - Vermilion Alvus Uncus 2 - Orange Cuspis Moles 3 - Light-yellow Murex Cortis 4 - Bright-green Lappa Penates 5 - Bright-blue Iter Oliva 6 - Light-grey Lares Tigris 7 - Indian-red Crux Orcus 8 - Yellow-ochre Sal Testudo 9 - Buff Cista Sector + (plus) - Wood Tessaract Tessara - (minus) - Brown-green Tholus Troja ± (plus or minus) - Sage-green Margo Lacerta × (multiplied by) - Reddish Callis Tibicen ÷ (divided by) - Chocolate Velum Sacerdos = (equal to) - French-grey Far Scena ≠ (not equal to) - Brown Arctos Ostrum > (greater than) - Dark-slate Daps Aer < (less than) - Dun Portica Clipeus ∶ } signs - Orange-vermilion Talus Portio ∷ } of proportion - Stone Ops Thyrsus · (decimal point) - Quaker-green Felis Axis ∟ (factorial) - Leaden Semita Merces ∥ (parallel) - Dull-green Mappa Vulcan ∦ (not parallel) - Indigo Lixa Postis π⁄2 (90°) (at right - angles) - Dull-blue Pagus Verbum log. base 10 - Dark-purple Mensura Nepos sin. (sine) - Pale-pink Vena Tabula cos. (cosine) - Dark-blue Moena Bidens tan. (tangent) - Earthen Mugil Angusta ∞ (infinity) - 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 α - Crimson Bolus Sypho β - Ochre Limbus Mica γ - Purple Solia Arcus δ - Leaf-green Luca Securis ε - Turquoise Ancilla Vinculum ζ - Dark-grey Orca Colus η - Fawn Nugæ Saltus θ - Smoke Limus Sceptrum ι - Light-buff Mala Pallor κ - Dull-purple Sors Vestis λ - Rich-red Lucta Cortex μ - Green-blue Pator Flagellum ν - Burnt-sienna Silex Luctus ξ - Sea-blue Lorica Lacus ο - Peacock-blue Passer Aries π - Deep-brown Meatus Hydra ρ - Dark-pink Onager Anguis σ - Dark Lensa Laurus τ - Dark-stone Pluvium Cudo υ - Silver Spira Cervix φ - Gold Corvus Urna χ - Deep-yellow Via Spicula ψ - Dark-green Calor Segmen ω - - -APPENDIX E. - -A THEOREM IN FOUR-SPACE. - -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 base; then the sides of these two triangles, if produced -in pairs, will meet in three points which are in a straight line, -namely, the line of intersection of the sectional plane and the plane of -the base. - -Let A B C D be a pyramid on a triangular base A B C, and let a b c be a -section such that A B, B C, A C, are respectively not parallel to a b, -b c, a c. It must be understood that a is a point on A D, b is a point -on B D, and c a point on C D. Let, A B and a b, produced, meet in m. B C -and b c, produced, meet in n; and A C and a c, produced, meet in o. -These three points, m, n, o, are in the line of intersection of the two -planes A B C and a b c. - -Now, let the line a b be projected on to the plane of the base, by -drawing lines from a and b at right angles to the base, and meeting it -in a′ b′; the line a′ b′, produced, will meet A B produced in m. If the -lines b c and a c be projected in the same way on to the base, to the -points b′ c′ and a′ c′; then B C and b′ c′ produced, will meet in n, and -A C and a′ c′ produced, will meet in o. The two triangles A B C and -a′ b′ c′ are such, that the lines joining A to a′, B to b′, and C to c′, -will, if produced, meet in a point, namely, the point on the base A B C -which is the projection of D. Any two triangles which fulfil this -condition are the possible base and projection of the section of a -pyramid; therefore the sides of such triangles, if produced in pairs, -will meet (if they are not parallel) in three points which lie in one -straight line. - -A four-dimensional pyramid may be defined as a figure bounded by a -polyhedron of any number of sides, and the same number of pyramids whose -bases are the sides of the polyhedron, and whose apices meet in a point -not in the space of the base. - -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 -base; then the sides of these two tetrahedra, if produced in pairs, will -meet in lines which all lie in one plane, namely, the plane of -intersection of the space of the base and the space of the section. - -If now the sectional tetrahedron be projected on to the base (by drawing -lines from each point of the section to the base at right angles to it), -there will be two tetrahedra fulfilling the condition that the line -joining the angles of the one to the angles of the other will, if -produced, meet in a point, which point is the projection of the apex of -the four-dimensional pyramid. - -Any two tetrahedra which fulfil this condition, are the possible base -and projection of a section of a four-dimensional pyramid. Therefore, in -any two such tetrahedra, where the sides of the one are not parallel to -the sides of the other, the sides, if produced in pairs (one side of the -one with one side of the other), will meet in four straight lines which -are all in one plane. - - -APPENDIX F. - - -EXERCISES ON SHAPES OF THREE DIMENSIONS. - -The names used are those given in Appendix B. - -Find the shapes from the following projections: - - 1. Syce projections: Ratis, Caput, Castrum, Plagua. - - Alvus projections: Merum, Oculus, Fulmen, Pruinus. - - Moena projections: Miles, Ventus, Navis. - - 2. Syce: Dies, Tuba, Lituus, Frons. - - Alvus: Sagitta, Regnum, Tellus, Fulmen, Pruinus. - - Moena: Tibia, Tunica, Robur, Finis. - - 3. Syce: Nemus, Sidus, Vertex, Nix, Cerva. - - Alvus: Lignum, Haedus, Vultus, Nemus, Humerus. - - Moena: Dexter, Princeps, Equus, Dux, Urbs, Pullis, Gens, Monstrum, - Miles. - - 4. Syce: Amphora, Castrum, Myrtus, Rota, Palma, Meta, Trabs, Ratis. - - Alvus: Dexter, Princeps, Moena, Aes, Merum, Oculus, Littus, Civis, - Fulmen. - - Moena: Gens, Ventus, Navis, Finis, Monstrum, Cursus. - - 5. Syce: Castrum, Plagua, Nix, Vertex, Aper, Caput, Cerva, Venator. - - Alvus: Triumphus, Tellus, Caterva, Lignum, Haedus, Pruinus, Fulmen, - Civis, Humerus, Vultus. - - Moena: Pharetra, Cursus, Miles, Equus, Dux, Navis, Monstrum, Gens, - Urbs, Dexter. - - -ANSWERS. - -The shapes are: - - 1. Umbra, Aether, Ver, Carina, Flos. - - 2. Pontus, Custos, Jaculum, Pratum, Arator, Agna. - - 3. Focus, Omus, Haedus, Tabema, Vultus, Hostis, Figura, Ales, Sidus, - Augur. - - 4. Tempus, Campus, Finis, Navis, Ventus, Pelagus, Notus, Cohors, - Aether, Carina, Res, Templum, Rex, Gens, Monstrum. - - 5. Portus, Arma, Sylva, Lucrum, Ornus, Onus, Os, Facies, Chorus, - Carina, Flos, Nox, Ales, Clamor, Res, Pugna, Ludus, Figura, Augur, - Humerus. - - -FURTHER EXERCISES IN SHAPES OF THREE DIMENSIONS. - -The Names used are those given in Appendix C; and this set of exercises -forms a preparation for their use in space of four dimensions. All are -in the 27 Block (Urna to Syrma). - - 1. Syce: Moles, Frenum, Plebs, Sypho. - - Alvus: Urna, Frenum, Uncus, Spicula, Comes. - - Moena: Moles, Bidens, Tibicen, Comes, Saltus. - - 2. Syce: Urna, Moles, Plebs, Hama, Remus. - - Alvus: Urna, Frenum, Sector, Ala, Mars. - - Moena: Urna, Moles, Saltus, Bidens, Tibicen. - - 3. Syce: Moles, Plebs, Hama, Remus. - - Alvus: Uma, Ostrum, Comes, Spicula, Frenum, Sector. - - Moena: Moles, Saltus, Bidens, Tibicen. - - 4. Syce: Frenum, Plebs, Sypho, Moles, Hama. - - Alvus: Urna, Frenum, Uncus, Sector, Spicula. - - Moena: Urna, Moles, Saltus, Scena, Vestis. - - 5. Syce: Urna, Moles, Plebs, Hama, Remus, Sector. - - Alvus: Urna, Frenum, Sector, Uncus, Spicula, Comes, Mars. - - Moena: Urna, Moles, Saltus, Bidens, Tibicen, Comes. - - 6. Syce: Uma, Moles, Saltus, Sypho, Remus, Hama, Sector. - - Alvus: Comes, Ostrum, Uncus, Spicula, Mars, Ala, Sector. - - Moena: Urna, Moles, Saltus, Scena, Vestis, Tibicen, Comes, Ostrum. - - 7. Syce: Sypho, Saltus, Moles, Urna, Frenum, Sector. - - Alvus: Urna, Frenum, Uncus, Spicula, Mars. - - Moena: Saltus, Moles, Urna, Ostrum, Comes. - - 8. Syce: Moles, Plebs, Hama, Sector. - - Alvus: Ostrum, Frenum, Uncus, Spicula, Mars, Ala. - - Moena: Moles, Bidens, Tibicen, Ostrum. - - 9. Syce: Moles, Saltus, Sypho, Plebs, Frenum, Sector. - - Alvus: Ostrum, Comes, Spicula, Mars, Ala. - - Moena: Ostrum, Comes, Tibicen, Bidens, Scena, Vestis. - - 10. Syce: Urna, Moles, Saltus, Sypho, Remus, Sector, Frenum. - - Alvus: Urna, Ostrum, Comes, Spicula, Mars, Ala, Sector. - - Moena: Urna, Ostrum, Comes, Tibicen, Vestis, Scena, Saltus. - - 11. Syce: Frenum, Plebs, Sypho, Hama. - - Alvus: Frenum, Sector, Ala, Mars, Spicula. - - Moena: Urna, Moles, Saltus, Bidens, Tibicen. - - -ANSWERS. - -The shapes are: - - 1. Moles, Plebs, Sypho, Pallor, Mora, Tibicen, Spicula. - - 2. Urna, Moles, Plebs, Hama, Cortis, Merces, Remus. - - 3. Moles, Bidens, Tibicen, Mora, Plebs, Hama, Remus. - - 4. Frenum, Plebs, Sypho, Tergum, Oliva, Moles, Hama. - - 5. Urna, Moles, Plebs, Hama, Remus, Pallor, Mora, Tibicen, Mars, - Merces, Comes, Sector. - - 6. Ostrum, Comes, Tibicen, Vestis, Scena, Tergum, Oliva, Tyro, Aer, - Remus, Hama, Sector, Merces, Mars, Ala. - - 7. Sypho, Saltus, Moles, Urna, Frenum, Uncus, Spicula, Mars. - - 8. Plebs, Pallor, Mora, Bidens, Merces, Cortis, Ala. - - 9. Bidens, Tibicen, Vestis, Scena, Oliva, Mora, Spicula, Mars, Ala. - - 10. Urna, Ostrum, Comes, Spicula, Mars, Tibicen, Vestis, Oliva, Tyro, - Aer, Remus, Sector, Ala, Saltus, Scena. - - 11. Frenum, Plebs, Sypho, Hama, Cortis, Merces, Mora. - - -APPENDIX G. - - -EXERCISES ON SHAPES OF FOUR DIMENSIONS. - -The Names used are those given in Appendix C. The first six exercises -are in the 81 Set, and the rest in the 256 Set. - - 1. Mala projection: Urna, Moles, Plebs, Pallor, Cortis, Merces. - - Lar projection: Urna, Moles, Plebs, Cura, Penates, Nepos. - - Pluvium projection: Urna, Moles, Vitta, Cudo, Luctus, Troja. - - Vesper projection: Urna, Frenum, Crates, Ocrea, Orcus, Postis, Arcus. - - 2. Mala: Urna, Frenum, Uncus, Pallor, Cortis, Aer. - - Lar: Urna, Frenum, Crates, Cura, Lacus, Arvus, Angusta. - - Pluvium: Urna, Thyrsus, Cardo, Cudo, Malleus, Anguis. - - Vesper: Urna, Frenum, Crates, Ocrea, Pilum, Postis. - - 3. Mala: Comes, Tibicen, Mora, Pallor. - - Lar: Urna, Moles, Vitta, Cura, Penates. - - Pluvium: Comes, Tibicen, Mica, Troja, Luctus. - - Vesper: Comes, Cortex, Praeda, Laurus, Orcus. - - 4. Mala: Vestis, Oliva, Tyro. - - Lar: Saltus, Sypho, Remus, Arvus, Angusta. - - Pluvium: Vestis, Flagellum, Aries. - - Vesper: Comes, Spicula, Mars, Ara, Arcus. - - 5. Mala: Mars, Merces, Tyro, Aer, Tergum, Pallor, Plebs. - - Lar: Sector, Hama, Lacus, Nepos, Angusta, Vulcan, Penates. - - Pluvium: Comes, Tibicen, Mica, Troja, Aries, Anguis, Luctus, Securis. - - Vesper: Mars, Ara, Arcus, Postis, Orcus, Polus. - - 6. Mala: Pallor, Mora, Oliva, Tyro, Merces, Mars, Spicula, Comes, - Tibicen, Vestis. - - Lar: Plebs, Cura, Penates, Vulcan, Angusta, Nepos, Telum, Polus, - Cervix, Securis, Vinculum. - - Pluvium: Bidens, Cudo, Luctus, Troja, Axis, Aries. - - Vesper: Uncus, Ocrea, Orcus, Laurus, Arcus, Axis. - - 7. Mala: Hospes, Tribus, Fragor, Aer, Tyro, Mora, Oliva. - - Lar: Hospes, Tectum, Rumor, Arvus, Angusta, Cera, Apis, Lapis. - - Pluvium: Acus, Torus, Malleus, Flagellum, Thorax, Aries, Aestas, - Capella. - - Vesper: Pardus, Rostrum, Ardor, Pilum, Ara, Arcus, Aestus, Septum. - - 8. Mala: Pallor, Tergum, Aer, Tyro, Cortis, Syrma, Ursa, Fama, Naxos, - Erisma. - - Lar: Plebs, Cura, Limen, Vulcan, Angusta, Nepos, Cera, Papaver, - Pignus, Messor. - - Pluvium: Bidens, Cudo, Malleus, Anguis, Aries, Luctus, Capella, Rheda, - Rapina. - - Vesper: Uncus, Ocrea, Orcus, Postis, Arcus, Aestus, Cussis, Dolium, - Alexis. - - 9. Mala: Fama, Conjux, Reus, Torus, Acus, Myrrha, Sypho, Plebs, - Pallor, Mora, Oliva, Alpis, Acies, Hircus. - - Lar: Missale, Fortuna, Vita, Pax, Furor, Ira, Vulcan, Penates, Lapis, - Apis, Cera, Pignus. - - Pluvium: Torus, Plenum, Pax, Thorax, Dolus, Furor, Vinculum, Securis, - Clavis, Gurges, Aestas, Capella, Corbis. - - Vesper: Uncus, Spicula, Mars, Ocrea, Cardo, Thyrsus, Cervix, Verbum, - Orcus, Polus, Spes, Senex, Septum, Porrum, Cussis, Dolium. - - -ANSWERS. - -The shapes are: - - 1. Urna, Moles, Plebs, Cura, Tessara, Lacerta, Clipeus, Ovis. - - 2. Urna, Frenum, Crates, Ocrea, Tessara, Glans, Colus, Tabula. - - 3. Comes, Tibicen, Mica, Sacerdos, Tigris, Lacerta. - - 4. Vestis, Oliva, Tyro, Pluma, Portio. - - 5. Mars, Merces, Vomer, Ovis, Portio, Tabula, Testudo, Lacerta, - Penates. - - 6. Pallor, Tessara, Lacerta, Tigris, Segmen, Portio, Ovis, Arcus, - Laurus, Axis, Troja, Aries. - - 7. Hospes, Tribus, Arista, Pellis, Colus, Pluma, Portio, Calathus, - Turtur, Sepes. - - 8. Pallor, Tessara, Domitor, Testudo, Tabula, Clipeus, Portio, - Calathus, Nux, Lectrum, Corymbus, Circaea, Cordax. - - 9. Fama, Conjux, Reus, Fera, Thorax, Pax, Furor, Dolus, Scala, Ira, - Vulcan, Penates, Lapis, Palus, Sepes, Turtur, Diota, Drachma, Python. - - -APPENDIX H. - - -SECTIONS OF CUBE AND TESSARACT. - -There are three kinds of sections of a cube. - -1. The sectional plane, which is in all cases supposed to be infinite, -can be taken parallel to two of the opposite faces of the cube; that is, -parallel to two of the lines meeting in Corvus, and cutting the third. - -2. The sectional plane can be taken parallel to one of the lines meeting -in Corvus and cutting the other two, or one or both of them produced. - -3. The sectional plane can be taken cutting all three lines, or any or -all of them produced. - -Take the first case, and suppose the plane cuts Dos half-way between -Corvus and Cista. Since it does not cut Arctos or Cuspis, or either of -them produced, it will cut Via, Iter, and Bolus at the middle point of -each; and the figure, determined by the intersection of the Plane and -Mala, is a square. If the length of the edge of the cube be taken as the -unit, this figure may be expressed thus: - - Z X Y - 0 . 0 . ¹⁄₂ - -showing that the Z and X lines from Corvus are not cut at all, and that -the Y line is cut at half-a-unit from Corvus. - -Sections taken - - Z X Y - 0 . 0 . ¹⁄₄ - -and - - Z X Y - 0 . 0 . 1 - -would also be squares. - -Take the second case. - -Let the plane cut Cuspis and Dos, each at half-a-unit from Corvus, and -not cut Arctos or Arctos produced; it will also cut through the middle -points of Via and Callis. The figure produced, is a rectangle which has -two sides of one unit, and the other two are each the diagonal of a -half-unit squared. - -If the plane cuts Cuspis and Dos, each at one unit from Corvus, and is -parallel to Arctos, the figure will be a rectangle which has two sides -of one unit in length; and the other two the diagonal of one unit -squared. - -If the plane passes through Mala, cutting Dos produced and Cuspis -produced, each at one-and-a-half unit from Corvus, and is parallel to -Arctos, the figure will be a parallelogram like the one obtained by the -section - - Z X Y - 0 . ¹⁄₂ . ¹⁄₂. - -This set of figures will be expressed - - Z X Y Z X Y Z X Y - 0 . ¹⁄₂ . ¹⁄₂ 0 . 1 . 1 0 . 1¹⁄₂ . 1¹⁄₂ - -It will be seen that these sections are parallel to each other; and that -in each figure Cuspis and Dos are cut at equal distances from Corvus. - -We may express the whole set thus:-- - - Z X Y - O . I . I - -it being understood that where Roman figures are used, the numbers do -not refer to the length of unit cut off any given line from Corvus, but -to the proportion between the lengths. Thus - - Z X Y - O . I . II - -means that Arctos is not cut at all, and that Cuspis and Dos are cut, -Dos being cut twice as far from Corvus as is Cuspis. - -These figures will also be rectangles. - -Take the third case. - -Suppose Arctos, Cuspis, and Dos are each cut half-way. This figure is an -equilateral triangle, whose sides are the diagonal of a half-unit -squared. The figure - - Z X Y - 1 . 1 . 1 - -is also an equilateral triangle, and the figure - - Z X Y - 1¹⁄₂ . 1¹⁄₂ . 1¹⁄₂ - -is an equilateral hexagon. - -It is easy for us to see what these shapes are, and also, what the -figures of any other set would be, as - - Z X Y - I . II . II - -or - - Z X Y - I . II . III - -but we must learn them as a two-dimensional being would, so that we may -see how to learn the three-dimensional sections of a tessaract. - -It is evident that the resulting figures are the same whether we fix the -cube, and then turn the sectional plane to the required position, or -whether we fix the sectional plane, and then turn the cube. Thus, in the -first case we might have fixed the plane, and then so placed the cube -that one plane side coincided with the sectional plane, and then have -drawn the cube half-way through, in a direction at right angles to the -plane, when we should have seen the square first mentioned. In the -second case - - (Z X Y) - (O . I . I) - -we might have put the cube with Arctos coinciding with the plane and -with Cuspis and Dos equally inclined to it, and then have drawn the cube -through the plane at right angles to it until the lines (Cuspis and Dos) -were cut at the required distances from Corvus. In the third case we -might have put the cube with only Corvus coinciding with the plane and -with Cuspis, Dos, and Arctos equally inclined to it (for any of the -shapes in the set - - Z X Y) - I . I . I) - -and then have drawn it through as before. The resulting figures are -exactly the same as those we got before; but this way is the best to -use, 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 round, with regard to the cube. - -We have already seen (p. 117) how a two-dimensional being would observe -the sections of a cube when it is put with one plane side coinciding -with his space, and is then drawn partly through. - -Now, suppose the cube put with the line Arctos coinciding with his -space, and the lines Cuspis and Dos equally inclined to it. At first he -would only see Arctos. If the cube were moved until Dos and Cuspis were -each cut half-way, Arctos still being parallel to the plane, Arctos -would disappear at once; and to find out what he would see he would have -to take the square sections of the cube, and find on each of them what -lines are given by the new set of sections. Thus he would take Moena -itself, which may be regarded as the first section of the square set. -One point of the figure would be the middle point of Cuspis, and since -the sectional plane is parallel to Arctos, the line of intersection of -Moena with the sectional plane will be parallel to Arctos. The required -line then cuts Cuspis half-way, and is parallel to Arctos, therefore it -cuts Callis half-way. - -[Illustration: Fig. 21.] - -Next he would take the square section half-way between Moena and Murex. -He knows that the line Alvus of this section is parallel to Arctos, and -that the point Dos at one of its ends is half-way between Corvus and -Cista, so that this line itself is the one he wants (because the -sectional plane cuts Dos half-way between Corvus and Cista, and is -parallel to Arctos). In Fig. 21 the two lines thus found are shown. a b -is the line in Moena, and c d the line in the section. He must now find -out how far apart they are. He knows that from the middle point of -Cuspis to Corvus is half-a-unit, and from the middle point of Dos to -Corvus is half-a-unit, and Cuspis and Dos are at right angles to each -other; 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 length. -This diagonal may be written d (¹⁄₂)². He would also see that from the -middle point of Callis to the middle point of Via is the same length; -therefore the figure is a parallelogram, having two of its sides, each -one unit in length, and the other two each d (¹⁄₂)². - -He could also see that the angles are right, because the lines a c and -b d are made up of the X and Y directions, and the other two, a b and -d, are purely Z, and since they have no tendency in common, they are at -right angles to each other. - -[Illustration: Fig. 22.] - -If he wanted the figure made by - - Z X Y - 0 . 1¹⁄₂ . 1¹⁄₂ - -it would be a little more difficult. He would have to take Moena, a -section halfway between Moena and Murex, Murex and another square which -he would have to regard as an _imaginary_ section half-a-unit further Y -than Murex (Fig. 22). He might now draw a ground plan of the sections; -that is, he would draw Syce, and produce Cuspis and Dos half-a-unit -beyond Nugæ and Cista. He would see that Cadus and Bolus would be cut -half-way, so that in the half-way section he would have the point a -(Fig. 23), and in Murex the point c. In the imaginary section he would -have g; but this he might disregard, since the cube goes no further than -Murex. From the points c and a there would be lines going Z, so that -Iter and Semita would be cut half-way. - -[Illustration: _Groundplan of Sections shown in Fig. 22._ - -Fig. 23.] - -He could find out how far the two lines a b and c d (Fig. 22) are apart -by referring d and b to Lama, and a and c to Crus. - -In taking the third order of sections, a similar method may be followed. - -[Illustration: Fig. 24.] - -Suppose the sectional plane to cut Cuspis, Dos, and Arctos, each at one -unit from Corvus. He would first take Moena, and as the sectional plane -passes through Ilex and Nugæ, the line on Moena would be the diagonal -passing through these two points. Then he would take Murex, and he would -see that as the plane cuts Dos at one unit from Corvus, all he would -have is the point Cista. So the whole figure is the Ilex to Nugæ -diagonal, and the point Cista. - -Now Cista and Ilex are each one inch from Corvus, and measured along -lines at right angles to each other; therefore, they are d (1)² from -each other. By referring Nugæ and Cista to Corvus he would find that -they are also d (1)² apart; therefore the figure is an equilateral -triangle, whose sides are each d (1)². - -Suppose the sectional plane to pass through Mala, cutting Cuspis, Dos, -and Arctos each at unit from Corvus. To find the figure, the plane-being -would have to take Moena, a section half-way between Moena and Murex, -Murex, and an imaginary section half-a-unit beyond Murex (Fig. 24). He -would produce Arctos and Cuspis to points half-a-unit from Ilex and -Nugæ, and by joining these points, he would see that the line passes -through the middle points of Callis and Far (a, b, Fig. 24). In the last -square, the imaginary section, there would be the point m; for this is -1¹⁄₂ unit from Corvus measured along Dos produced. There would also be -lines in the other two squares, the section and Murex, and to find these -he would have to make many observations. He found the points a and b -(Fig. 24) by drawing a line from r to s, r and s being each 1¹⁄₂ unit -from Corvus, and simply seeing that it cut Callis and Far at the middle -point of each. He might now imagine a cube Mala turned about Arctos, so -that Alvus came into his plane; he might then produce Arctos and Dos -until they were each unit long, and join their extremities, when he -would see that Via and Bucina are each cut half-way. Again, by turning -Syce into his plane, and producing Dos and Cuspis to points 1¹⁄₂ unit -from Corvus and joining the points, he would see that Bolus and Cadus -are cut half-way. He has now determined six points on Mala, through -which the plane passes, and by referring them in pairs to Ilex, Olus, -Cista, Crus, Nugæ, Sors, he would find that each was d (¹⁄₂)² from the -next; so he would know that the figure is an equilateral hexagon. The -angles he would not have got in this observation, and they might be a -serious difficulty to him. It should be observed that a similar -difficulty does not come to us in our observation of the sections of a -tessaract: for, if the angles of each side of a solid figure are -determined, the solid angles are also determined. - -There is another, and in some respects a better, way by which he might -have found the sides of this figure. If he had noticed his plane-space -much, he would have found out that, if a line be drawn to cut two other -lines which meet, the ratio of the parts of the two lines cut off by the -first line, on the side of the angle, is the same for those lines, and -any other two that are parallel to them. Thus, if a b and a c (Fig. 25) -meet, making an angle at a, and b c crosses them, and also crosses a′ b′ -and a′ c′, these last two being parallel to a b and a c, then a b ∶ -a c ∷ a′ b′ ∶ a′ c′. - -[Illustration: Fig. 25] - -If the plane-being knew this, he would rightly assume that if three -lines meet, making a solid angle, and a plane passes through them, the -ratio of the parts between the plane and the angle is the same for those -three lines, and for any other three parallel to them. - -In the case we are dealing with he knows that from Ilex to the point on -Arctos produced where the plane cuts, it is half-a-unit; and as the Z, -X, and Y lines are cut equally from Corvus, he would conclude that the X -and Y lines are cut the same distance from Ilex as the Z line, that is -half-a-unit. He knows that the X line is cut at 1¹⁄₂ units from Corvus; -that is, half-a-unit from Nugæ: so he would conclude that the Z and Y -lines are cut half-a-unit from Nugæ. He would also see that the Z and X -lines from Cista are cut at half-a-unit. He has now six points on the -cube, the middle points of Callis, Via, Bucina, Cadus, Bolus, and Far. -Now, looking at his square sections, he would see on Moena a line going -from middle of Far to middle of Callis, that is, a line d (¹⁄₂)² long. -On the section he would see a line from middle of Via to middle of Bolus -d (1)² long, and on Murex he would see a line from middle of Cadus to -middle of Bucina, d (¹⁄₂)² long. Of these three lines a b, c d, e f, -(Fig. 24)--a b and e f are sides, and c d is a section of the required -figure. He can find the distances between a and c by reference to -Ilex, between b and d by reference to Nugæ, between c and e by reference -to Olus, and between d and f by reference to Crus; and he will find that -these distances are each d (¹⁄₂)². - -Thus, he would know that the figure is an equilateral hexagon with its -sides d (¹⁄₂)² long, of which two of the opposite points (c and d) are d -(1)² apart, and the only figure fulfilling all these conditions is an -equilateral and equiangular hexagon. - -Enough has been said about sections of a cube, to show how a plane-being -would find the shapes in any set as in - - Z X Y - I . II . II - -or - - Z X Y - I . I . II. - -He would always have to bear in mind that the ratio of the lengths of -the Z, X, and Y lines is the same from Corvus to the sectional plane as -from any other point to the sectional plane. Thus, if he were taking a -section where the plane cuts Arctos and Cuspis at one unit from Corvus -and Dos at one-and-a-half, that is where the ratio of Z and of X to Y is -as two to three, he would see that Dos itself is not cut at all; but -from Cista to the point on Dos produced is half-a-unit; therefore from -Cista, the Z and X lines will be cut at ²⁄₃ of ¹⁄₂ unit from Cista. - -It is impossible in writing to show how to make the various sections of -a tessaract; and even if it were not so, it would be unadvisable; for -the value of doing it is not in seeing the shapes themselves, so much as -in the concentration of the mind on the tessaract involved in the -process of finding them out. - -Any one who wishes to make them should go carefully over the sections of -a cube, not looking at them as he himself can see them, or determining -them as he, with his three-dimensional conceptions, can; but he must -limit his imagination to two dimensions, and work through the problems -which a plane-being would have to work through, although to his higher -mind they may be self-evident. Thus a three-dimensional being can see at -a glance, that if a sectional plane passes through a cube at one unit -each way from Corvus, the resulting figure is an equilateral triangle. - -If he wished to prove it, he would show that the three bounding lines -are the diagonals of equal squares. This is all a two-dimensional being -would have to do; but it is not so evident to him that two of the lines -are the diagonals of squares. - -Moreover, when the figure is drawn, we can look at it from a point -outside the plane of the figure, and can thus see it all at once; but -he who has to look at it from a point in the plane can only see an edge -at a time, or he might see two edges in perspective together. - -Then there are certain suppositions he has to make. For instance, he -knows that two points determine a line, and he assumes that three points -determine a plane, although he cannot conceive any other plane than the -one in which he exists. We assume that four points determine a solid -space. Or rather, we say that _if_ this supposition, together with -certain others of a like nature, are true, we can find all the sections -of a tessaract, and of other four-dimensional figures by an infinite -solid. - -When any difficulty arises in taking the sections of a tessaract, the -surest way of overcoming it is to suppose a similar difficulty occurring -to a two-dimensional being in taking the sections of a cube, and, step -by step, to follow the solution he might obtain, and then to apply the -same or similar principles to the case in point. - -A few figures are given, which, if cut out and folded along the lines, -will show some of the sections of a tessaract. But the reader is -earnestly begged not to be content with _looking_ at the shapes only. -That will teach him nothing about a tessaract, or four-dimensional -space, and will only tend to produce in his mind a feeling that “the -fourth dimension” is an unknown and unthinkable region, in which any -shapes may be right, as given sections of its figures, and of which any -statement may be true. While, in fact, if it is the case that the laws -of spaces of two and three dimensions may, with truth, be carried on -into space of four dimensions; then the little our solidity (like the -flatness of a plane-being) will allow us to learn of these shapes and -relations, is no more a matter of doubt to us than what we learn of two- -and three-dimensional shapes and relations. - -There are given also sections of an octa-tessaract, and of a -tetra-tessaract, the equivalents in four-space of an octahedron and -tetrahedron. - -A tetrahedron may be regarded as a cube with every alternate corner cut -off. Thus, if Mala have the corner towards Corvus cut off as far as the -points Ilex, Nugæ, Cista, and the corner towards Sors cut off as far as -Ilex, Nugæ, Lama, and the corner towards Crus cut off as far as Lama, -Nugæ, Cista, and the corner towards Olus cut off as far as Ilex, Lama, -Cista, what is left of the cube is a tetrahedron, whose angles are at -the points Ilex, Nugæ, Cista, Lama. In a similar manner, if every -alternate corner of a tessaract be cut off, the figure that is left is a -tetra-tessaract, which is a figure bounded by sixteen regular -tetrahedrons. - -[Illustration: (i) - -Fig. 26. - -Fig. 27. - -Fig. 27. - -Fig. 26.] - -[Illustration: (ii) - -Fig. 28. - -Fig. 29. - -Fig. 30.] - -[Illustration: (iii) - -Fig. 31. - -Fig. 32.] - -[Illustration: (iv) - -Fig. 33. - -Fig. 34. - -Fig. 35.] - -[Illustration: (v) - -Fig. 36. - -Fig. 37. - -Fig. 38.] - -[Illustration: (vi) - -Fig. 39. - -Fig. 40. - -Fig. 41.] - -The octa-tessaract is got by cutting off every corner of the tessaract. -If every corner of a cube is cut off, the figure left is an octa-hedron, -whose angles are at the middle points of the sides. The angles of the -octa-tessaract are at the middle points of its plane sides. 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 -figures. It will be seen that there is much to learn of them, as--How -many planes and lines there are in each, How many solid sides there are -round a point in each. - - -A DESCRIPTION OF FIGURES 26 TO 41. - - Z X Y W - Z X Y W {26 is a section taken 1 . 1 . 1 . 1 - I . I . I . I {27 „ „ „ 1¹⁄₂ . 1¹⁄₂ . 1¹⁄₂ . 1¹⁄₂ - {28 „ „ „ 2 . 2 . 2 . 2 - - Z X Y W - Z X Y W {29 is a section taken 1 . 1 . 1 . ¹⁄₂ - II . II . II . I {30 „ „ „ 1¹⁄₂ . 1¹⁄₂ . 1¹⁄₂ . ³⁄₄ - {31 „ „ „ 2 . 2 . 2 . 1 - 32 „ „ „ 2¹⁄₂ . 2¹⁄₂ . 2¹⁄₂ . 1¹⁄₄ - -The above are sections of a tessaract. Figures 33 to 35 are of a -tetra-tessaract. The tetra-tessaract is supposed to be imbedded in a -tessaract, and the sections are taken through it, cutting the Z, X and Y -lines equally, and corresponding to the figures given of the sections of -the tessaract. - -Figures 36, 37, and 38 are similar sections of an octa-tessaract. - -Figures 39, 40, and 41 are the following sections of a tessaract. - - Z X Y W - Z X Y W {39 is a section taken 0 . ¹⁄₂ . ¹⁄₂ . ¹⁄₂ - O . I . I . I {40 „ „ „ 0 . 1 . 1 . 1 - {41 „ „ „ 0 . 1¹⁄₂ . 1¹⁄₂ . 1¹⁄₂ - -It is clear that there are four orders of sections of every -four-dimensional figure; namely, those beginning with a solid, those -beginning with a plane, those beginning with a line, and those beginning -with a point. There should be little difficulty in finding them, if the -sections of a cube with a tetra-hedron, or an octa-hedron enclosed in -it, are carefully examined. - - -APPENDIX K. - -[Illustration: MODEL 1. MALA.] - - COLOURS: MALA, LIGHT-BUFF. - - _Points_: Corvus, Gold. Nugæ, Fawn. Crus, Terra-cotta. Cista, Buff. - Ilex, Light-blue. Sors, Dull-purple. Lama, Deep-blue. Olus, Red. - - _Lines_: Cuspis, Orange. Bolus, Crimson. Cadus, Green-grey. Dos, Blue. - Arctos, Brown. Far, French-grey. Daps, Dark-slate. Bucina, Green. - Callis, Reddish. Iter, Bright-blue. Semita, Leaden. Via, Deep-yellow. - - _Surfaces_: Moena, Dark-blue. Proes, Blue-green. Murex, Light-yellow. - Alvus, Vermilion. Mel, White. Syce, Black. - -[Illustration: MODEL 2. MARGO.] - - COLOURS: MARGO, SAGE-GREEN. - - _Points_: Spira, Silver. Ancilla, Turquoise. Mugil, Earthen. Panax, - Blue-tint. Felis, Quaker-green. Passer, Peacock-blue. Talus, - Orange-vermilion. Solia, Purple. - - _Lines_: Luca, Leaf-green. Mappa, Dull-green. Mensura, Dark-purple. - Opex, Purple-brown. Pagus, Dull-blue. Onager, Dark-pink. Vena, - Pale-pink. Lixa, Indigo. Tholus, Brown-green. Calor, Dark-green. - Livor, Pale-yellow. Lensa, Dark. - - _Surfaces_: Silex, Burnt-sienna. Sal, Yellow-ochre. Portica, Dun. - Crux, Indian-red. Lares, Light-grey. Lappa, Bright-green. - -[Illustration: MODEL 3. LAR.] - - COLOURS: LAR, BRICK-RED. - - _Points_: Spira, Silver. Ancilla, Turquoise. Mugil, Earthen. Panax, - Blue-tint. Corvus, Gold. Nugæ, Fawn. Crus, Terra-cotta. Cista, Buff. - - _Lines_: Luca, Leaf-green. Mappa, Dull-green. Mensura, Dark-purple. - Opex, Purple-brown. Ops, Stone. Limus, Smoke. Offex, Magenta. Lis, - Light-green. Cuspis, Orange. Bolus, Crimson. Cadus, Green-grey. Dos, - Blue. - - _Surfaces_: Lotus, Azure. Olla, Rose. Lorica, Sea-blue. Lua, - Bright-brown. Syce, Black. Lappa, Bright-green. - -[Illustration: MODEL 4. VELUM.] - - COLOURS: VELUM, CHOCOLATE. - - _Points_: Felis, Quaker-green. Passer, Peacock-blue. Talus, - Orange-vermilion. Solia, Purple. Ilex, Light-blue. Sors, Dull-purple. - Lama, Deep-blue. Olus, Red. - - _Lines_: Tholus, Brown-green. Calor, Dark-green. Livor, Pale-yellow. - Lensa, Dark. Lucta, Rich-red. Pator, Green-blue. Libera, Sea-green. - Orsa, Emerald. Callis, Reddish. Iter, Bright-blue. Semita, Leaden. - Via, Deep-yellow. - - _Surfaces_: Limbus, Ochre. Meatus, Deep-brown. Mango, Deep-green. - Croeta, Light-red. Mel, White. Lares, Light-grey. - -[Illustration: MODEL 5. VESPER.] - - COLOURS: VESPER, PALE-GREEN. - - _Points_: Spira, Silver. Corvus, Gold. Cista, Buff. Panax, Blue-tint. - Felis, Quaker-green. Ilex, Light-blue. Olus, Red. Solia, Purple. - - _Lines_: Ops, Stone. Dos, Blue. Lis, Light-green. Opex, Purple-brown. - Pagus, Dull-blue. Arctos, Brown. Bucina, Green. Lixa, Indigo. Lucta, - Rich-red. Via, Deep-yellow. Orsa, Emerald. Lensa, Dark. - - _Surfaces_: Pagina, Yellow. Alvus, Vermilion. Camoena, Deep-crimson. - Crux, Indian-red. Croeta, Light-red. Lua, Light-brown. - -[Illustration: MODEL 6. IDUS.] - - COLOURS: IDUS, OAK. - - _Points_: Ancilla, Turquoise. Nugæ, Fawn. Crus, Terra-cotta. Mugil, - Earthen. Passer, Peacock-blue. Sors, Dull-purple. Lama, Deep-blue. - Talus, Orange-vermilion. - - _Lines_: Limus, Smoke. Bolus, Crimson. Offex, Magenta. Mappa, - Dull-green. Onager, Dark-pink. Far, French-grey. Daps, Dark-slate. - Vena, Pale-pink. Pator, Green-blue. Iter, Bright-blue. Libera, - Sea-green. Calor, Dark-green. - - _Surfaces_: Pactum, Yellow-green. Proes, Blue-green. Orca, Dark-grey. - Sal, Yellow-ochre. Meatus, Deep-brown. Olla, Rose. - -[Illustration: MODEL 7. PLUVIUM.] - - COLOURS: PLUVIUM, DARK-STONE. - - _Points_: Spira, Silver. Ancilla, Turquoise. Nugæ, Fawn. Corvus, Gold. - Felis, Quaker-green. Passer, Peacock-blue. Sors, Dull-purple. Ilex, - Light-blue. - - _Lines_: Luca, Leaf-green. Limus, Smoke. Cuspis, Orange. Ops, Stone. - Pagus, Dull-blue. Onager, Dark-pink. Far, French-grey. Arctos, Brown. - Tholos, Brown-green. Pator, Green-blue. Callis, Reddish. Lucta, - Rich-red. - - _Surfaces_: Silex, Burnt-Sienna. Pactum, Yellow-green. Moena, - Dark-blue. Pagina, Yellow. Limbus, Ochre. Lotus, Azure. - -[Illustration: MODEL 8. TELA.] - - COLOURS: TELA, SALMON. - - _Points_: Panax, Blue-tint. Mugil, Earthen. Crus, Terra-cotta. Cista, - Buff. Solia, Purple. Talus, Orange-vermilion. Lama, Deep-blue. Olus, - Red. - - _Lines_: Mensura, Dark-purple. Offex, Magenta. Cadus, Green-grey. Lis, - Light-green. Lixa, Indigo. Vena, Pale-pink. Daps, Dark-slate. Bucina, - Green. Livor, Pale-yellow. Libera, Sea-green. Semita, Leaden. Orsa, - Emerald. - - _Surfaces_: Portica, Dun. Orca, Dark-grey. Murex, Light-yellow. - Camoena, Deep-crimson. Mango, Deep-green. Lorica, Sea-blue. - -[Illustration: MODEL 9. SECTION BETWEEN MALA AND MARGO.] - - COLOURS: INTERIOR OR TESSARACT, WOOD. - - _Points_ (_Lines_): Ops, Stone. Limus, Smoke. Offex, Magenta. Lis, - Light-green. Lucta, Rich-red. Pator, Green-blue. Libera, Sea-green. - Orsa, Emerald. - - _Lines_ (_Surfaces_): Lotus, Azure. Olla, Rose. Lorica, Sea-blue. Lua - Bright-brown. Pagina, Yellow. Pactum, Yellow-green. Orca, Dark-grey. - Camoena, Deep-crimson. Limbus, Ochre. Meatus, Deep-brown. Mango, - Deep-green. Croeta, Light red. - - _Surfaces_ (_Solids_): Pluvium, Dark-stone. Idus, Oak. Tela, Salmon. - Vesper, Pale-green. Velum, Chocolate. Lar, Brick-red. - -[Illustration: MODEL 10. SECTION BETWEEN LAR AND VELUM.] - - COLOURS: INTERIOR OR TESSARACT, WOOD. - - _Points_ (_Lines_): Pagus, Dull-blue. Onager, Dark-pink. Vena, - Pale-pink. Lixa, Indigo. Arctos, Brown. Far, French-grey. Daps, - Dark-slate. Bucina, Green. - - _Lines_ (_Surfaces_): Silex, Burnt-sienna. Sal, Yellow-ochre. Portica, - Dun. Crux, Indian-red. Pagina, Yellow. Pactum, Yellow-green. Orca, - Dark-grey. Camoena, Deep-crimson. Moena, Dark-blue. Proes, Blue-green. - Murex, Light-yellow. Alvus, Vermilion. - - _Surfaces_ (_Solids_): Pluvium, Dark-stone. Idus, Oak. Tela, Salmon. - Vesper, Pale-green. Mala, Light-buff. Margo, Sage-green. - -[Illustration: MODEL 11. SECTION BETWEEN VESPER AND IDUS.] - - COLOURS: INTERIOR OR TESSARACT, WOOD. - - _Points_ (_Lines_): Luca, Leaf-green. Cuspis, Orange. Cadus, - Green-grey. Mensura, Dark-purple. Tholus, Brown-green. Callis, - Reddish. Semita, Leaden. Livor, Pale-yellow. - - _Lines_ (_Surfaces_): Lotus, Azure. Syce, Black. Lorica, Sea-blue. - Lappa, Bright-green. Silex, Burnt-sienna. Moena, Dark-blue. Murex, - Light-yellow. Portica, Dun. Limbus, Ochre. Mel, White. Mango, - Deep-green. Lares, Light-grey. - - _Surfaces_ (_Solids_): Pluvium, Dark-stone. Mala, Light-buff. Tela, - Salmon. Margo, Sage-green. Velum, Chocolate. Lar, Brick-red. - -[Illustration: MODEL 12. SECTION BETWEEN PLUVIUM AND TELA.] - - COLOURS: INTERIOR OR TESSARACT, WOOD. - - _Points_ (_Lines_): Opex, Purple-brown. Mappa, Dull-green. Bolus, - Crimson. Dos, Blue. Lensa, Dark. Calor, Dark-green. Iter, Bright-blue. - Via, Deep-yellow. - - _Lines_ (_Surfaces_): Lappa, Bright-green. Olla, Rose. Syce, Black. - Lua, Bright-brown. Crux, Indian-red. Sal, Yellow-ochre. Proes, - Blue-green. Alvus, Vermilion. Lares, Light-grey. Meatus, Deep-brown. - Mel, White. Croeta, Light-red. - - _Surfaces_ (_Solids_): Margo, Sage-green. Idus, Oak. Mala, Light-buff. - Vesper, Pale-green. Velum, Chocolate. Lar, Brick-red. - - - - - Transcriber’s Notes - - - Lay-out and formatting have been optimised for browser html (available - at www.gutenberg.org); some versions and narrow windows may not - display all elements of the book as intended, depending on the hard- - and software used and their settings. For this text file, several - diagrams and tables have been split or otherwise re-arranged in order - to fit the available width. Wherever possible, these splits and - re-arrangements have been done so that the meaning of the lay-out has - been retained. The contents of Tables A through D have been replaced - with place holders; their meanings are listed below the diagrams. - - Inconsistencies in spelling (Mœnas v. Moenas; Praeda v. Proeda), - hyphenation (Deep-blue v. Deep blue, etc.) have been retained. - - Page 197, row starting Sophos: the last letter of Blue has been - assumed. - - Changes made: - - Footnotes, tables, diagrams and illustrations have been moved outside - text paragraphs. Indications for the location of illustrations (To - face p. ...) have been removed; the illustrations concerned have been - moved to where they are discussed. - - Some minor obvious typographical errors have been corrected silently. - - Page 42: ... the flat, being ... changed to ... the flat being ... - - Page 127: Cube itself: considered to be the table header rather than a - table element - - Page 175: is all Ana our space changed to is all Ana in our space - - Page 187: Clipens changed to Clipeus; legend Y added to right-hand - side grid axes - - Page 219: Part II. Appendix K. changed to Appendix K. cf. other - Appendices. - - - - - -End of Project Gutenberg's A New Era of Thought, by Charles Howard Hinton - -*** END OF THIS PROJECT GUTENBERG EBOOK A NEW ERA OF THOUGHT *** - -***** This file should be named 60607-0.txt or 60607-0.zip ***** -This and all associated files of various formats will be found in: - http://www.gutenberg.org/6/0/6/0/60607/ - -Produced by Chris Curnow, Harry Lame and the Online -Distributed Proofreading Team at http://www.pgdp.net (This -file was produced from images generously made available -by The Internet Archive) - - -Updated editions will replace the previous one--the old editions will -be renamed. - -Creating the works from print editions not protected by U.S. copyright -law means that no one owns a United States copyright in these works, -so the Foundation (and you!) can copy and distribute it in the United -States without permission and without paying copyright -royalties. 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