TERTttJM ORGANUM A-KEY'TOTHE ENIGMA5-OF THEWORLD PDOlWPENfKY THE LIBRARY- OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES IN MEMORY OF EDWIN CORLE PRESENTED BY JEAN CORLE TERTIUM ORGANUM ; f , их** "D TERTIUM ORGANUM (THE THIRD ORGAN OF THOUGHT) A Key to the Enigmas of the World P. D. OUSPENSKY TRANSLATED FROM THE RUSSIAN BY NICHOLAS BESSARABOFF AND CLAUDE BRAGDON — WITH AN INTRODUCTION BY CLAUDE BRAGDON The Mystery of Space and Time. Shadows and Reality. Occultism and Love. Animated Nature. Voices of the Stones. Mathematics of the Infinite. The Logic of Ecstasy. Mystical Theosophy. Cosmic Consciousness. The New Morality. Birth of the Superman. MANAS PRESS, ROCHESTER, N.Y. 1920 COPYRIGHT, 1920 BY THE MANAS PRESS Copyrighted in Great Britain and Colonies INTRODUCTION TO THE ENGLISH TRANSLATION In the spring of 1918, a young Russian, Nicholas Bessaraboff, appeared at my door bearing in his hand Tertium Organum, a precious gift to the mind and to the spirit, but shrouded in the seven-fold veil of the to me incomprehensible Russian tongue. With ardent enthusiasm and admirable patience the young man outlined to me the nature and content of the book. I took fire at once, for I saw that the author, Ouspensky, was the Columbus of that uncharted ocean of thought in which I and others had indeed adventured, haunted by dreams of rich argosies from virgin con- tinents. Some authors gain only readers: others, more fortunate, win disciples, and Ouspensky is of this latter class. Bessaraboff was a disciple, and I found that I had become one without knowing it. So in a spirit of true discipleship we set to work to make Tertium Organum known to the English speaking world. The method we adopted was for him to make a somewhat rigidly literal transla- tion, into which I then infused only so much of literary form as seemed necessary for lucidity and ease. Faithfulness to the orig- inal was the aim held piously by both of us from beginning to end. The reader has the right to know something of our fitness for this service. For my own less arduous and important part in the translation, I can only say that Ouspensky 's thought is so curious- ly parallel to the movement of my own mind as expressed in Four Dimensional Vistas, that I could be accused of plagarism by any- one unaware of the fact that my book was published before I had read his. Mr. Bessaraboff studied mechanical engineering at the Petro- grad Polytechnic Institute, whose student body is composed of the honor men from various Russian schools. While a student, his absorption in mathematics and mechanics had been complete, but the reading of Tertium Organum so awakened his interest in gener- al science, philosophy and mysticism that he embarked upon a collateral course of study that made him familiar with practically every phase of thought, in its broad outlines, dealt with by Ous- pensky in his book. His knowledge of English, though it does not extend to all the niceties of construction and literary expression, 20374 94 TERTIUM ORGANUM is sufficient for him to read the language with perfect understand- ing, and to express himself with great precision; while his vocabu- lary, as so often is the case with foreigners who have learned Eng- lish out of books, is larger than that of the average American uni- versity graduate. As for Ouspensky himself he is an accomplished mathematician, magister of pure mathematics, and he holds the position of in- structor of mathematics in the Petrograd Institute of Engineers of Ways of Communication, one of the oldest of the Russian techni- cal schools. He is by now thirty-eight years old, has travelled ex- tensively, visiting England, Italy, Egypt and India; he has con- tributed to mathematical text-books, and is the author of several works other than Tertium Organum. This latter is now in its sec- ond edition in Russia. The present translation was made from this second edition, the date on the title page being 1916. In naming his book "Tertium Organum" Ouspensky reveals at a stroke that astounding audacity which characterizes his thought throughout — an audacity which we are accustomed to associate with the Russian mind in all its phases. Such a title says, in effect: "Here is a book which will reorganize all knowledge. The Organon of Aristotle formulated the laws under which the subject thinks; the Novum Organum of Bacon, the laws under which the object may be known. Behold! I give you a Third Organ which shall guide and govern human thought henceforth." How passing strange, in this era of negative thinking, of timid philosophizing, does such a challenge sound. And yet it has the echo in it of something heard before, — what but the title of an- other volume, Hinton's A New Era of Thought. Ouspensky's Tertium Organum, and Hinton's A New Era of Thought present substantially the same philosophy (though Hin- ton's book only sketchily), arrived at by the same route — mathe- matics. Here is food for thought. In the words of Philip Henry Wynne, "Mathematics possesses the most potent and perfect symbolism the intellect knows; and this symbolism has offered for genera- tions certain concepts (of which hyper-dimensionality is only one) whose naming and envisagement by the human intellect is per- haps its loftiest achievement. Mathematics presents the highest INTRODUCTION certitudes known to the intellect, and is becoming more and more the final arbiter and interpreter in physics, chemistry and astron- omy. Like Aaron's rod it threatens to swallow all other know- ledges as fast as they assume organized form. Mathematics has already taken possession of great provinces of logic and psychol- ogy, — will it embrace ethics, religion and philosophy?" In Tertium Organum mathematics enters and pervades the field of philosophy; but so adroitly, so silently as it were, that one hardly knows that it is "there." It dwells more in Ouspensky's method than in his matter, because for the most part the math- ematical ideas necessary for an understanding of his thesis are such as any intelligent high school student can comprehend. The author puts to himself and to the reader certain questions, pro- pounds certain problems, which have baffled the human mind for thousands of years — the problems of space, time, motion, causal- ity, of free will and determination — and he deals with them ac- cording to the mathematical method : that is all. He has sensed the truth that the problem of mathematics is the problem of the world order, and as such must deal with every aspect of human life. Mathematics is a terrible word to those whose taste and train- ing have led them into other fields, so lest the non-mathematical reader should be turned back at the very threshold, deciding too hastily that the book is not for him, let me dwell rather on its richly humanistic aspect. To such as ask no "key to the enigmas of the world," but only some light to live by, some mitigation of the daily grind, some glimpse of some more enlightened polity than that which rules the world today, this book should have an appeal. The author has thrown overboard all the jargon of all the schools; he uses the language of common sense, and of every day; his illustrations and figures of speech are homely, taken from the life of every day. He simply says to the reader, "Come let us reason together," and leads him away from the haunted jungle of philosophical systems and metaphysical theories, out into the light of day, there to con- template and to endeavor to understand those primal mysteries which puzzle the mind of a child or of a savage no less than that of the sophisticated and super-subtle ponderer on the enigmas of the world. Not that Ouspensky is a trafficker in the obvious — far from it: those who know most, think most, feel most, will get most out of his book — but a great sanity pervades his pages, and he TERTIUM ORGANUM never leads away into labyrinths where guide and follower alike lose their way and fail to come to any end. Leaving the average reader out of account for the moment, there are certain others whom the book should particularly inter- est — if only in the way of repulsion. First of all come the mathematicians and the theoretical phys- icists, for they already, without knowing it, have invaded that "dark backward and abysm of time" which the Ouspenskian philosophy lights up — and are by way of losing themselves there. That is to say, in certain of their calculations they are em- ploying four mutually interchangeable co-ordinates, three of space and one of time. In other words, they use time as though it were a dimension of space. Ouspensky tells them the reason they are able to do this. — Time is the fourth dimension of space imper- fectly sensed — apprehended by consciousness successively, and thereby creating the temporal illusion. Moreover, mathematicians are perforce concerning themselves with magnitudes to which the ordinary logic no longer applies. Ouspensky presents a new logic — the logic of intuition — remov- ing at a stroke all of the nightmare aspects, the preposterous para- doxes of the new mathematics, which by reason of its extraordin- ary development has shattered the old logic, as a growing oak shatters the containing jar. It is from the philosophic camp, no doubt, that the book will receive its sharpest criticism, on account of the author's lese- majeste toward so many of the crowned kings of philosophic thought, and his devastating assault on positivism — that inevit- able by-product of our materialistic way of looking at the world. His attempt to prove the Kantian problem — the subjectivity of space and time — doubtless will be acutely challenged, and with some chance of success, because the two chapters devoted to this are the least convincing of the book. But no one heretofore has even attempted to absolutely demonstrate or successfully con- trovert the staggering proposition advanced by Kant regarding space and time as forms of consciousness. Whatever the verdict of the philosophical pundits of the day and hour, whether favorable or otherwise, Ouspensky is sure of a place in the hierarchy of philosophers, for he has essayed to solve INTRODUCTION the most profound problems of human existence by the aid of the binocular vision of a born mathematician and an intuitive mystic. Starting from the irreducible minimum of knowledge, he has car- ried philosophy into regions not hitherto explored. To persons of an artistic or devotional bent the book will be as water in the desert. These, always at a disadvantage among the purely practical-minded, by whom they are outnumbered twenty to one, will find in Ouspensky a champion whose weapon is math- ematical certitude, the very thing by which the practical minded swear. These, their enemies, he puts to rout, holds up to ridicule. He applauds their efforts to escape into the "world of the won- drous," and justifies the faith that is in them. But most of all, Ouspensky will be loved by all true lovers, for his chapter on the subject of love. We have had Schopenhauer on love, and Freud on love, but what dusty answers do they give to the soul of a lover! Edward Carpenter comes much nearer the mark, but Ouspensky penetrates to its very center. It is because our loves are so dampened by our egotisms, our cynicisms and our cowardices that we rot and smoulder instead of bursting into puri- fying flame. Just as Goethe's Werther, with its sex-sentimentality, is said to have provoked an epidemic of suicides, so may Tertium Organum — which restores love to that high heaven from whence descend every beauty and benison — inaugurate a renascence of love and joy. From one point of view this is a terrible book : there is a revolu- tion in it — a revolution of the very poles of thought. Some it will rob of their dearest illusions, it will cut the very ground from be- neath their feet, it will consign them to the Abyss. It is a great destroyer of complacency. Yes, this is a dangerous book — but then, life is like that. It is beyond the province of this Introduction either to outline the Ouspenskian philosophy at any length, or to discuss it criti- cally; but some slight indication of its drift may be of assistance to the reader. The book might have appropriately been called A Study of Con- sciousness, for Ouspensky comes early to the conclusion that all other methods of approach to an understanding of the "enigmas of the world" are vain. Chapters I to VII, inclusive, deal with the TERTIUM ORGANUM problem of the world-order by the objective method. The author erects an elaborate scaffolding for his future edifice, and after it has served its purpose, throws it down. Aware of the deficiencies of the objective method, and having made the reader conscious of them too, he suddenly alters his system of attack. From chap- ter VIII onward, he undertakes the study of the world order from the standpoint of subjectivity — of consciousness. By a method both ingenious and new, he correlates the different grades of consciousness observable in nature — those of vegetable- animal, animal and man — with the space sense, showing that as consciousness changes and develops, the sense of space changes and develops too. That is to say, the dimensionality of the world depends on the development of consciousness. Man, having reached the third stage in that development, has a sense of three- dimensional space — and for no other reason. Ouspensky concludes that nothing except consciousness unfolds, develops, and as there appears to be no limit to this development he conceives of space as the multi-dimensional mirror of con- sciousness and of time and motion as illusion — what appears to be time and motion being in reality only the movement of conscious- ness upon a higher space. The problem of superior states of consciousness in which "there shall be time no longer" is thus directly opened up, and in dis- cussing their nature and method of attainment, he quotes freely from the rich literature of mysticism. Instead of attempting to rationalize these higher states of consciousness, as some authors do, he applies to them the logic of intuition — "Tertium Organum" — paradoxical from the standpoint of ordinary reason, but true in relation to the noumenal world. Joseph Conrad and Ford Madox Hueffer once wrote a novel called The Inheritors and by this they meant the people of the fourth dimension. Though there is small resemblance between Ouspensky 's "superman" and theirs, it is his idea also that those of this world who succeed in developing higher-dimensional, or "cosmic consciousness" will indeed inherit — will control and reg- ulate human affairs by reason of their superior wisdom and power. In this, and in this alone, dwells the "salvation"of the world. His superman is far removed from the "blond beast"of Nietzche: it is the "just man made perfect" of the Evangelist. This struggle for mastery between the blind and unconscious forces of materialism INTRODUCTION on the one hand, and the spiritually illumined on the other, is al- ready upon us, and all conflicts between nations, peoples and classes must now be interpreted in terms of this greater warfare between "two races" of men, in which the superior minority will either conquer or disappear. These people of the fourth dimension are in the world but not of it: their range is far wider than this slum of space. In them dor- mant faculties are alert. Like birds of the air, their fitting symbol, they are at home in realms which others cannot enter, even though already "there". Nor are these heavenly eagles confined to the narrow prison of the breast. Their bodies are as tools which they may take up or lay aside at will. This phenomenal world, which seems so real, is to them as insubstantial as the image of a land- scape in a lake. Such is the Ouspenskian superman. The entire book is founded upon a new generalization — new, that is, in philosophy, but already familiar to mathematicians and theoretical physicists. This generalization involves startling and revolutionary ideas in regard to space, time and motion, far removed from those of Euclidian geometry and classical physics. Ouspensky handles these new ideas in an absolutely original way, making them the basis of an entire philosophy of life. To the timid and purblind this philosophy will be nothing short of terrifying, but to the clear-eyed and steadfast watcher, shipwreck- ed on this shoal of time, these vistas, overflowing with beauty, strangeness, doubt, terror and divinity, will be more welcome than anything in life. "Fear not the new generalization" Ouspensky's clearness of thought is mirrored in a corresponding clarity of expression, with every aid to understanding of which his office of teacher of mathematics has given him command. He sometimes repeats the difficult and important passages in an al- tered form of words, he uses short sentences and short paragraphs, and italicizes significant phrases and significant words. He de- fines where definition is needed, and suggests collateral trains of thought with a skill which makes the reader who is intuitive a creator on his own account. Schopenhauer has said that it is al- ways a sign of genius to treat difficult matters simply, as it is a sign of dullness to make simple matters appear recondite. Ous- TERTIUM ORGANUM pensky exhibits this order of genius, and that other, mentioned by Schopenhauer, which consists in choosing always the apt illus- tration, the illuminating simile. The translators have tried to be rigidly true to the Russian or- iginal, as has been said, and they have been at great pains to verify every English quotation so far as has been possible. The only liberty they have taken with the text consists in the omission of a brief personal reference which might possibly give offense. Rochester, N. Y. Claude Bragdon August 1, 1919 CONTENTS CHAPTER I What do we know and what do we not know? Our data, and the things sought for. The unknown mistaken for the known. Matter and motion. What does positive philosophy come to? Identity of the unknown: x = y, y=x. What we really know. The existence of consciousness in us, and of the world outside of us. Dual- ism or monism? Subjective and objective knowledge. Where do the causes of the sensations lie? Kant's system. Time and space. Kant and the "ether." Mach's observation. With what does the physicist really deal? 1 CHAPTER II A new view of the Kantian problem. The ideas of Hinton. The "space sense" and its evolution. A system for the development of a sense of the fourth dimension by ex- ercises with colored cubes. The geometrical conception of space. Three perpen- diculars—why three? Can everything existing be measured by three perpendicu- lars? Facts physical and metaphysical. The indices of existence. Reality of ideas. Insufficient evidence of the existence of matter and motion. Matter and motion are only logical concepts, like "good" and "evil." 13 CHAPTER III What may we learn about the fourth dimension by a study of the geometrical relations within our space? What should be the relation between a three-dimensional body and one of four dimensions? The four-dimensional body as the tracing of the movement of a three-dimensional body in a direction which is not confined within it. A four-dimensional body as containing an infinite number of three-dimensional bodies. A three-dimensional body as a section of a four-dimensional one. Parts of bodies and entire bodies in three and in four dimensions. The incommensurability of a three-dimensional and a four-dimensional body. A material atom as a section of a four-dimensional line 25 CHAPTER IV In what direction may the fourth dimension lie? What is motion? Two kinds of mo- tion—motion in space and motion in time— which are contained in every movement. What is time? Two ideas contained in the conception of time. The new dimension of space, and motion upon that dimension. Time as the fourth dimension of space. Impossibility of understanding the fourth dimension without the idea of motion. The idea of motion and the "time sense." The time sense as a limit (surface) of the space sense. Hinton on the law of surfaces. The "ether" as a surface. Riemann's idea concerning the translation of time into space in the fourth dimension. Pres- ent, past and future. Why we do not see the past and the future. Life as a feeling of one' sway. Wundt on the subject of our sensuous knowledge 29 CHAPTER V Four-dimensional space. "Temporal body" — Linga Sharira. The form of a human body from birth to death. Incommensurability of three-dimensional and four- dimensional bodies. Newton's fluents. The unreality of constant quantities in our world. The right and the left hands in three-dimensional and in four-dimensional space. Differences between three-dimensional and four-dimensional space. Not two different spaces, but two different methods of receptivity of one and the same world 43 TERTIUM ORGANUM CHAPTER VI Methods of investigation of the problem of higher dimensions. The analogy between imaginary worlds of different dimensions. The one dimensional world on a line. "Space" and "time" of a one-dimensional being. The two-dimensional world on a plane. "Space" and "time," "ether," "matter" and "motion" of a two-dimension- al being. Reality and illusion on a plane. The impossibility of seeing an "angle." An angle as motion. The incomprehensibility to a two-dimensional being of the functions of things in our world. Phenomena and noumena of a two-dimensional being. How could a plane being comprehend the third dimension? .... 49 CHAPTER VII The impossibility of the mathematical definition of dimensions. Why does not mathe- matics sense dimensions? The entire conditionality of the representation of dimen- sions by powers. The possibility of representing all powers on a line. Kant and Lobachevsky. The difference between non-Euclidian geometry and metageom- etry. Where shall we find the explanation of the three-dimensionality of the world if Kant's ideas are true? Are not the conditions of the three-dimensionality of the world confined to our receptive apparatus, in our psyche? 65 CHAPTER VIII Our receptive apparatus. Sensation. Perception. Conception. Intuition. Art as the language of the future. To what extent does the three-dimensionality of the world depend upon the properties of our receptive apparatus? What might prove this interdependence? Where may we find the real affirmation of this interdepen- dence? The animal psyche. In what does it differ from the human? Reflex ac- tion. The irritability of the cell. Instinct,., Pleasure-pain. Emotional thinking. Absence of concepts. Language oLanimals. Logic of animals. Different degrees of psychic development in animals. The goose, the cat, the dog and the monkey. 71 CHAPTER IX The receptivity of the world by a man and by an animal. Illusions of the animal and its lack of control of the receptive faculties. The world of moving planes. Angles and curves considered as motion. The third dimension as motion. The animal's two-dimensional view of our three-dimensional world. The animal as a real two- dimensional being. Lower animals as one-dimensional beings. The time and space of a snail. The time sense as an imperfect space sense. The time and space of a dog. The change in the world coincident with a change in the psychic apparatus. The proof of Kant's problem. The three-dimensional world — an illusory percep- tion 89 CHAPTER X The spatial understanding of time. The angles and curves of the fourth dimension in our life. Does motion exist in the world or not? Mechanical motion and "life". Biological phenomena as the manifestation of motions going on in the higher di- mension. Evolution of the space sense. The growth of the space sense and the dim- inution of the time sense. The transformation of the time sense into the space sense. The difficulties of our language and of our concepts. The necessity for seeking a method of spatial expression for temporal concepts. Science in relation to the fourth dimension. The solid of four dimensions. The four-dimensional sphere 108 CHAPTER XI Science and the problem of the fourth dimension. The address of Prof. N. A. Oumoff before the Mendeleevsky Convention in 1911 — "The Characteristic Traits and Problems of Contemporary Scientific Thought." The new physics. The electro- TERTIUM ORGANUM magnetic theory. The principle of relativity. The works of Einstein and Min- kowsky. Simultaneous existence of the past and the future. The Eternal Now. Van Manen's book about occult experiences. The drawing of a four-dimensional figure CHAPTER XII 115 Analysis of phenomena. What defines different orders of phenomena for us? Methods and forms of the translation of one order of phenomena into another. Phenomena of motion. Phenomena of life. Phenomena of consciousness. The central ques- tion of our knowledge of the world : what order of phenomena is generic and pro- duces the others? Can the origin of everything lie in motion? The laws of the transformation of energy. Simple transformation and liberation of latent energy. Different liberating forces of the different orders of phenomena. The force of me- chanical energy, the force of a living cell, the force of an idea. Phenomena and nou- mena of our world !*• CHAPTER XIII The apparent and the hidden side of life. Positivism as the study of the phenomenal side of life. Of what does the "two-dimensionality" of positive philosophy con- sist? The regarding of everything upon a single plane, in one physical sequence. The streams which flow underneath the earth. What can the study of life as a phenomenon yield? The artificial world which science erects for itself. The un- reality of finished and isolated phenomena. The new apprehension of the world. . 135 CHAPTER XIV The voices of stones. The wall of a church and the wall of a prison. The mast of a ship and a gallows. The shadow of a hangman and of an ascetic. The soul of a hangman and of an ascetic. The different combinations of known phenomena in higher space. The relationship of phenomena which appear unrelated, and the diff- erence between phenomena which appear similar. How shall we approach the nou- menal world? The understanding of things outside the categories of space and time. The reality of many "figures of speech." The occult understanding of energy. The letter of a Hindu-occultist. Art as the knowledge of the noumenal world. What we see and what we do not see. Plato's dialogue about the cavern. . 151 CHAPTER XV Occultism and love. Love and death. Our different relation to the problems of death and to the problems of love. What is lacking in our understanding of love? Love as an every-day and merely psychological phenomenon. The possibility of a spiri- tual understanding of love. The creative force of love. The negation of love. Ma- terialism and asceticism. The flight from love. Love and mysticism. The "wond- rous" in love. Prof. Lutoslawsky. Leo Tolstoy. Nietzche and Edward Carpen- ter on love. "The Ocean of Sex." 161 CHAPTER XVI The phenomenal and the noumenal side of man. "Man-in-himself ." How do we know the inner side of man? Can we know of the existence of consciousness in condi- tions of space not analogical to ours? Brain and consciousness. Unity of the world. Logical impossibility of the simultaneous existence of spirit and matter. Either all spirit or all matter. Rational and irrational actions in nature and in the life of man. Can rational actions exist alongside of irrational? The world as an accidentally self-created mechanical toy. The impossibility of consciousness in a mechanical universe. The irreconcilability of mechanicalness with the existence of consciousness. The fact of human consciousness as destroying the mechanistic sys- tem. The consciousness of other sections of the world. How may we know about them? Kant concerning "ghosts." Spinoza on the knowledge of the invisible world. Necessity for the intellectual definition of that which can be, and that which cannot be in the noumenal world 17© TERTIUM ORGANUM CHAPTER XVII A conscious universe. Different forms of consciousness. Different lines of conscious- ness. Animated nature. The souls of stones and the souls of trees. The soul of a forest. The human "I" as a collective consciousness. Man as a complex being. Humanity as a being. The world's consciousness. The face of Mahadeva. Prof. James on the consciousness of the universe. Fechner's ideas. "Zendavesta." A living Earth 203 CHAPTER XVIII Consciousness and life. Life as knowledge. Consciousness as a realization of exist- ence. Intellect and emotions. Emotion as an organ of knowledge. The evolution of emotion from the standpoint of knowledge. Pure and impure emotions. Per- sonal and super-personal emotions. The elimination of self elements as a means of approach to true knowledge. "Be as little children." "Blessed are the pure in heart." The value of morals from the standpoint of knowledge. The defects of in- tellectualism. Dreadnaughts as the crown of intellectual culture. The dangers of morality. Moral esthetics. Religion and art as organized forms of emotional know- ledge. The knowledge of God and the knowledge of beauty 219 CHAPTER XIX The intellectual method. Objective and subjective knowledge. The study of the Not- I and the study of the I. Impossibility of the objective study of the I. The limits of objective knowledge. The possibility of the expansion of subjective knowledge. The absorption of all Not-I by the I. The ideas of Plotinus. Different forms of consciousness. Sleep (the potential state of consciousness). Dreams (conscious- ness enclosed in itself, reflected from itself). Waking consciousness (dualistic sen- sation of the world, the division of the I and the Not-I). Ecstasy (the liberation of theself). "Turiya" (the absolute consciousness of all, as of the self ) . "Thedewdrop slips into the shining sea." "Nirvana." 237 CHAPTER XX The sense of infinity. The neophyte's first ordeal. An intolerable sadness. The loss of everything real. What would an animal feel on becoming a man? The transi- tion to the new logic. Our logic as founded on the observation of the laws of the phenomenal world. Its invalidity for the study of the world of noumena. The necessity for another logic. Analogy between the axioms of logic and of mathema- tics. Two mathematics. The mathematics of real magnitudes (infinite and vari- able) ; and the mathematics of unreal, imaginary magnitudes (finite and constant) . Transfinite numbers — numbers lying beyond infinity. The possibility of differ- ent infinities 251 CHAPTER XXI Man's transition to a higher logic. The necessity for rejecting everything "real." "Poverty of the spirit." The recognition of the infinite alone as real. Laws of the infinite. Logic of the finite — the "Organon" of Aristotle and the "Novum Organ- urn' of Bacon. Logic of the infinite — Teriium Organum. The higher logic as an instrument of thought, as a key to the mysteries of nature, to the hidden side of life, to the world of noumena. A definition of the world of noumena on the basis of all the foregoing. The impression of the noumenal world on an unprepared con- sciousness. "The thrice unknown darkness in the contemplation of which all knowledge is resolved into ignorance." 263 TERTIUM ORGANUM CHAPTER XXII «Theosophy'ofMaxMiiller. Ancient India. Philosophy of the Vedanta Tattwam ^Knowledge by means of the expansion of ™ ousnes *^^^ cism of different ages and peoples. Unity of experiences. Tertium Organumasj. keTto mysScism. Signs of the noumenal world. Treatise о Plotinus On Intel- fi2b BeSy" as a misunderstood system of higher logic. Illuminations in Jacob Boehme "A harp of many strings, of which each string is a separate instrument, wht'he whole Гоп1у one W ? Mystics of "The Loveof*e ^J£$™ Dorotheus and others. Clement of Alexandria. Lao-Tzu and cnuang izu. "St on the Path." "The Voice of the Silence." Mohammedan mystics. Poet- r o( the Suffs. Mystical states under narcotics. The anaesthetic revelation^ Ex- S^nStaSftof^ames. Dostoyevsky on "time" (The Idiot). Influence of na- ^ ture on the soul of man CHAPTER XXIII "Cosmic Consciousness" of Dr. Bucke. The three forms of cons ^ io "^ff 4 acC g! ; f f ill g ^ Dr Bucke Simple consciousness, or the consciousness of animals belt con sc ousness?or the consciousness of man. Cosmic consciousness In what is t ex- pressed? Sensation, perception, concept, higher moral concept-creative intui- ST Men of cosmic consciousness. Adam's fall into sin. The ^edge оЩ and evil. Christ and the salvation of man. Commentary on Dr Bucke s book Srth of the new humanity. Two races. Superman. Table of the four forms of ^ the manifestation of consciousness EVOLUTION. OR THE CULTURE OF CONSCIOUSNESS 338 . . 340 CONCLUSION TABLE OF THE FOUR FORMS OF CONSCIOUSNESS TERTIUM ORGANUM "And swear . . . that there should be TIME NO LONGER." REVELATIONS, X. 6 " . . . . That ye, being rooted and grounded in love may be able to comprehend with al! saints what is the breadth and length and DEPTH AND HEIGHT" Paul the Apostle THE EPISTLE TO THE EPHESIANS. III. 18 CHAPTER I What do we know and what do we not know? Our data, and the things for which we seek. The unknown mistaken for the known. Matter and motion. What does the positive philosophy come to/ Identity of the unknown: x = y. y = x. What we really know The existence of consciousness in us, and of the world outside ot us. Dualism or monism? Subjective and objective knowledge. Where do the causes of the sensations lie? Kant s system. lime and Space. Kant and the "ether." Mach's observation. With what does the physicist really deal? "Learn to discern the real from the false" THE VOICE OF THE SILENCE H. P. B. HE most difficult thing is to know what we do know, and what we do not know. Therefore, desiring to know anything, we shall be- fore all else determine what we accept as given, and what as demanding definition and proof: that is, determine what we know already, and what we wish to know. In relation to the knowledge of the world and of ourselves, the conditions would be ideal could we venture to accept nothing as given, and count all as demanding definition and proof. In other words, it would be best to assume that we know nothing, and make this our point of departure. But unfortunately such conditions are impossible to create. Knowledge must start from some foundation, something must be recognized as known, or we shall be obliged always to define one unknown by means of another. Looking at the matter from another point of view, we shall hesitate to accept as the known things— as the given ones— those in the main completely unknown, only presupposed, and there- fore the things sought for. Should we do this, we are likely to fall into such a dilemma as that in which positive philosophy now finds itself. For a long time this was founded on the idea of the existence of matter (materialism), and now it is founded on the conception of the existence of energy: that is, of a force, or motion 2 TERTIUM ORGANUM (energeticism) , though in reality matter and motion were always the unknown x and y, and were defined by means of one another. It must be perfectly clear to everyone that it is impossible to ac- cept the thing sought as the given; and impossible to define one un- known by means of another. The result is nothing but the iden- tity of the unknown : x = у , у = x. This identity of the unknown is the ultimate conclusion to which positive philosophy comes. Matter is that in which proceed the changes called motion: and motions are those changes which proceed in matter. But what do we know ? We know that with the very first awakening of self -conscious- ness, man is confronted with two obvious facts : The existence of the world in which he lives; and the existence of consciousness in himself. Neither of these can he prove or disprove, but they are facts: they constitute reality for him. It is possible to meditate upon the mutual correlation of these facts. It is possible to try to reduce them to one; that is, to re- gard consciousness as a part, or function of the world, or the world as a part, or function of consciousness. But such a pro- cedure constitutes a departure from facts, and all such considera- tions of the world and of the self, to the ordinary non-philosophical mind, will not have the character of obviousness. On the contrary the sole obvious fact remains the antithesis of / and Not-I — con- sciousness and the world. Further on we shall return to this fundamental thesis. But thus far we have no basis on which to found a contradiction of the obvious fact of the existence of ourselves — i. е., of our conscious- ness — and of the world in which we live. This we shall therefore accept as the given. This however is the only thing that we have the right to accept as given: all the rest demands proof and definition in terms of these two given data. Space, with its extension; time, with the idea of before, now, after; quantity, mass, substantiality; number, equality and in- equality; identity and difference; cause and effect, the ether, atoms, electrons, energy, life, death — all things that form the TERTIUM ORGANUM 3 foundation of our so-called knowledge: these are the unknown things. The existence of consciousness in us, and the existence of the world outside of us — from these two fundamental data imme- diately proceed our common and clearly understood division of everything that we know into subjective and objective. Everything that we accept as a property of the world, we call objective; and everything that we accept as a property of con- sciousness, we call subjective. The subjective world we recognize directly: it is in ourselves — we are one with it. The objective world we picture to ourselves as existing some- where outside of us — we and it are different things. It seems to us that if we should close our eyes, then the objective world would continue to exist, such as we just saw it; and if con- sciousness were to cease, and our "I" to disappear, so would the subjective world disappear — yet the objective world would exist as before, as it existed at the time when we were not; when our subjective world was not. Our relation to the objective world is most exactly defined by the fact that we perceive it as existing in time and space; other- wise, out of these conditions, we can neither conceive nor imagine it. In general, we say that the objective world consists of things and phenomena, i. е., things and changes in states of things. The phenomena exist for us in time; the things, in space. But such a division of the subjective and objective world does not satisfy us. By means of reasoning we can establish the fact that in reality we know only our own sensations, perceptions and conceptions, and we cognize the objective world by projecting outside of our- selves the causes of our sensations, presupposing them to contain these causes. Then we find that our knowledge of the subjective and of the objective world as well, can be true and false, correct and incorrect. The criterion for the definition of correctness or incorrectness of our knowledge of the subjective world is the form of the rela- tions of one sensation to others, and the force and character of the sensation itself. In other words, the correctness of one sensation is verified by the comparison of it with another of which we are more sure, or by the intensity and "taste" of a given sensation. 4 TERTIUM ORGANUM The criterion for the definition of correctness or incorrectness of our knowledge of the objective world is the very same. It seems to us that we define the things and phenomena of the objective world by means of comparing them among themselves; and we think we find the laws of their existence outside of us, and inde- pendent of our perception of them. But it is an illusion. We know nothing about things separately from us; and we have no other means of verifying the correctness of our knowledge of the objective world but by sensations. Since the remotest antiquity the question of our relation to the true causes of our sensations constituted the main subject of philosophical research. Men have always had some discussion of this question, some answer for it. And these answers have vacillated between two poles, from the full negation of the causes themselves, and the assertion that the causes of sensa- tions are contained within ourselves and not in anything outside of us — up to the recognition that we know these causes, that they are embodied in the phenomena of the outer world, that these phenomena constitute the cause of sensations; and that the cause of all observed phenomena lies in the movement of "atoms", and the oscillations of the "ether". It is believed that if we cannot observe these motions and oscillations it is only because we have not sufficiently powerful instruments, and that when such instru- ments are at our disposal we shall be able to see the movements of atoms as well as we see, through powerful telescopes, stars the very existence of which were never guessed. In modern philosophy Kant's system occupies a middle posi- tion in relation to this problem of the causes of sensations, not sharing either of these extreme views. Kant proved that the causes of our sensations are in the outside world, but that we can- not know these causes through any sensuous approach — that is, by such means as we know phenomena — and that we can not know these causes, and will never know them. Kant established the fact that everything that is known through the senses is known in terms of time and space, and that out of time and space we cannot know anything by way of the TERTIUM ORGANUM о senses; that time and space are necessary conditions of sensuous receptivity (i. е., receptivity by means of the five organs of sense). Moreover, what is most important, he established the fact that extension in space and existence in time are not proper- ties appertaining to things, but just the properties of our sensuous receptivity; that in reality, apart from our sensuous knowledge of them, things exist independently of time and space, but we can never perceive them out of time and space, and perceiving things and phenomena thus sensuously, by virtue of it we impose upon them the conditions of time and space, as belonging to our form of perception. Thus space and time, defining everything that we cognize by sensuous means, are in themselves just forms of consciousness, categories of our intellect, the prism through which we regard the world — or in other words space and time do not represent properties of the world, but just properties of our knowledge of the world gained through our sensuous organism. Consequently the world, until by these means we come into relation to it, has neither extension in space nor existence in time; these are proper- ties which we add to it. Cognitions of space and time arise in our intellect during its touch with the external world by means of the organs of sense, and do not exist in the external world apart from our contact with it. . Space and time are categories of intellect, i. е., properties which are ascribed by us to the external world. They are signal posts, signs put up by ourselves because we cannot picture the external world without their help. They are graphics by which we repre- sent the world to ourselves. Projecting outside of ourselves the causes of our sensations, we are designing mentally (and only mentally) those causes in space, and we picture continuous reality to ourselves as a series of moments of time following one another. This is necessary for us because a thing having no definite extension in space, not occupying a certain part of space and not lasting a certain length of time does not exist for us at all. That is, a thing not in space, divorced from the idea of space, and not included in the category of space, will not differ from some other thing in any particular; it will occupy the very same place, will coincide with it. Also, all phenomena not in time, divorced from the idea of time, not taken in this or that fashion from the 6 TERTITJM ORGANUM standpoint of before, now, after, would proceed for us as though they were simultaneously moving among themselves, and our weak intellect would not be able to distinguish one moment in the infinite variety. Therefore our consciousness segregates out of a chaos of im- pressions, separate groups, as we construct in space and time the perceptions of things according to these groups of impressions. It is necessary for us to divide things somehow, and we divide them into the categories of space and time. But we should remember that these divisions exist only in us, in our knowledge of things, and not in the things themselves; that we do not know the true relations of things among them- selves, and the real things we do not know, but only phantoms, visions of things — we do not know the relations existing among the things in reality. At the same time we quite definitely know that our division of things into the categories of space and time does not at all correspond to the division of things in themselves, inde- pendently of our receptivity of them; and we quite definitely know that if there exists any division at all among things in themselves, it will in no case be a division in terms of space and time, because these are not a property of things, but of our knowledge of things gained through the senses. Moreover, we do not know if it is even possible to distinguish those divisions which we see, i. е., in space and time, if things are looked at not through human eyes, not from the human standpoint. In point of fact we do not know but that our world would present an entirely different aspect for a differently built organism. We cannot perceive things as images outside of the categories of space and time, but we constantly think of them outside of space and time. When we say that table, we picture the table to ourselves in space and time; but when we say an object made of wood, not meaning any definite thing, but speaking generally, it will relate to all things made of wood throughout the world, and in all ages. An imaginative person could conceive that we are referring to some great thing made of wood, composed of all objects whenever and wherever wooden things existed, these forming its constituent atoms, as it were. We do not comprehend all these matters quite clearly, but in general it is plain that we think in space and time by perceptions TERTIUM ORGANUM ' only; but by concepts we think independently of space and time. Kant named his views critical idealism, in contradistinction to dogmatic idealism, of which Berkeley was a representative. According to dogmatic idealism, all the world, all things— i е., the true causes of our sensations— do not exist except in our con- sciousness: they exist only so far as we know them. The entire world perceived by us is just a reflection of ourselves. Kantian idealism recognizes a world of causes outside ot us, but asserts that we cannot know the world by means of sensuous per- ception, and everything that we perceive, generally speaking, is of our own creation— the product of a cognizing being. So, according to Kant, everything that we find in things is put in them by ourselves. Independently of ourselves, we do not know what the world is like. And our cognition of things has nothing in common with the things as they are outside of us— that is, in themselves. Furthermore, and most important, our ignor- ance of things in themselves does not depend upon our insufficient knowledge, but it is due to the fact that by means of sensuous perception we cannot know the world correctly at all. lnat is to say, we cannot truly declare that although now we perhaps know little, presently we shall know more, and at length shall come to a correct understanding of the world. It is not true be- cause our experimental knowledge is not a confused perception of a real world. It is a very acute perception of an entirely unreal world appearing round about us at the moment of our contact with the world of true causes, to which we cannot find the way because we are lost in an unreal "material" world.— For this reason the extension of the objective sciences does not brmg us any nearer to the knowledge of things in themselves, or ot true causes. In "A Critique of Pure Reason" Kant affirms that: Nothing which is intuited in space is a thing in itself, and space is not a form which belongs as a property to things; but objects are quite un- known to us in themselves, and what we call outward objects are nothing else but mere representations of our sensibility, whose form is space, but whose real correlated thing in itself is not known by means of these representations, nor ever can be, but respecting which, in experience, no inquiry is ever made. 8 TERTIUM ORGANUM The things which we intuit are not in themselves the same as our rep- resentation of them in intuition, nor are their relations in themselves so constituted as they appear to us; and if we take away the subject, or even only the subjective constitution of our senses in general, then not only the nature and relations of objects in space and time disappear, but even space and time themselves. What may be the nature of objects considered as things in themselves and without reference to the receptivity of our sensibility is quite un- known to us. We know nothing more than our own mode of perceiving them, which is peculiar to us and which though not of necessity pertain- ing to every animated being, is so to the whole human race. Supposing that we should carry our empirical intuition even to the very highest degree of clearness we should not thereby advance one step nearer to the constitution of objects as things in themselves. To say then that our sensibility is nothing but the confused repre- sentation of things containing exclusively that which belongs to them as things in themselves, and this under an accumulation of characteristic marks and partial representations which we cannot distinguish in con- sciousness, is a falsification of the conception of sensibility and phe- nominization, which renders our whole doctrine thereof empty and useless. The difference between a confused and clear representation is merely logical, and has nothing to do with content. Up to the present time Kant's propositions have remained in the very form that he left them. Despite the multiplicity of new philosophical systems which appeared during the nineteenth cen- tury, and despite the number of philosophers who have particu- larly studied, commented upon, and interpreted Kant's writings Kant's principal propositions have remained quite undeveloped primarily because most people do not know how to read Kant at all, and they therefore dwell upon the unimportant and non- essential, ignoring the substance. Yet really Kant only just put the question, threw to the world the problem, demanding the solution but not pointing the way toward it. This fact is usually omitted when speaking of Kant. He pro- pounded the riddle, but did not give the solution of it. And to the present day we repeat Kant's propositions, we con- sider them incontrovertible, but in the main we represent them to our understanding very badly, and they are not correlated with other departments of our knowledge. All our positive science — physics (with chemistry) and biology — is built upon hypotheses contradictory to Kant's propositions. TERTIUM ORGANUM 9 Moreover, we do not realize how we ourselves impose upon the world the properties of space, i. е., extension; nor do we realize how the world — earth, sea, trees, men — cannot possess such ex- tension. We do not understand how we can see and measure that exten- sion if it does not exist — nor what the world represents in itself, if it does not possess extension. But does the world really exist? — Or, as a logical conclusion from Kant's ideas, shall we recognize the validity of Berkeley's idea, and deny the existence of the world itself except in imagina- tion? Positive philosophy stands in a very ambiguous relation to Kant's views. It accepts them and it does not accept them: it ac- cepts, and considers them correct in their relation to the direct experience of the organs of sense — what we see, hear, touch. That is, positive philosophy recognizes the subjectivity of our recep- tivity, and recognizes everything that we perceive in objects as imposed upon them by ourselves — but this in relation to the di- rect experience of the senses only. When it concerns itself with "scientific experience" however, in which precise instruments and calculations are used, positive philosophy evidently considers Kant's view in relation to that invalid, assuming that "scientific experience" makes known to us the very substance of things, the true causes of our sensations — or if it does not do so now, it brings us closer to the truth of things, and can inform us later. Such dualism in the fundamental ideas of knowledge moves the physicist, for example, to recognize the subjectivity of those color impressions by which we perceive the world by means of the eye — i. е., sensuously — at the same time that he attributes a real ex- istence to the vibrations of the ether, and calculates the number of vibrations corresponding to this or that color. The fact of eth- eric vibrations — a definite number of vibrations for every €0101* — seems to him as established quite independently of the sensuous receptivity of colors by means of the eye, its affiliated nerves, and so on. Consequently, green light, as it is perceived by the eye, is regarded as subjective, i. е., as the product of a perceiving person; but the very same green light, investigated by the physicist, who calculates the number of etheric vibrations corresponding to green 10 TERTIUM ORGANUM light, is considered as existing really and objectively. The physi- cist is sure that a certain number of etheric vibrations produces the subjective sensation of the color green, and is entirely unwill- ing to allow that the sole reality in all this concatenation is that very subjective sensation of green color, and that the definition of green as an etheric vibration is nothing less than the solution of an equation containing two unknown quantities: color, and green, with the help of two other unknown quantities: ether, and vibra- tion. By such a method, of course, it is easliy possible to solve any equation whatsoever: but the method can only be called a change of variables. All "positivism" is in substance the substitution of one set of variables by another. Nevertheless, contrary to Kant, the positivists are sure that "more clear knowledge of phenomena makes them acquainted with things-in-themselves. ' ' They think that looking upon phys- ical phenomena as the motions of the ether, or electrons, and cal- culating their motions, they begin to know the very substance of things; that is, they believe exactly in the possibility of what Kant denied — the comprehension of the true substance of things by means of the investigation of phenomena. Moreover many physicists do not consider it necessary even to know Kant; and they could not themselves exactly define in what relation they stand toward him. Of course it is possible not to know Kant, but it is impossible to controvert him. Every description of physical phenomena, by its every word, is related to the problems set forth by Kant — remains in this or that relation to them. For to accept the theory of etheric vibration, or the activity of electrons, it is necessary to recognize space and time as existing outside of us, to recognize them as real properties of the world and not alone as properties of our sensuous receptivity; to assume that space and time are not imposed upon the world by us, but are per- ceived by us from without as something inherent in the world. In general, the position of "science" in regard to this question of "subjectively imposed" or "objectively cognized" is more than tottering, and in order to form its conclusions "science" is forced to accept many purely hypothetical suppositions as things known — as indubitable data, not demanding proof. This fact is usually lost sight of, and the definition of physical phenomena as the vibrations of the ether, or the development of electronic energy, has come into such universal use that we count TERTIUM ORGANUM 11 it almost as a fact, and we forget that everything is just hypothesis all the way through. We are so used to the "ether" and its "vibra- tions" or oscillations; to "electrons" and their energy, that we cannot dispense with them, and even forget to examine into the relation these hypotheses bear to the problem of space and time as set forth by Kant. We are simply "not thinking" that one ex- cludes the other, and that these hypotheses— i. е., hypotheses of the "ether" or electrons — and Kant's hypothesis are impossible when taken in conjunction. Moreover, physicists forget one very significant fact: in his book, "Analysis of Sensations" Mach says: In the investigation of purely physical processes we generally employ concepts of so abstract a character that as a rule we think only cursorily, or not at all, of the sensations (elements) that lie at their base. . . The foundation of all purely physical operations is based upon an almost unending series of sensations, particularly if we take into consideration the adjustment of the apparatus which must precede the actual experi- ment. Now it can easily happen to the physicist who does not study the psychology of his operations, that he does not (to reverse a well-known saying) see the trees for the wood, that he overlooks the sensory ele- ment at the foundation of his work. . . Psychological analysis has taught us that this is not surprising, since the physicist is always oper- ating with sensations.* Mach here calls attention to a very important thing. Physi- cists do not consider it necessary to know psychology and to deal with it in their conclusions. But when they know psychology and take it into consideration, then they hold the most fantastic duality of opinion, as in the case of the man of orthodox belief who tries to reconcile the dogmas of faith with the arguments of reason, and who is obliged to believe simultaneously in the creation of the world in seven days, seven thousand years ago, and in geological periods hundreds of thous- ands of years long, and in the evolutionary theory. He is thus forced to resort to sophisms, and demonstrate that by seven days is meant seven periods. But why seven, exactly, he is unable to explain. For physicists the role of the "creation of the world" is played by the atomic theory and the ether, with its wave-like vi- brations, and further by the electrons, and the energetic, or electro-magnetic theory of the world. Or sometimes it is even worse, for the physicist in the depth of his soul knows where the truth lies — knows what all atomic and ♦Open Court Publishing Co's edition of Mach'a work. 1914, pages 41, 42, and 43. 12 ТЛЖТШМ ORGANUM energetic theories are worth in reality, but fears to hang in the air, as it were; to take refuge in mere negation. He has no definite system in place of that whose falsity he already knows; he is afraid to make a plunge into mere emptiness. Lacking sufficient cour- age to declare that he believes in nothing at all, he accoutres himself in all materialistic theories, as in an official uniform, only because with this uniform are bound up certain rights and priveleges, outer as well as inner, consisting of a certain confidence in himself and in his surroundings, to forego which he has no strength and deter- mination. The "unbelieving materialist" — this is the tragic fig- ure of our times, analogous to the "atheist" or "unbelieving priest" of the times of Voltaire. Out of this abhorrence of a vacuum come all dualistic theories which recognize "spirit" and "matter" existing simultaneously and independently of one another. In general, to a disinterested observer, the state of our con- temporary science should be of great psychological interest. In all branches of scientific knowledge we are absorbing an enormous number of facts destructive of the harmony of existing systems. And these systems can maintain themselves only by reason of the heroic attempts of scientific men who are trying to close their eyes to a long series of new facts which threatens to submerge every- thing in an irresistible stream. If in reality we were to collect these system-destroying facts they would be so numerous in every department of knowledge as to exceed those upon which existing systems are founded. The systematization of that which we do not know may yield us more for the true understanding of the world and the self than the systematization of that which in the opinion of "exact science" we do know. CHAPTER II \ new view of the Kantian problem. The Ideas of Hinton. The"space Tense" and its evolution. A system for the development of a sense of the fourth dimension by exercises with colored cubes. The lometrkal conception of space. Three perpendiculars-why thrS Can everything existing be measured by three perpendicu- ars? Facts physical and metaphysical. The indices of existence^ Reality of ideas. Insufficient evidence of the existence of matter and motion. ^ Matter and motion are only logical concepts, like "good" and "evil." S already stated, Kant propounded the problem, but gave no solution of it, nor did he point the way to a solution. And not one of the known commenta- tors, interpreters, followers or adversaries of Kant has found a solution, nor the way to it. I find the first flashes of a right understanding of the Kantian problem, and the first suggestions in regard to a possible way toward its solution in the writings of С. H. Hinton, author of the books, "A New Era of Thought" and "The Fourth Dimension. These books contain interesting synopses of many things pre- viously written about problems of higher dimensions, together with ideas of the author's own which have a bearing upon the sub- ject under discussion here. m Hinton notes that in commenting upon Kantian ideas, only their negative side is usually insisted upon, namely, the fact that we can cognize things in a sensuous way, in terms of space and time only, is regarded as an obstacle, hindering us from seeing what things in themselves really are, preventing the possibility of cognizing them as they are, imposing upon them that which is not inherent in them, shutting them off from us. But [says Hinton] if we take Kant's statement simply as it is— not seeing in the spatial conception a hindrance to right receptivity— that we apprehend things by means of space— then it is equally allowable to con- sider our space sense not as a negative condition, hindering our percep- tion of the world, but as a positive means by which the mind grasps its experiences, i. е., by which we cognize the world. 13 14 TERTIUM ORGANUM 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. 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. Now according to Kant [Hinton goes on to say] 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 thorough-going education of the space sense. It is left to be organized by accident. Yet the special development of the space sense makes us acquainted with a whole series of new conceptions. Fichte, Schelling, Hegel, have developed certain tendencies and have written remarkable books, but the true successors of Kant are Gauss and Lobachevsky. For if our intuition of space is the means whereby 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 colored, so to speak, by the conditions (of psychical activity) of the being which uses it. 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 P ' Л New Era of Thought. Hinton invented a complicated system for the education and development of the space sense by means of exercises with groups of cubes of different colors. The books above mentioned are devoted to the exposition of this system. In my opinion Hinton's exercises are interesting from a theoretical standpoint, but they are practically valuable only for such as have the same turn of mind as Hinton's own. Exercises of the mind according to his system must first of all lead to the development of the ability to imagine objects, not as the eye sees them, i. е., in perspective, but as they are geometric- ally — to learn to imagine the cube, for example, simultaneously from all sides. Moreover, such a development of the imagination * Mr Ouspensky does not quote authors verbatim, aa a rule, but sometimes condenses, sometimes develops their thoughts. A comparison of pp. 2, 3, and 4 of Hinton's book with the quotation will indicate his method. Traml. TERTIUM ORGANUM 15 as overcomes the illusions of perspective results in the expansion of the limits of consciousness, thus creating new conceptions and augmenting the faculty for perceiving analogies. Kant established the fact that the development of knowledge under the existing conditions of receptivity will not bring us any closer to things in themselves. But Hinton asserts that it is possi- ble, if desired, to change the very conditions of receptivity, and thus to approach the true substance of things. 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 dimen- sions, 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 which we can measure 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 dimen- sions* 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? There is really no more difficulty in conceiving four-dimensional shapes, when we go about it in the right way, than in conceiving the idea of solid shapes, nor is there any mystery at all about it. When the faculty to apprehend in four dimensions 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. * Italics by P. D. Ouspensky. Transl. 16 TERTIUM ORGANUM The next step after having formed this power of conception in ampler space, is to investigate nature and see what phenomena are to be ex- plained by four-dimensional relations. The thought of past ages has used the conception of a three-dimensional space, and by that means has classified many phenomena and has ob- tained 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 apprehen- sion. . . For development of knowledge it is necessary to separate the self elements, i. е., the personal element which we put in everything cognized by us from that which is cognized, in order that our attention may not be distracted (upon ourselves) from the properties which we, in substance, perceive. Only by getting rid of the self elements in our receptivity do we put ourselves in a position in which we can propound sensible questions. Only by getting rid of the notion of a circular motion of the sun around the earth (i. е., around us — self-element) do we prepare our way to study the sun as it really is. But the worst about a self element is that its presence is never dreamed of till it is got rid of. In order to understand what the self element in our receptivity means, imagine ourselves to be translated suddenly to another part of the uni- verse, and to find there intelligent beings and to hold conversation with them. If we told them that we came from this world, and were to describe the sun to them, saying that it was a bright, hot body which moved around us, they would reply: "You have told us something about the sun, but you have also told us something about yourselves." Therefore, desiring to tell something about the sun, we shall first of all get rid of the self element which is introduced into our knowledge of the sun by the movement of the earth, upon which we are, round it. . . One of our serious pieces of work will be to get rid of the self elements in the knowledge of the arrangement of objects. The relations of our universe or our space with regard to the wider universe of four-dimensional space are 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 seems to us now. I would divide studies of arrangement 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 un- fortunately, 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. . . TERTIUM ORGANUM 17 Of the possible units which will serve for the study of arrangement, 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. . . Our work then will 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. Thus we will bring our minds into contact with nature.* A Ney) Em of Thought I shall return again to Hinton's books several times, but mean- while it is necessary to establish our relation to the ideas which Kant's problem touches. What is space? Taken as object, that is, perceived by our consciousness, space is for us the form of the universe or the form of the matter in the universe. Space possesses an infinite extension in all directions. But it can be measured in only three directions independent of one another; in length, breadth, and height; these directions we call the dimensions of space, and we say that our space has three dimensions: it is three-dimensional. By independent direction we mean in this case a line at right angles to another line. Our geometry, (or the science of measurement of the earth, or matter in space) knows only three such lines, which are mutually at right angles to one another and not parallel among them- selves. Should we mean by independent direction the line which is not at right angles, i. е., which does not form with the others an angle of 90 degrees, but an angle, say, of 30 degrees, then we would have the number of dimensions not three, but nine. It is seen from this that the three-dimensionality of our space is simply a geometrical condition, and depends upon the fact that we are using right angles as a unit of measurement. But at the same time, in our space and our universe we know only three perpendiculars, i. е., only three independent right angles. But why three only, and not ten or fifteen? This we do not know. * The entire quotation ia compiled by Mr. Ouspensky, conveying Hinton's ideas and omitting all non-essentials. Transl. 18 TERTIUM ORGANUM And here is another very significant fact: either because of some mysterious property of the universe, or because of some mental limitation, we cannot even imagine to ourselves more than three independent directions. But we speak of the universe as infinite, and because the first condition of infinity is infinity in all directions and in all possible relations, so we must presuppose in space an infinite number of dimensions: that is, we must presuppose an infinite number of lines perpendicular and not parallel to each other : and yet out of these lines we know, for some reason, only three. It is usually in some such guise that the question of higher dimensionality appears to normal human consciousness. Since we cannot construct more than three mutually inde- pendent perpendiculars, and if the three-dimensionality of our space is conditional upon this, we are forced to admit the indubit- able fact of the limitedness of our space in relation to geometrical possibilities : though of course if the properties of space are created by some limitation of consciousness, then the limitedness lies in ourselves. No matter what this limitedness depends on, it is a fact that it exists. A given point can be the vertex of only eight independent tetrahedrons. Through a given point it is possible to draw only three perpendicular and not parallel straight lines. Upon this as a basis, we define the dimensionality of space by the number of lines it is possible to draw in it which are mutually at right angles one with another. The line upon which there cannot be a perpendicular, that is, another line, constitutes linear, or one-dimensional space. Upon the surface two perpendiculars are possible. This is superficial, or two-dimensional space. In "space" three perpendiculars are possible. This is solid, or three-dimensional space. The idea of the fourth dimension arose from the assumption that in addition to the three dimensions known to our geometry there exists still a fourth, for some reason unknown and inaccessi- ble to us, i. е., that in addition to the three known to us, a mys- terious fourth perpendicular is possible. This assumption is practically founded on the consideration that there are things and phenomena in the world undoubtedly TERTIUM ORGANUM 19 really existing, but quite incommensurable in terms of length, breadth and thickness, and lying as it were outside of three- dimensional space. By really existing we understand that which produces definite action, which possesses certain functions, which appears to be the cause of something else. That which does not exist cannot produce any action, has no function, cannot be a cause. But there are different modes of existence. There is physical existence, recognized by certain sorts of actions and functions, and there is metaphysical existence, recognized by its actions and its functions. A house exists, and the idea of good and evil exists. But they do not exist in like manner. One and the same method of proof of existence does not suffice for the proof of the existence of a house and for the proof of the existence of an idea. A house is a physical fact, an idea is a metaphysical fact. Physical and metaphysical facts exist, but they exist differently. In order to prove the idea of a divison into good and evil, i. е., a metaphysical fact, I have only to prove its possibility. This is already sufficiently established. But if I should prove that a house, i. е., a physical fact, may exist, it does not at all mean that it exists really. If I prove that a man may own the house it is no proof that he owns it. Our relation to an idea and to a house are quite different. It is possible by a certain effort to destroy a house — to burn, to wreck it. The house will cease to exist. But suppose you attempt to destroy, by an effort, an idea. The more you try to contest, argue, refute, ridicule, the more the idea is likely to spread, grow, strengthen. And contrary wise, silence, oblivion, non-action, " non- resistance" will exterminate, or in any case will weaken the idea. Silence, oblivion, will not wreck a house, will not hurt a stone. It is clear that the existence of a house and that of an idea are quite different existences. Of such different existences we know very many. A book exists, and also the contents of a book. Notes exist, and so does the music that the notes combine to make. A coin exists, and so does the pur- chasing value of a coin. A word exists, and the energy which it contains. 20 TERTIUM ORGANUM We discern on the one hand, a whole series of physical facts, and on the other hand, a series of metaphysical facts. As facts of the first kind exist, so also do facts of the second kind exist, but differently. From the usual positivist point of view it will seem naive in the highest degree to speak of the purchasing value of a coin separately from the coin: of the energy of a word separately from the word: of the contents of a booh separately from the book, and so on. We all know that these are only "what people say," that in reality purchasing value, energy of a word, and contents of a book do not exist, that by these conceptions we only denote a series of phenomena in some way linked with coin, word, book, but in substance quite separate from them. But is it so? We decided to accept nothing as given, consequently we shall not negate anything as given. We see in things, in addition to what is external, something in- ternal. We know that this internal element in things constitutes a continuous part of things, usually their principal substance. And quite naturally we ask ourselves, where is this internal element, and what does it represent in and by itself. We see that it is not embraced within our space. We begin to conceive of the idea of a "higher space" possessing more dimensions than ours. Our space then appears to be somehow a part of higher space, i. е., we begin to believe that we know, feel, and measure only part of space, that part which is measureable in terms of length, width and height. As was said before, we usually regard space as a form of the universe, or as a form of the matter of the universe. To make this clear it is possible to say that a "cube" is the form of the matter in a cube; a "sphere" is the form of the matter in a sphere; "space" — an infinite sphere — is the form of the entire matter of the universe. H. P. Blavatsky, in "The Secret Doctrine" has this to say about space: The superficial absurdity of assuming that Space itself is measurable in any direction is of little consequence. The familiar phrase (the fourth dimension of space) can only be an abbreviation of the fuller form— the "Fourth dimension of Matter in Space" . . The progress TERTIUM ORGANUM 21 of evolution may be destined to introduce us to new characteristics of matter. . ." * But the formula defining "space" as "the form of matter in the universe" suffers from this deficiency, that there is introduced in it the concept of "matter," i. е., the unknown. I have already spoken of that "dead end siding," x =y, у =x, to which all attempts at the physical definition of matter inevitably lead. Psychological definitions lead to the same thing. In a well known book, "The Physiology of the Soul," A. I. Gerzen says: We call matter everything which directly or indirectly offers resist- ance to motion, directly or indirectly produced by us, manifesting a remarkable analogy with our passive states. And we call force (motion) that which directly or indirectly com- municates movement to us or to other bodies, thus manifesting the greatest similitude to our active states. Consequently, "matter" and "motion" are something like pro- jections of our active and passive states. It is clear that it is possible to define the passive state only in terms of the active, and the active in terms of the passive — again two unknowns, de- fining one another. E. Douglas Fawcett, in an article entitled "Idealism and the Problem of Nature" in "The Quest" (April, 1910), discusses matter from this point of view. Matter (like force) does not give us any trouble. We know all about it, for the very good reason that we invented it. By "matter" we think of sensuous objects. It is mental change of concrete, but too complicated facts, which are difficult to deal with. Strictly speaking, matter exists only as a concept. Truth to tell, the character of matter, even when treated only as a conception, is so un- obvious that the majority of persons are unable to tell us exactly what they mean by it. An important fact is here brought to light : matter and force are just logical concepts, i. е., only words accepted for the designation of a lengthy series of complicated facts. It is difficult for us, edu- cated almost exclusively along physical lines, to understand this clearly, but in substance it may be stated as follows: Who has seen matter and force, and when? We see things, see phenomena. Matter, independently of the substance from which a given thing is made, or of which it consists, we have never seen and never * "The Secret Doctrine," The Theosophical Publishing Society. Third Edition, p. 271, vol. I. 22 TERTIUM ORGANUM will see; but the given substance is not quite matter, this is wood, or iron or stone. Similarly, we shall never see force separately from motion. What does this mean? It means that "matter" and "force" are just such abstract conceptions, as "value" or "labor," as "the purchasing value of a coin" or the "contents" of a book; it means that matter is "such stuff as dreams are made of." And because we can never touch this "stuff" and can see it only in dreams, so we can never touch physical matter, nor see, nor hear, nor photograph it, separately from the object. We cognize things and phenomena which are bad or good, but we never cognize "matter" and "force" separately from things and phe- nomena. Matter is as much an abstract conception as are truth, good and evil. It is as impossible to put matter or any part of matter into a chemical retort or crucible as it is impossible to sell "Egyptian darkness" in vials. However as it is said that "Egyptian darkness" is sold as a black powder in Athos, therefore perhaps even matter, somewhere, by some one, has been seen.* In order to discuss questions of this order a certain preparation is necessary, or a high degree of intuition; but unfortunately it is customary to consider fundamental questions of cosmogony very lightly. A man easily admits his incompetency in music, dancing, or higher mathematics, but he always maintains the privilege of having an opinion and being a judge of questions relating to "first principles." It is difficult to discuss with such men. For how will you answer a man who looks at you in perplexity, knocks on the table with his finger and says, " This is matter. I know it ; feel ! How can it be an abstract conception ? " To answer this is as difficult as to answer the man who says: "I see that the sun rises and sets!" Returning to the consideration of space, we shall under no cir- cumstances introduce unknown quantities in the definition of it. We shall define it only in terms of those two data which we decided to accept at the very beginning. The world and consciousness are the facts which we decided to recognize as existing. * This is irony which the English speaking may easily fail to understand. Some unscrupulous monks of the monastery of Athos, famous throughout Greece and Russia, made a practice, it is said, of selling "Egyptian darkness" in little vials, thus making capital out of the credulity and piety of the illiterate Russian pilgrims who were wont to visit this monastery in great numbers. Transl. TERTIUM ORGANUM 23 By the world we mean the combination of unknown causes of our sensations. By the material world we mean the combination of unknown causes of a definite series of sensations, those of sight, hearing, touch, smell, taste, sensations of weight, and so on. Space is either a property of the world or a property of our knowledge of the world. Three-dimensional space is either a property of the material world or a property of our receptivity of the material world. Our inquiry is confined to the problem : how shall we approach the study of space? CHAPTER III What may we learn about the fourth dimension by a study of the geo- metrical relations within our space? What should be the relation between a three-dimensional body and one of four dimensions ? The four-dimensional body as the tracing of the movement of a three- dimensional body in the direction which is not confined within it. A four-dimensional body as containing an infinite number of three- dimensional bodies. A three-dimensional body as a section of a four- dimensional one. Parts of bodies and entire bodies in three and in four dimensions. The incommensurability of a three-dimensional and a four-dimensional body. A material atom as a section of a four-dimensional line. |N another of his books, "The Fourth Dimension," Hinton makes an interesting remark about the method by which we may approach the question of the higher dimensions. This is what he says : Our space itself bears within it relations through which we can establish relations to other (higher) spaces. For within space are given the conception of point and line, line and plane, which really involve the relation of space to a higher space. If we concentrate upon this thought, and consider the very great difference between the point and the line, between the line and the surface, surface and solid, we shall indeed come to under- stand how much of the new and inconceivable the fourth dimen- sion holds for us. As in the point it is impossible to imagine the line and the laws of the line; as in the line it is impossible to imagine the sur- face and the laws of the surface; as in the surface it is impossible to imagine the solid and the laws of the solid, so in our space it is impossible to imagine the body having more than three dimensions, and impossible to understand the laws of the exist- ence of such a body. But studying the mutual relations between the point, the line, the surface, the solid, we begin to learn something about the fourth dimension, i. е., of four-dimensional space. We begin to 25 26 TERTIUM ORGANUM learn what it can be in comparison with our three-dimensional space, and what it cannot be. This last we learn first of all. And it is especially important, because it saves us from many deeply inculcated illusions, which are very detrimental to right knowledge. We learn what cannot be in four-dimensional space, and this permits us to set forth what can be there. Let us consider these relations within our space, and let us see what conclusions we can derive from their investigation. We know that our geometry regards the line as a tracing of the movement of a point; the surface as a tracing of the movement of a line; and the solid as a tracing of the movement of a surface. On these premises we put to ourselves this question: Is it not possible to regard the "four-dimensional body" as a tracing of the movement of a three-dimensional one? But what is this movement, and in what direction? The point, moving in space, and leaving the tracing of its move- ment, a line, moves in a direction not contained in it, because in a point there is no direction whatsoever. The line, moving in space, and leaving the tracing of its move- ment, the surface, moves in a direction not contained in it because, moving in a direction contained in it, a line will continue to be a line. The surface, moving in space, and leaving a tracing of its move- ment, the solid, moves also in a direction not contained in it. If it should move otherwise, it would remain always the surface. In order to leave a tracing of itself as a "solid," or three-dimensional figure, it must set off from itself, move in a direction which in itself it has not. In analogy with all this, the solid, in order to leave as the tracing of its movement, the four-dimensional figure (hypersolid) shall move in a direction not confined in it; or in other words it shall come out of itself, set off from itself, move in a direction which is not present in it. Later on it will be shown in what manner we shall understand this. But for the present we can say that the direction of the move- ment in the fourth dimension lies out of all those directions which are possible in a three-dimensional figure. We consider the line as an infinite number of points; the surface as an infinite number of lines; the solid as an infinite number of surfaces. TERTIUM ORGANUM 27 In analogy with this it is possible to consider that it is necessary to regard a four-dimensional body as an infinite number of three- dimensional ones, and four-dimensional space as an infinite num- ber of three-dimensional spaces. Moreover, we know that the line is limited by points, that the surface is limited by lines, that the solid is limited by surfaces. It is possible that a four-dimensional body is limited by three- dimensional bodies. Or it is possible to say that the line is a distance between two points; the surface a distance between two lines; the solid — be- tween two surfaces. Or again, that the line separates two points or several points from one another (for the straight line is the shortest distance between two points) ; that the surface separates two or several lines from each other; that the solid separates several surfaces one from another; so the cube separates six flat surfaces one from another — its faces. The line binds several separate points into a certain whole (the straight, the curved, the broken line) ; the surface binds several lines into something whole (the quadrilateral, the triangle); the solid binds several surfaces into something whole (the cube, the pyramid) . , It is possible that four-dimensional space is the distance between a group of solids, separating these solids, yet at the same time binding them into some to us inconceivable whole, even though they seem to be separate from one another. Moreover, we regard the point as a section of a line; the line as a section of a surface; the surface as a section of a solid. By analogy, it is possible to regard the solid (the cube, sphere, pyramid) as a section of a four-dimensional body, and our entire three-dimensional space as a section of a four-dimensional space. If every three-dimensional body is the section of a four- dimen- sional one, then every point of a three-dimensional body is the section of a four-dimensional line. It is possible to regard an "atom" of a physical body, not as something material, but as an intersection of a four-dimensional line by the plane of our con- sciousness. The view of a three-dimensional body as the section of a four- dimensional one leads to the thought that many (for us) separ- ate bodies may be the sections of parts of one four-dimensional body. 28 TERTIUM ORGANUM A simple example will clarify this thought. If we imagine a horizontal plane, intersecting the top of a tree, and parallel to the surface of the earth, then upon this plane the sections of branches will seem separate, and not bound to one another. Yet in our space, from our standpoint, these are sections of branches of one tree, comprising together one top, nourished from one root, casting one shadow. Or here is another interesting example expressing the same idea, given by Mr. Leadbeater, the theosophical writer, in one of his books. If we touch the surface of a table with our finger tips, then upon the surface will be just five circles, and from this plane pre- sentment it is impossible to construe any idea of the hand, and of the man to whom this hand belongs. Upon the table's surface will be five separate circles. How from them is it possible to imagine a man, with all the richness of his physical and spiritual life? It is impossible. Our relation to the four-dimensional world will be analogous to the relation of that consciousness which sees five circles upon the table to a man. We see just "finger tips;" to us the fourth dimension is inconceivable. We know that it is possible to represent a three-dimensional body upon a plane, that it is possible to draw a cube, a polyhedron or a sphere. This will not be a real cube or a real sphere, but the pro- jection of a cube or of a sphere on a plane. We may conceive of the three-dimensional bodies of our space somewhat in the nature of images in our space of to us incomprehensible four-dimensional bodies. CHAPTER IV In what direction may the fourth dimension lie? What is motion? Two kinds of motion — motion in space and motion in time— ;which are contained in every movement. What is time? Two ideas con- tained in the conception of time. The new dimension of space, and motion upon that dimension. Time as the fourth dimension of space. Impossibility of understanding the fourth dimension with- out the idea of motion. The idea of motion and the "time sense." The time sense as a limit (surface) of the "space sense." Hinton on the law of surfaces. The "ether" as a surface. Riemann's idea concerning the translation of time into space in the fourth dimen- sion. Present, past, and future. Why we do not see the past and the future. Life as a feeling of one's way. Wundt on the sub- ject of our sensuous knowledge. E have established by a comparison of the relation of lower dimensional figures to higher dimensional ones that it is possible to regard a four-dimen- sional body as the tracing of the motion of a three-dimensional body upon the dimension not contained in it; i. е., that the direction of the motion upon the fourth dimension lies outside of all the directions which are possible in three-dimensional space. But in what direction is it? In order to answer this question it will be necessary to discover whether we do not know some motion not confined in three- dimensional space. We know that every motion in space is accompanied by that which we call motion in time. Moreover, we know that everything existing, even if not moving in space, moves eternally in time. And equally in all cases, whether speaking of motion or absence of motion, we have in mind an idea of what was before, what now becomes, and what will follow after. In other words, we have in mind the idea of time. The idea of motion of any kind, also the idea of absence of motion is indissolubly bound up with the idea of time. Any motion or absence of motion proceeds in time and cannot proceed out of time. Consequently, before speaking of what motion is, we must answer the question, what is time? 29 30 TERTIUM ORGANUM Time is the most formidable and difficult problem which con- fronts humanity. Kant regards time as he does space: as a subjective form of our receptivity; i. е., he says that we create time ourselves, as a function of our receptive apparatus, for convenience in perceiving the out- side world. Reality is continuous and constant, but in order to make possible the perception of it, we must dissever it into sepa- rate moments ; imagine it as an infinite series of separate moments out of which there exists for us only one. In other words, we perceive reality as though through a narrow slit, and what we are seeing through this slit we call the present; what we did see and now do not see — the past, and what we do not quite see but are expecting — the future. Regarding each phenomenon as an effect of another, or others, and this in its turn as a cause of a third; that is, regarding all phenomena in functional interdependence one upon another, by this very act we are contemplating them in time, because we picture to ourselves quite clearly and precisely first a cause, then an effect; first an action, then its function, and cannot contem- plate them otherwise. Thus we may say that the idea of time is bound up with the idea of causation and functional interdepend- ence. Without time causation cannot exist, just as without time motion or the absence of motion cannot exist. But our perception concerning our "being in time" is entangled and misty up to improbability. First of all let us analyze our relation toward the past, present and future. Usually we think that the past already does not exist. It has passed, disappeared, altered, transformed itself into some- thing else. The future also does not exist — it does not exist yet. It has not arrived, has not formed. By the present we mean the moment of transition of the future into the past, i. е., the moment of transition of a phenomenon from one non-existence into another one. Only for that short moment does the phenomenon exist for us in reality; before, it existed in potentiality, afterward it will exist in remembrance. But this short moment is in substance only a fiction: it has no measurement. We have a full right to say that the present does not exist. We can never catch it. That which we did catch is always the past! TERTIUM ORGANUM 31 If we are to stop at that we must admit that the world does not exist, or exists only in some phantasmagoria of illusions, flashing and disappearing. д , .i nt Usually we take no account of this, and do not reflect that our usual view of time leads to utter absurdity. Let us imagine a stupid traveller going from one city to another and half way between these two cities. A stupid traveller thinks that the city from which he has departed last week does not exist now only the memory of it is left; the walls are ruined, the towers fallen, the inhabitants have either died or gone away. Also that city at which he is destined to arrive in several days does not exist now either, but is being hurriedly built for his arrival, and on the day of that arrival will be ready, populated, and set in order, and on the day after his departure will be destroyed just as was the first one. We are thinking of things in time exactly m this way— every- thing passes away, nothing returns! The spring has passed, it does not exist still. The autumn has not come, it does not exist yet. But what does exist? The present. . . . , But the present is not a seizable moment, it is continuously transitory into the past. So, strictly speaking, neither the past, nor the present, nor the future exists for us. Nothing exists! And yet we are living, teel- ing, thinking— and something surrounds us. Consequently, in our usual attitude toward time there exists some mistake. Inis error we shall endeavor to detect. We accepted in the very beginning that something exists. We called that something the world. How then can the world exist it it is not existing in the past, in the present, in the future.'' That conception of the world which we deduced from our usual view of time makes the world appear like a continuously gushing out igneous fountain of fireworks, each spark of which flashes lor a moment and disappears, never to appear any more. Flashes are going on continuously, following one after another, there are an infinite number of sparks, and everything together produces the impression of a flame, though it does not exist in reality. The autumn has not come yet. It will be, but it does not^ exist now And we give no thought to how that can appear which is not. 32 TERTIUM ORGANUM We are moving upon a plane, and recognize as really existing only the small circle lighted by our consciousness. Every thing out of this circle, which we do not see, we negate, we do not like to admit that it exists. We are moving upon the plane in one direc- tion. This direction we consider as eternal and infinite. But the direction at right angles to it, those lines which we are intersecting, we do not like to recognize as eternal and infinite. We imagine them as going into non-existence at once, as soon as we have passed them, and that the lines before us have not yet risen out of non-existence. If, presupposing that we are moving upon a sphere, upon its equator or one of its parallels, then it will appear that we recognize as really existing only one meridian: those which are behind us have disappeared and those ahead of us have not appeared yet. We are going forward like a blind man, who feels paving stones and lanterns and walls of houses with his stick and believes in the real existence of only that which he touches now, which he feels now. That which has passed has disappeared and will never re- turn! That which has not yet been does not exist. The blind man remembers the route which he has traversed; he expects that ahead the way will continue, but he sees neither forward nor back- ward because he does not see anything, because his instrument of knowledge — the stick — has a definite, and not very great length, and beyond the reach of his stick non-existence begins. Wundt, in one of his books, called attention to the fact that our famous five organs of sense are in reality just feelers by which we feel the world around us. We live groping about. We never see anything. We are always just feeling everything. With the help of the microscope and the telescope, the telegraph and the telephone we are extending our feelers a little, so to speak, but we are not beginning to see. To say that we are seeing would be possible only in case we could know the past and the future. But we do not see, and because of this we can never assure ourselves of that which we cannot feel. This is the reason why we count as really existing only that circle which our feelers grasp at a given moment. Beyond that — darkness and non-existence. But have we any right to think in this way? Let us imagine a consciousness that is not bound by the condi- tions of sensuous receptivity. Such a consciousness can rise above the plane upon which we are moving; it can see far beyond the TERTIUM ORGANUM 33 limits of the circle enlightened by our usual consciousness; it can see that not only does the line upon which we are moving exist, but also all lines perpendicular to it which we are intersecting, which we have ever intersected, and which we shall intersect. After rising above the plane this consciousness can see the plane, can convince itself that it is really a plane, and not a single line. Then it can see the past and the future, lying together and exist- ing simultaneously. That consciousness which is not bound by the conditions of sensuous receptivity can outrun the stupid traveler, ascend the mountain to see in the distance the town to which he is going, and be convinced that this town is not being built anew for his arrival, but exists quite independently of the stupid traveler. And that consciousness can look off and see on the horizon the towers of that city where that traveler had been, and be convinced that those towers have not fallen, that the city continues to stay and live just as it stayed and lived before the traveler's advent. It can rise above the plane of time and see the spring behind and the autumn ahead, see simultaneously the budding flowers and ripening fruits. It can make the blind man recover his sight and see the road along which he passed and that which still lies before him. The past and the future cannot not exist, because if they do not exist then neither does the present exist. Unquestionably they exist somewhere together, but we do not see them. The present, compared with the past and the future, is the most unreal of all unrealities. We are forced to admit that the past, the present and the future do not differ in anything, one from another: there exists just one 'present — the Eternal Now of Hindu philosophy. But we do not perceive this, because in every given moment we experi- ence just a little bit of that present, and this alone we count as existent, denying a real existence to everything else. If we admit this, then our view of everything with which we are surrounded will change very considerably. Usually we regard time as an abstraction, made by us during the observation of really existing motion. That is, we think that observing motion, or changes of relations between things and comparing the relations which existed before, which exist now, 34 TERTIUM ORGANUM and which may exist in the future, that we are deducing the idea of time. We shall see later on how far this view is correct. Thus the idea of time is composed of the conception of the past, of that of the present, and of that of the future. Our conceptions of the past and present, though not very clear, are yet very much alike. As to the future there exists a great variety of views. It is necessary for us to analyze the theories of the future as they exist in the mind of contemporary man. There are in existence two theories — that of the preordained future, and that of the free future. Preordination is established in this way: we say that every future event is the result of those which happened before, and is created such as it will be and not otherwise as a consequence of a definite direction of forces which are contained in preceding events. This means, in other words, that future events are wholly con- tained in preceding ones, and if we could know the force and direc- tion of all events which have happened up to the present moment, i. е., if we knew all the past, by this we could know all the future. And sometimes, knowing the present moment thoroughly, in all its details, we may really foretell the future. If the prophecy is not fulfilled, we say that we did not know all that had been, and we dis- cover in the past some cause which had escaped our observation. The idea of the free future is founded upon the possibility of voluntary action and accidental new combinations of causes. The future is regarded as quite indefinite, or defined only in part, because in every given moment new forces, and new events and new phenomena are born which lie in a potential state, not causeless, but so incommensurable with causes — as the firing of a city from one spark — that it is impossible to detect or measure them. This theory affirms that one and the same action can have different results; one and the same cause, different effects; and it introduces the hypothesis of quite arbitrary volitional actions on the part of a man, bringing about profound changes in the subse- quent events of his own life and the lives of others. Supporters of the preordination theory contend on the con- trary that volitional, involuntary actions depend also upon causes, making them necessary and unavoidable at a given moment; that there is nothing accidental, and that there cannot be; that we call accidental only those things the causes of which we do not see by TERTIUM ORGANUM 35 reason of our limitations; and that different effects of causes seem- ingly the same occur becauses the causes are different in reality and only seem similar for the reason that we do not understand them well enough nor see them sufficiently clear. The dispute between the theory of the preordained future and that of the free future is an infinite dispute. Neither of these theories can say anything decisive. This is so because both theories are too literal, too inflexible, too material, and one repu- diates the other: both say, "either this or the other." In the one case there results a complete cold predestination; that which will be, will be, nothing can be changed— that which will befall to- morrow was predestined tens of thousands of years ago. There results in the other case a life upon some sort of needle-point called the present, which is surrounded on all sides by an abyss of non-existence, a journey in a country which does not yet exist, a life in a world which is born and dies every moment, in which nothing ever returns. And both these opposite views are equally untrue, because the truth, in the given case, as in so many others, is con- tained in a union of two opposite understandings in one. In every given moment all the future of the world is predestined and is existing, but is predestined conditionally, i. е., it will be such or another future according to the direction of events at a given moment, unless there enters a new fact, and a new fact can enter only from the side of consciousness and the will resulting from it. It is necessary to understand this, and to master it. Besides this we are hindered from a right conception of the relation of the present toward the future by our misunderstand- ing of the relation of the present to the past. The difference of opinion exists only concerning the future; concerning the past all agree that it has passed, that it does not exist now — and that it was such as it has been. In this last lies the key to the under- standing of the incorrectness of our views of the future. As a matter of fact, in reality our relation both to the past and to the future is far more complicated than it seems to us. In the past, behind us, lies not only that which really happened, but that which could have been. In the same way, in the future lies not only that which will be, but everything that may be. The past and the future are equally undetermined, equally exist in all their possibilities, and equally exist simultaneously with the present. 36 TERTIUM ORGANUM By time we mean the distance separating events in the order of their succession and binding them in different wholes. This dis- tance lies in a direction not contained in three-dimensional space, therefore it will be the new dimension of space. This new dimension satisfies all possible requirements of the fourth dimension on the ground of the preceding reasoning. It is incommensurable with the dimensions of three-dimensional space, as a year is incommensurable with St. Petersburg. It is perpendicular to all directions of three-dimensional space and is not parallel to any of them. As a deduction from all the preceding we may say that time (as it is usually understood) includes in itself two ideas : that of a cer- tain to us unknown space (the fourth dimension), and that of a motion upon this space. Our constant mistake consists in the fact that in time we never see two ideas, but see always only one. Usually we see in time the idea of motion, but cannot say from whence, where, whither, nor upon what space. Attempts have been made heretofore to unite the idea of the fourth dimension with the idea of time. But in those theories which have attempted to combine the idea of time with the idea of the fourth dimension appeared always the idea of some spatial element as existing in time, and along with it was admitted motion upon that space. Those who were constructing these theories evidently did not understand that leaving out the possibility of motion they were advancing the demand for a new time, because motion cannot proceed out of time. And as a result time goes ahead of us, like our shadow, receding according as we approach it. All our per- ceptions of motion have become confused. If we imagine the new dimension of space and the possibility of motion upon this new dimension, time will still elude us, and declare that it is unex- plained, exactly as it was unexplained before. It is necessary to admit that by one term, time, we designated, properly, two ideas — "a certain space" and "motion upon that space." This motion does not exist in reality, and it seems to us as existing only because we do not see the spatiality of time. That is, the sensation of motion in time, (and motion out of time does not exist) arises in us because we are looking at the world as though through a narrow slit, and are seeing the lines of in- tersection of the time-plane with our three-dimensional space only. TERTIUM ORGANUM 37 Therefore it is necessary to declare how profoundly incorrect is our usual theory that the idea of time is deduced by us from the observation of motion, and is really nothing more than the idea of that succession which is observed by us in motion. It is necessary to recognize quite the reverse: that the idea of motion is deduced by us out of an incomplete sensation of time, or of the time-sense, i. е., out of a sense or sensation of the fourth dimension, but out of an incomplete sensation. This incomplete sensation of time (of the fourth dimension) — the sensation through the slit — gives us the sensation of motion, that is, creates an illusion of motion which does not exist in reality, but instead of which there exists in reality only the extension upon a direction inconceivable to us. One other aspect of the question has very great significance. The fourth dimension is bound up with the ideas of "time" and "motion." But up to this point we shall not be able to under- stand the fourth dimension unless we shall understand the fifth dimension. Attempting to look at time as at an object, Kant says that it has one dimension: i. е., he imagines time as a line extending from the infinite future into the infinite past. Of one point of this line we are conscious — always only one point. And this point has no dimension because that which in the usual sense we call the present, is the recent past, and sometimes also the near future. This would be true in relation to our illusory perception of time. But in reality eternity is not the infinite dimension of time, but the one perpendicular to time; because, if eternity exists, then every moment is eternal. We can discover in time two dimensions. The second dimension of time, i. е., eternity, will be the fifth dimen- sion of space. The line of the first dimension of time extends in that order of succession of phenomena which are in causal inter- dependence — first the cause, then the effect: before, now, after. The line of the second dimension of time — the line of eternity — ex- tends perpendicularly to that line. It is impossible to understand the idea of time without con- ceiving to ourselves the idea of eternity; it is likewise impossible to understand space if we have no idea of eternity. From the standpoint of eternity, time does not differ in any- thing from the other lines and dimensions of space — length, 38 TERTIUM ORGANUM breadth, and height. This means that just as in space exist the things that we do not see, or speaking differently, not alone that which we see, so in time "events" exist before our consciousness has touched them, and they still exist after our consciousness has left them behind. Consequently, extension in time is extension into unknown space, and therefore time is the fourth dimension of space. But as has been shown already, time is not a simple, but a com- plex conception. And we shall have this in view — it consists of a conception of unknown space, vanishing in the past and future, and of illusory motion upon this space. It is necessary for us to regard time as a spatial conception con- sidered with relation to our two data — the universe and con- sciousness. The idea of time appears when consciousness comes in contact with the world through sensuous receptivity. It has been already shown that because of the properties of sensuous receptivity, con- sciousness sees the world as through a narrow slit. Out of this the following questions arise: 1. Why does there exist in the world illusionary motion? That is, why does not consciousness see through this slit the same thing at all times? Why, behind the slit, do changes proceed creating the illusion of motion, i. е., in what manner, and how does the focus of our consciousness run over the world of phenomena? In addition to all this it is necessary to remember that through the very same slit through which it sees the world, consciousness observes itself as part of the world, and sees in itself changes sim- ilar to the changes in the rest of things. 2. Why cannot consciousness extend that slit? We shall endeavor to answer these questions. First of all we shall remark that within the limits of our usual observation consciousness is always in the same conditions and cannot escape these conditions. In other words, it is as it were chained to some plane above which it cannot rise. These condi- tions or that plane we call matter. Our consciousness lives, so to speak, upon the very plane, and never rises above it. If conscious- ness could rise above this plane, so undoubtedly it would see underneath itself simultaneously, a far greater number of events TERTIUM ORGANUM 39 than it usually sees while on a plane. Just as a man, ascending a mountain, or going up in a balloon, begins to see simultaneously and at once many things which it is impossible to see simultaneous- ly and at once from below: the movement of two trains toward one another between which a collision will occur; the approach of an enemy detachment to a sleeping camp ; two cities divided by a ridge, etc. — so consciousness rising above the plane in which it usually functions, must see simultaneously the events divided for ordinary consciousness by periods of time. These will be the events which ordinary consciousness never sees together, as cause and effect: the work, and the payment; the crime and the punishment; the movement of trains toward one another and their collision; the approach of the enemy and the battle; the sunrise and the sunset; the morning and the evening; the day and the night; spring, autumn, summer and winter; the birth and the death of a man. The angle of vision will enlarge during such an ascent, the moment will expand. If we imagine a consciousness higher than our consciousness, possessing a broader angle of view, then this consciousness will be able to grasp, as something simultaneous, i. е., as a moment, all that is happening for us during a certain length of time — minutes, hours, a day, a month. Within the limits of its moment such a consciousness will not be in a position to discriminate between before, now, after, all this will be for it now. Now will expand. But in order for this to happen it would be necessary for us to liberate ourselves from matter, because matter is nothing more than the conditions of space and time in which we dwell. Thence arises the question: can consciousness leave the conditions of material existence without itself undergoing fundamental changes or without disappearing altogether, as men of positivistic views would affirm. This is a debatable question, and later I shall give examples and proofs, speaking on behalf of the idea that our consciousness can leave the conditions of materiality. For the present I wish to establish purely theoretically what must proceed during this leaving. There would ensue the expansion of the moment, i. е., all that we are apprehending in time would become something like a single moment, in which the past, the present, and the future would be 40 TERTIUM ORGANUM seen at once. This shows the relativity of motion, as depending for us upon the limitation of the moment, which includes only a very small part of the moments of life perceived by us. We have a perfect right to say, not that "time" is deduced from "motion," but that motion is sensed because of the time-sense. We have that sense, therefore we sense motion. The time-sense is the sensation of changing moments. If we did not have this time-sense we could not feel motion. The "time-sense" is itself, in substance, the limit or the surface of our "space-sense." Where the "space-sense" ends, there the "time-sense" begins. It has been made clear that "time" is identical in its properties with "space," i. е., it has all the signs of space extension. However, we do not feel it as spatial extension, but we feel it as time, that is, as something specific, inexpressible, in other words, uninter- ruptedly bound up with "motion." This inability to sense time spatially has its origin in the fact that the time-sense is a misty space-sense; by means of our time-sense we feel obscurely the new characteristics of space, which emerge from the sphere of three dimensions. But what is the time-sense and why does there arise the illusion of motion? To answer this question at all satisfactorily is possible only by studying our consciousness, our I. "I" is a complicated quantity, and within itself goes on a con- tinuous motion. About the nature of this motion we shall speak later, but this very motion inside of our I creates the illusion of motion around us, motion in the material world. The noted mathematician Riemann understood that when higher dimensions of space are in question time, by some means, translates itself into space, and he regarded the material atom as the entrance of the fourth dimension into three-dimensional space. In one of his books Hinton writes very interestingly about "surface tensions." The relationship of a surface to a solid or of a solid 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. TERTIUM ORGANUM 41 If our space is in the same co-relation with higher space as is the surface to our space, then it may be that our space is really the surface, that is the place of contact, of two higher-dimensional spaces. 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 behavior of the surfaces of liquids is governed. And it may well be that the laws of our universe are the surface ten- sions of a higher universe. If the surface be regarded as a medium lying between bodies, then indeed it will have no weight, but be a powerful means of transmitting vibrations. Moreover, it would be unlike any other substance, and it would be impossible to get rid of it. However perfect a vacuum be made, there would be in this vacuum just as much of this unknown medium (i. е., of that surface) as there was before. Matter would pass freely through this medium. . . vibrations of this medium would tear asunder portions of matter. And involun- tarily the conclusion would be drawn that this medium was unlike any ordinary matter. . . These would be very different properties to reconcile in one and the same substance. Now is there anything in our experience which corresponds to this medium? . . . 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 per- fect, which penetrates all bodies, is weightless, and yet can never be laid hold of. The "substance" which possesses all these qualities is called the "ether." . . The properties of the ether are a perpetual object of investigation in science. . . But taking into consideration the ideas expressed before it would be interesting to look at the world supposing that we are not in it but on the ether; where the "ether" is the surface of contact of two bodies of higher dimensions.* Hinton here expresses an unusually interesting thought, and brings the idea of the "ether" nearer to the idea of time. The materialistic, or even the energetic understanding of contemporary physics of the ether is perfectly fruitless — a dead-end siding. For Hinton the ether is not a substance but only a "surface," the "boundary" of something. But of what? Again not that of a substance, but the boundary, the surface, the limit of one form of receptivity and the beginning of another In one sentence the walls and fences of the materialistic dead- end siding are broken down and before our thought open wide horizons of regions unexplored. * Hinton, "A New Era of Thought," pp. 52, 56, 57. ч pfiT Ъооъ r o !*1 So ? A K CHAPTER V Four-dimensional space. "Temporal body"— Linga Sharira. The form of a human body from birth to death. Incommensurabil- ity of three-dimensional and four-dimensional bodies. Newton s fluents. The unreality of constant quantities in our world. The right and the left hands in three-dimensional and in four-dimen- sional space. Difference between three-dimensional and four- dimensional space. Not two different spaces but different meth- ods of receptitivity of one and the same world. OUR-DIMENSIONAL space, if we try to imagine it to ourselves, will be the infinite repetition of our space, of our infinite three-dimensional sphere, as a line is the infinite repetition of a point. Many things that have been said before will become much clearer to us when we dwell on the fact that the fourth dimension must be sought for in time. It will become clear what is meant by the fact that it is possible to regard a four-dimensional body as the tracing of the movement in space of a three-dimensional body in a direction not confined within that space. Now the direction not confined in three- dimensional space in which any three-dimensional body moves— this is the direction of time. Any three-dimensional body, exist- ing, is at the same time moving in time and leaves as a tracing of its movement the temporal, or four-dimensional body. We never see nor feel this body, because of the limitations of our re- ceptive apparatus, but we see the section of it only, which section we call the three-dimensional body. Therefore we are in error in thinking that the three-dimensional body is in itself something real. It is the projection of the four-dimensional body — its pic- ture — the image of it on our plane. The four-dimensional body is the infinite number of three- dimensional ones. That is, the four-dimensional body is the in- finite number of moments of existence of the three-dimensional one— its states and positions. The three-dimensional body which we see appears as a single figure — one of a series of pictures on a cinematographic film as it were. 43 44 TERTIUM ORGANUM Four-dimensional space — time — is really the distance between forms, states, and positions, of one and the same body (and different bodies, i. е., those seeming different to us). It separates those states, forms, and positions each from the other, and it binds them also into some to us incomprehensible whole. This incomprehensible whole can be formed in time out of one physical body — and out of different bodies. It is easier for us to imagine the temporal whole as related to one physical body. If we consider the physical body of a man, we will find in it besides its "matter" something, it is true, changing, but undoubt- edly one and the same from birth until death. This something is the Linga-Sharira of Hindu philosophy, i. е., the form on which our physical body is moulded. (H. P. Blavatsky : "The Secret Doctrine.") Eastern philosophy regards the phy- sical body as something impermanent, which is in a condition of perpetual interchange with its surroundings. The particles come and go. After one second the body is already not absolutely the same as it was one second before. To-day it is in a con- siderable degree not that which it was yesterday. After seven years it is a quite different body. But despite all this, something always persists from birth to death, changing its aspect a little, but remaining the same. This is the Linga-Sharira. The Linga-Sharira is the form, the image, it changes, but re- mains the same. That image of a man which we are able to repre- sent to ourselves is not the Linga-Sharira. But if we try to repre- sent to ourselves mentally the image of a man from birth to death, with all the particularities and traits of childhood, man- hood and senility, as though extended in time, then it will be the Linga-Sharira. Form pertains to all things. We say that everything consists of matter and form. Under the category of "matter," as already stated, the cause of a lengthy series of mixed sensations is predi- cated, but matter without form is not comprehensible to us; we cannot even think of matter without form. But we can think and imagine form without matter. The thing, i. е., the union of form and matter, is never constant; it always changes in the course of time. This idea afforded Newton the possibility of building his theory of fluents and fluxions. TERTIUM ORGANUM 45 Newton came to the conclusion that constant quantities do not exist in Nature. Variables do exist — flowing, fluents only. The velocities with which different fluents change were called by Newton fluxions. From the standpoint of this theory all things known to us— men, plants, animals, planets— are fluents, and they differ by the magnitude of their fluxions. But the thing, changing continu- ously in time, sometimes very much, and quickly, as in the case of a living body for example, still remains one and the same. The body of a man in youth, the body of a man in senil- ity — these are one and the same, though we know that in the old body there is not one atom left that was in the young one. The matter changes, but something remains one under all changes, this something is the Linga-Sharira. Newton's theory is valid for the three-dimensional world existing in time. In this world there is nothing constant. All is variable because every consecu- tive moment the thing is already not that which it was before. We never see the Linga-Sharira, we see always its parts, and they appear to us variable. But if we observe more attentively we shall see that it is an illusion. Things of three dimensions are un- real and variable. They cannot be real because they do not exist in reality, just as the imaginary sections of a solid do not exist. Four-dimensional bodies alone are real. In one of the lectures contained in the book, "A Pluralistic Universe," Prof. James calls attention to Prof. Bergson's remark that science studies always the t of the universe only, i. е., not the universe in its entirety, but the moment, the "temporal section" of the universe. The properties of four-dimensional space will become clearer to us if we compare in detail three-dimensional space with the surface, and discover the difference existing between them. Hinton, in his book, "A New Era of Thought," examines these differences very attentively. He represents to himself, on a plane, two equal rectangular triangles, cut out of paper, the right angles of which are placed in opposite directions. These triangles will be equal, but for some reason quite different. The right angle of one is directed to the right, that of the other to the left. If anyone wants to make them quite similar, it is possible to do so only with 46 TERTIUM ORGANUM the help of three-dimensional space. That is, it is necessary to take one triangle, turn it over, and put it back on the plane. Then they will be two equal, and exactly similar triangles. But in order to effect this, it was necessary to take one triangle from the plane into three-dimensional space, and turn it over in that space. If the triangle is left on the plane, then it will never be possible to make it identical with the other, keeping the same relation of angles of the one to those of the other. If the triangle is merely rotated in the plane this similarity will never be established. In our world there are figures quite analogous to these two triangles. 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 represent a complex case of a symmetrical similarity. 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 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 such an operation were possible, the right hand would be turned into an exact model of the left hand.* But such an operation would be possible in the higher dimen- sional space only, just as the overturning of the triangle is possible only in a space relatively higher than the plane. Even granting the existence of four-dimensional space it is possible that the turning of the hand inside out and the pulling of it through itself is a practical impossibility on account of causes independent of geometrical conditions. But this does not diminish its value as an example. Things like the turning of the hand inside out are possible theoretically in four-dimensional space because in this space different, and even distant points of our space and time touch, or have the possibility of contact. All points of a sheet of paper lying on a table are separated one from another, but by taking the sheet from the table it is possible to fold it in such a way as to bring together any given points. If on one corner is written St. Petersburg, and on another Madras, nothing prevents the putting together of these corners. And if on the third corner is written the year 1812, and on the fourth 1912, these corners can touch each other too. If on one corner the year is written in red * C. H. Hinton, "A New Era of Thought," p. 44. TERTIUM ORGANUM 47 ink, and the ink has not yet dried, then the figures may imprint themselves on the other corner. And if afterwards the sheet is straightened out and laid on the table, it will be perfectly incom- prehensible, to a man who has not followed the operation, how the figure from one corner could transfer itself to another corner. For such a man the possibility of the contact of remote points of the sheet will be incomprehensible, and it will remain incom- prehensible so long as he thinks of the sheet in two-dimensional space only. The moment he imagines the sheet in three-dimen- sional space this possibility will become real and obvious to him. In considering the relation of the fourth dimension to the three known to us, we must conclude that our geometry is obviously insufficient for the investigation of higher space. As before stated, a four-dimensional body is as incommensur- able with a three-dimensional one as a year is incommensurable with St. Petersburg. It is quite clear why this is so. The four-dimensional body consists of an infinitely great number of three-dimensional ones; accordingly, there cannot be a common measure for them. The three-dimensional body, in comparison with the four-dimen- sional one is equivalent to the point in comparison with the line. And just as the point is incommensurable with the line, so is the line incommensurable with the surface; as the surface is incom- mensurable with the solid body, so is the three-dimensional body incommensurable with the four-dimensional one. It is clear also why the geometry of three dimensions is insuffi- cient for the definition of the position of the region of the fourth dimension in relation to three-dimensional space. Just as in the geometry of one dimension, that is, upon the line, it is impossible to define the position of the surface, the side of which constitutes the given line; just as in the geometry of two dimensions, i. е., upon the surface, it is impossible to define the position of the solid, the side of which constitutes the given sur- face, so in the geometry of three dimensions, in three-dimensional space, it is impossible to define a four-dimensional space. Briefly speaking, as planimetry is insufficient for the investigation of the problems of stereometry, so is stereometry insufficient for four- dimensional space. As a conclusion from all of the above we may repeat that every point of our space is the section of a line in higher space, or as 48 TERTIUM ORGANUM В. Riemann expressed it: the material atom is the entrance of the fourth dimension into three-dimensional space. For a nearer approach to the problem of higher dimensions and of higher space it is necessary first of all to understand the consti- tion and properties of the higher dimensional region in comparison with the region of three dimensions. Then only will appear the possibility of a more exact investigation of this region, and a classi- fication of the laws governing it. What is it that it is necessary to understand? It seems to me that first of all it is necessary to understand that we are considering not two regions spatially different, and not two regions of which one (again spatially, "geometrically") constitutes a part of the other, but two methods of receptivity of one and the same unique world of a space which is unique. Furthermore it is necessary to understand that all objects known to us exist not only in those categories in which they are perceived by us, but in an infinite number of others in which we do not and cannot sense them. And we must learn first to think things in other categories, and then so far as we are able, to im- agine them therein. Only after doing this can we possibly develop the faculty to apprehend them in higher space — and to sense "higher" space itself. Or perhaps the first necessity is the direct perception of every- thing in the outside world which does not fit into the frame of three dimensions, which exists independently of the categories of time and space — everything that for this reason we are accustomed to consider as non-existent. If variability is an indication of the three-dimensional world, then let us search for the constant and there- by approach to an understanding of the four-dimensional world. We have become accustomed to count as really existing only that which is measurable in terms of length, breadth and height, but as has been shown it is necessary to expand the limits of the really existing. Mensurability is too rough an indication of exist- ence, because mensurability itself is too conditioned a concep- tion. We may say that for any approach to the exact investiga- tion of the higher dimensional region the certainty obtained by the immediate sensation is probably indispensable, that much that is immeasurable exists just as really as, and even more really than, much that is measurable. CHAPTER VI Methods of investigation of the problem of higher dimensions. The analogy between imaginary worlds of different dimensions. The one-dimensional world on a line. "Space" and "time" of a one- dimensional being. The two-dimensional world on a plane. "Space" and "time," "ether," "matter" and "motion" of a two- dimensional being. Reality and illusion on a plane. The impossi- bility of seeing an "angle." An angle as motion. The incompre- hensibility to a two-dimensional being of the functions of things in our world. Phenomena and noumena of a two-dimensional being. How could a plane being comprehend the third dimension? SERIES of analogies and comparisons are used for the definition of that which can be, and that which cannot be, in the region of the higher dimension. Fechner, Hinton, and many others employ this method. They imagine "worlds" of one, and of two dimensions, and out of the relations of lower-dimensional worlds to higher ones they deduce possible relations of our world to one of four dimen- sions; just as out of the relations of points to lines, of lines to surfaces, and of surfaces to solids we deduce the relations of our solids to four-dimensional ones. Let us try to investigate everything that this method of analogy can yield. ^>G Let us imagine a world of one dimension. It will be a line. Upon this line let us imagine living beings. Upon this line, which represents the universe for them, they will be able to move forward and backward only, and these beings will be as the points, or segments of a line. Nothing will exist for them outside their line — and they will not be aware of the line upon which they are living and moving. For there will exist only two points, ahead and behind, or may be just one point ahead. Noticing the change in states of these points, the one-dimensional being will call these changes phenomena. If we suppose the line upon which the one-dimensional being lives to be passing through the different objects of our world, then of all these objects the 49 50 TERTIUM ORGANUM one-dimensional being will perceive one point only; if different bodies intersect his line, the one-dimensional being will sense them only as the appearance, the more or less prolonged existence, and the disappearance of a point. This appearance, existence, and disappearance of a point will constitute a phenomenon. Phe- nomena, according to the character and properties of passing ob- jects and the velocity and properties of their motions, for the one- dimensional being will be constant or variable, long or short- timed, periodical or unperiodical. But the one-dimensional being will be absolutely unable to understand or explain the con- stancy or variability, the duration or brevity, the periodicity or unperiodicity of the phenomena of his world, and will regard them simply as properties pertaining to them. The solids inter- secting his line may be different, but for the one-dimensional being all phenomena will be absolutely identical — just the appearance or the disappearance of a point — and phenomena will differ only in duration and greater or less periodicity. Such strange monotony and similarity of the diverse and hetero- geneous phenomena of our world will be the characteristic pecu- liarity of the one-dimensional world. Moreover, if we assume that the one-dimensional being pos- sesses memory, it is clear that recalling all the points seen by him as phenomena, he will refer them to time. The point which was: this is the phenomenon already non-existent, and the point which may appear tomorrow: this is the phenomenon which does not exist yet. All of our space except one line will be in the category of time, i. е., something wherefrom phenomena come and into which they disappear. And the one-dimensional being will de- clare that the idea of time arises for him out of the observation of motion, that is to say, out of the appearance and disappearance of points. These will be considered as temporal phenomena, be- ginning at that moment when they become visible, and ending — ceasing to exist — at that moment when they become invisible. The one-dimensional being will not be in a position to imagine that the phenomenon goes on existing somewhere, though invisi- bly to him; or he will imagine it as existing somewhere on his line, far ahead of him. We can imagine this one-dimensional being more vividly. Let us take an atom, hovering in space, or simply a particle of dust, carried along by the air, and let us imagine that this atom or par- TERTIUM ORGANUM 61 tide of dust possesses a consciousness, i. е., separates himself from the outside world, and is conscious only of that which lies in the line of his motion, and with which he himself comes in contact. He will then be a one-dimensional being in the full sense of the word. He can fly and move in all directions, but it will always seem to him that he is moving upon a single line; outside of this line will be for him only great Nothingness — the whole universe will appear to him as one line. He will feel none of the turns and angles of his line, for to feel an angle it is necessary to be conscious of that which lies to right or left, above or below. In all other respects such a being will be absolutely identical with the before-described imaginary being living upon the imaginary line. Everything that he comes in contact with, that is, everything that he is conscious of, will seem to him to be emerging from time, i. е., from nothing, and vanishing into time, i. е., into nothing. This nothing will be all our world. All our world except one line will be called time and will be counted as actually non-existent. Let us next consider the two-dimensional world, and the being living on a plane. The universe of this being will be one great plane. Let us imagine beings on this plane having the shape of points, lines, and flat geometrical figures. The objects and "solids" of that world will have the shape of flat geometrical figures too. In what manner will a being living on such a plane universe cognize his world? First of all we can affirm that he will not feel the plane upon which he lives. He will not do so because he will feel the objects, i. е., figures, which are on this plane. He will feel the lines which limit them, and for this reason he will not feel his plane, for in that case he would not be in a position to discern the lines. The lines will differ from the plane in that they produce sensations; there- fore they exist. The plane does not produce sensations; therefore it does not exist. Moving on the plane, the two-dimensional being, feeling no sensations, will declare that nothing now exists. After having encountered some figure, having sensed its lines, he will say that something appeared. But gradually, by a process of reasoning, the two-dimensional being will come to the conclusion that the figures he encounters exist on something, or in something. 52 TERTIUM ORGANUM Thereupon he may name such a plane (he will not know, indeed, that it is a plane) the "ether." Accordingly he will declare that the "ether" fills all space, but differs in its qualities from "mat- ter." By "matter" he will mean lines. Having come to this con- clusion the two-dimensional being will regard all processes as happening in his "ether," i. е., in his space. He will not be in a position to imagine anything outside of this ether, that is, out of his plane. If anything, proceeding out of his plane, comes in con- tact with his consciousness, then he will either deny it, or regard it as something subjective, the creation of his own imagination, or else he will believe that it is proceeding right on the plane, in the ether, as are all other phenomena. Sensing lines only, the plane being will not sense them as we do. First of all, he will see no angle. It is extremely easy for us to verify this by experiment. If we will hold before our eyes two matches, inclined one to the other in a horizontal plane, then we shall see one line. To see the angle we shall have to look from above. The two-dimensional being cannot look from above and therefore cannot see the angle. But measuring the distance between the lines of different "solids" of his world, the two-dimensional being will come continually in contact with the angle, and he will regard it as a strange property of the line, which is sometimes manifest and sometimes is not. That is, he will refer the angle to time, he will regard it as a temporary, evanescent phenomenon, a change in the state of a "solid," or as motion. It is difficult for us to un- derstand this. It is difficult to imagine how the angle can be re- garded as motion. But it must be absolutely so, and cannot be otherwise. If we try to represent to ourselves how the plane being studies the square, then certainly we shall find that for the plane being the square will be a moving body. Let us imagine that the plane being is opposite one of the angles of the square. He does not see the angle — before him is a line, but a line possess- ing very curious properties. Approaching this line, the two- dimensional being observes that a strange thing is happening to the line. One point remains in the same position, and other points are withdrawing back from both sides. We repeat, that the two-dimensional being has no idea of an angle. Apparently the line remains the same as it was, yet something is happening to it, without a doubt. The plane being will say that the line is moving, but so rapidly as to be imperceptible to sight. If the plane being TERTIUM ORGANUM 53 goes away from the angle and follows along a side of the square, then the side will become immobile. When he comes to the angle, he will notice the motion again. After going around the square several times, he will establish the fact of regular, periodical mo- tions of the line. Quite probably in the mind of the plane being the square will assume the form of a body possessing the property of periodical motions, invisible to the eye, but producing definite physical effects (molecular motion) — or it will remain there as a perception of periodical moments of rest and motion in one com- plex line, and still more probably it will seem to be a rotating body. Quite possibly the plane being will regard the angle as his own subjective perception, and will doubt whether any objective reality corresponds to this subjective perception. Nevertheless he will reflect that if there is action, yielding to measurement, so must there be the cause of it, consisting in the change of the state of the line, i. е., in motion. The lines visible to the plane being he may call matter, and the angles — motion. That is, he may call the broken line with an angle, moving matter. And truly to him such a line by reason of its properties will be quite analogous to matter in motion. If a cube were to rest upon the plane upon which the plane being lives, then this cube will not exist for the two-dimensional being, but only the square face of the cube in contact with the plane will exist for him — as a line, with periodical motions. Cor- respondingly, all other solids lying outside of his plane, in contact with it, or passing through it, will not exist for the plane being. The planes of contact or cross-sections of these bodies will alone be sensed. But if these planes or sections move or change, then the two-dimensional being will think, indeed, that the cause of the change or motion is in the bodies themselves, i.e., right there on his plane. As has been said, the two-dimensional being will regard the straight lines only as immobile matter; irregular lines and curves will seem to him as moving. So far as really moving lines are concerned, that is, lines limiting the cross sections or planes of contact passing through, or moving along the plane, these will be for the two-dimensional being something inconceivable and in- commensurable. It will be as though there were in them the presence of something independent, depending upon itself only, 54 TERTIUM ORGANUM animated. This effect will proceed from two causes: He can measure the immobile angles and curves, the properties of which the two-dimensional being calls motion, for the reason that they are immobile; moving figures, on the contrary, he cannot measure, because the changes in them will be out of his control. These changes will depend upon the properties of the whole body and its motion, and of that whole body the two-dimensional being will know only one side or section. Not perceiving the existence of this body, and contemplating the motion pertaining to the sides and sections he probably will regard them as living beings. He will affirm that there is something in them which differentiates them from other bodies: vital energy, or even soul. That something will be regarded as inconceivable, and really will be inconceivable to the two-dimensional being, because to him it is the result of an incomprehensible motion of inconceivable solids. If we imagine an immobile circle upon the plane, then for the two-dimensional being it will appear as a moving line with some very strange and to him inconceivable motions. The two-dimensional being will never see that motion. Perhaps he will call such motion molecular motion, i.e., the movement of minutest invisible particles of "matter." Moreover, a circle rotating around an axis passing through its center for the two-dimensional being will differ in some incon- ceivable way from the immobile circle. Both will appear to be moving, but moving differently. For the two-dimensional being a circle or a square, rotating around its center, on account of its double motion will be an inex- plicable and incommensurable phenomenon, like a phenomenon of life for a modern physicist. Therefore, for a two-dimensional being, a straight line will be immobile matter; a broken or a curved line — matter in motion; and a moving line — living matter. The center of a circle or a square will be inaccessible to the plane being, just as the center of a sphere or of a cube made of solid matter is inaccessible to us — and for the two-dimensional being even the idea of a center will be incomprehensible, since he possesses no idea of a center. Having no idea of phenomena proceeding outside of the plane — that is, out of his "space" — the plane being will think of all phe- nomena as proceeding on his plane as has been stated. And all TERTIUM ORGANUM 55 phenomena which he regards as proceeding on his plane, he will consider as being in causal interdependence one with another: that is, he will think that one phenomenon is the effect of another which has happened right there, and the cause of a third which will happen right on the same plane. If a multi-colored cube passes through the plane, the plane being will perceive the entire cube and its motion as a change in the color of lines lying in the plane. Thus, if a blue line replaces a red one, then the plane being will regard the red line as a past event. He will not be in a position to realize the idea that the red line is still existing somewhere. He will say that the line is single, but that it becomes blue as a consequence of certain causes of a physical character. If the cube moves backward so that the red line appears again after the blue one, then for the two-dimensional being this will constitute a new phenomenon. He will say that the line became red again. For the being living on a plane, everything above and below (if the plane be horizontal), and on the right or left (if the plane be vertical) will be existing in time, in the past and in the future : that which in reality is located outside of the plane will be re- garded as non-existent, either as that which is already past, i.e., as something which has disappeared, ceased to be, will never return, or as in the future, i. е., as not existent, not manifested, as a thing in potentiality. Let us imagine that a wheel with the spokes painted different colors is rotating through the plane upon which the plane-being lives. To such a being all the motion of the wheel will appear as a variation of the color of the line of intersection of the wheel and the plane. The plane being will call this variation of the color of the line a phenomenon, and observing these phenomena he will notice in them a certain succession. He will know that the black line is followed by the white one, the white by the blue, the blue by the red, and so on. If simultaneously with the appearance of the white line some other phenomenon occurs— say the ringing of a bell— the two-dimensional being will say that the white line is the cause of that ringing. The change of the color of the lines, in the opinion of the two-dimensional being, will depend on causes lying right in his plane. Any presupposition of the possibility of the existence of causes lying outside of the plane he will characterize 56 TERTIUM ORGANUM as fantastic and entirely unscientific. It will seem so to him be- cause he will never be in a position to represent the wheel to himself, i. е., the parts of the wheel on both sides of the plane. After a rough study of the color of the lines, and knowing the order of their sequence, the plane being, perceiving one of them, say the blue one, will think that the black and the white ones have already passed, i. е., disappeared, ceased to exist, gone into the past; and that those lines which have not yet appeared — the yel- low, the green, and so on, and the new white and black ones still to come — do not yet exist, but lie in the future. Therefore, though not conceiving the form of his universe, and regarding it as infinite in all directions, the plane being will never- theless involuntarily think of the past as situated somewhere at one side of all, and of the future as somewhere at the other side of this totality. In such manner will the plane being conceive of the idea of time. We see that this idea arises because the two-dimen- sional being senses only two out of three dimensions of space; the third dimension he senses only after its effects become manifest upon the plane, and therefore he regards it as something different from the first two dimensions of space, calling it time. Now let us imagine that through the plane upon which the two- dimensional being lives, two wheels with multi-colored spokes are rotating and are rotating in opposite directions. The spokes of one wheel come from above and go below; the spokes of the other come from below and go above. The plane being will never notice it. He will never notice that where for one line (which he sees) there lies the past — for another line there lies the future. This thought will never even come into his head, because he will con- ceive of the past and the future very confusedly, regarding them as concepts, not as actual facts. But at the same time he will be firmly convinced that the past goes in one direction, and the future in another. Therefore it will seem to him a wild absurdity that on one side something past and something future can lie together, and on another side — and also beside these two — something future and something past. To the plane being the idea that some phenomena come whence others go, and vice versa, will seem equally absurd. He will tenaciously think that the future is that wherefrom every- TERTIUM ORGANUM 57 thing comes, and the past is that whereto everything goes and wherefrom nothing returns. He will be totally unable to under- stand that events may arise from the past just as they do from the future. Thus we see that the plane being will regard the changes of color of the lines lying on the plane very naively. The appear- ance of different spokes he will regard as the change of color of one and the same line, and the repeated appearance of the same colored spoke he will regard every time as a new appearance of a given color. But nevertheless, having noticed periodicity in the change of the color of the lines upon the surface, having remembered the order of their appearance, and having learned to define the "time" of the appearance of certain spokes in relation to some other more constant phenomenon, the plane being will be in a posi- tion to foretell the change of the line from one color to another. Thereupon he will say that he has studied this phenomenon, that he can apply to it "the mathematical method" — can "calculate it." If we ourselves enter the world of plane-beings, then its in- habitants will sense the lines limiting the sections of our bodies. These sections will be for them living beings; they will not know from whence they appear, why they alter, or whither they disap- pear in such a miraculous manner. So also, the sections of all our in- animate but moving objects will seem independent living beings. If the consciousness of a plane being should suspect our exist- ence, and should come into some sort of communion with our con- sciousness, then to him we would appear as higher, omniscient, possibly omnipotent, but above all incomprehensible beings of a quite inconceivable category. We could see his world just as it is, and not as it seems to him. We could see the past and the future; could foretell, direct and even create events. We could know the very substance of things — could know what "matter" (the straight line) is, what "motion" (the broken line, the curve, the angle) is. We could see an angle, and we could see a center. All this would give us an enormous advantage over the two-dimensional being. 58 TERTIUM ORGANUM In all of the phenomena of the world of the two-dimensional being we could see considerably more than he sees — or could see quite other things than he. And we could tell him very much that was new, amazing, and unexpected about the phenomena of his world — provided, indeed, that he could hear us and understand us. First of all we could tell him that what he regards as phe- nomena — angles and curves, for instance — are properties of higher figures; that other "phenomena" of his world are not phenomena, but only "parts" or "sections" of phenomena; that what he calls "solids" are only sections of solids, — and many more things be- sides. We should be able to tell him that on both sides of his plane (i. е., of his space or ether) lies infinite space (which the plane being calls time) ; and that in this space lie the causes of all his phenomena, and the phenomena themselves, the past as well as the future ones; moreover, we might add that "phenomena" them- selves are not something happening and then ceasing to be, but combinations of properties of higher solids. But we should experience considerable difficulty in explaining anything to the plane being; and it would be very difficult for him to understand us. First of all it would be difficult because he would not have the concepts corresponding to our concepts. He would lack necessary "words." For instance, "section" — this would be for him a quite new and inconceivable word; then " angle"— again an inconceivable word; "center" — still more inconceivable; the third perpendicular — something incomprehensible, lying outside of his geometry. The fallacy of his conception of time would be the most difficult thing for the plane being to understand. He could never under- stand that that which has passed and that which is to be are ex- isting simultaneously on the lines perpendicular to his plane. And he could never conceive the idea that the past is identical with the future, because phenomena come from both sides and go in both directions. But the most diflScult thing for the plane being would be to conceive the idea that "time" includes in itself two ideas: the idea of space, and the idea of motion upon this space. We have shown that what the two-dimensional being living on the plane calls motion has for us quite a different aspect. TERTIUM ORGANUM 59 In his book "The Fourth Dimension," under the heading "The First Chapter in the History of Four-space," Hinton writes: Parmenides, and the Asiatic thinkers with whom he is in close affinity, propound a theory of existence which is in close accord with a conception of a possible relation between a higher and lower dimensional space. . . It is one which in all ages has had a strong attraction for pure intellect, and is the natural mode of thought for those who refrain from projecting their own volition into nature under the guise of causality. According to Parmenides of the school of Elea the all is one, unmov- ing and unchanging. The permanent amid the transient — that foothold for thought, that solid ground for feeling, on the discovery of which depends all our life — is no phantom; it is the image amidst deception of true being, the eternal, the unmoved, the one. Thus says Parmenides. But how is it possible to explain the shifting scene, these mutations of things? "Illusion," answers Parmenides. Distinguishing between truth and error, he tells of the true doctrine of the one — the false opinion of a changing world. He is no less memorable for the manner of his advo- cacy than for the cause he advocates. Can the mind conceive a more delightful intellectual picture than that of Parmenides pointing to the one, the true, the unchanging, and yet on the other hand ready to discuss all manner of false opinion ! . . In support of the true opinion he proceeded by the negative way of showing the self-contradictions in the ideas of change and motion. . . To express his doctrine in the ponderous modern way we must make the statement that motion is phenomenal, not real. Let us represent his doctrine. Imagine a sheet of still water into which a slanting stick is being lowered with a motion vertically downwards. Let 1, 2, 3, (Fig. 1), be three consecutive positions of the stick. А, В, С will be three con- nective positions of the meeting of the stick with the surface of the water. As the stick passes down, the meeting will move from A on to В and C. Suppose now all the water to be removed except a film. At the meeting of the film and the stick there will be an interruption of the film. If we suppose the film to have a property, like that of a soap bub- ble, of closing up round any penetrating object, then as the stick goes vertically downwards the interruption in the film will move on. If we pass a spiral through the film the intersection will give a point moving in a circle (shown by the dotted lines in Fig. 2). For the plane being such a point, moving in a circle in its plane, would probably constitute a cosmical phenomenon, something like the motion of a planet in its orbit. Suppose now the spiral to be still and the film to move vertically upward, the whole spiral will be represented in the film in the con- secutive positions of the point of intersection.* * С. H. Hinton, "The Fourth Dimension," pp. 23, 24 and 25. t Ibid. 60 TERTIUM ORGANUM If instead of one spiral we take a complicated construction consisting of spirals, inclined, and straight lines, broken and curved lines, and if the film move vertically upwards we shall have an entire universe of moving points the movements of which will appear to the plane being as original. Fig. 1 Fig. 2 The plane being will explain these movements as depending one upon another, and indeed he will never happen to think that these move- ments are fictitious and are dependent upon the spirals and other lines lying outside his space.f Returning to the plane being and his perception of the world, and analyzing his relations to the three-dimensional world, we see that for the two-dimensional or plane being it will be very diffi- cult to understand all the complexity of the phenomena of our world, as it appears to us. He (the plane being) is accustomed to perceive the world as being too simple. Taking into consideration the sections of figures instead of the figures themselves, the plane being will compare them in relation to their length and their greater or lesser curvature, i. е., their for him more or less rapid motion. The differences between the objects of our world, as they exist for us he would not understand. The functions of the objects of our world would be completely mysterious to his mind — incom- prehensible, "supernatural." Let us imagine that a coin, and a candle the diameter of which is equal to that of the coin, are on the plane upon which the two- dimensional being lives. To the plane being they will seem two equal circles, i. е., two moving, and absolutely identical lines; he will TERTIUM ORGANUM 61 never discover any difference between them. The functions of the coin and of the candle in our world — these are for him absolutely a terra incognita. If we try to imagine what an enormous evolu- tion the plane being must pass through in order to understand the function of the coin and of the candle and the difference between these functions, we will understand the nature of the division be- tween the plane world and the world of three dimensions, and the complete impossibility of even imagining, on the plane, anything at all like the three-dimensional world, with its manifoldness of function. The properties of the phenomena of the plane world will be extremely monotonous; they will differ by the order of their ap- pearance, their duration, and their periodicity. Solids, and the things of this world will be flat and uniform, like shadows, i. е., like the shadows of quite different solids, which seem to us uniform. Even if the plane being could come in contact with our conscious- ness, he would never be in a position to understand all the mani- foldness and richness of the phenomena of our world and the variety of function of the things of that world. Plane beings would not be in a position to master our most ordinary concepts. It would be extremely difficult for them to understand that phenomena, identical for them, are in reality different; and on the other hand, that phenomena quite separate for them are in reality parts of one great phenomenon, and even of one object or one being. This last will be one of the most difficult things for the plane being to understand. If we imagine our plane being to be inhabit- ing a horizontal plane, intersecting the top of a tree, and parallel to the surface of the earth, then for such a being each of the vari- ous sections of the branches will appear as a quite separate phe- nomenon or object. The idea of the tree and its branches will never occur to him. Generally speaking, the understanding of the most fundamental and simple things of our world will be infinitely long and difficult to the plane being. He would have to entirely reconstruct his concepts of space and time. This would be the first step. Unless it is taken, nothing is accomplished. Until the plane being will imagine all our universe as existing in time, i. е., until he refers to time everything lying on both sides of his plane, he will never 62 TERTIUM ORGANUM understand anything. In order to begin to understand "the third dimension" the inhabitant of the plane must conceive of his time concepts spatially, that is, translate his time into space. To achieve even the spark of a true understanding of our world he will have to reconstruct completely all his ideas — to r evaluate all values, to revise all concepts, to dissever the uniting concepts, to unite those which are dissevered; and, what is most important, to create an infinite number of new ones. If we put down the five fingers of one hand on the plane of the two-dimensional being they will be for him five separate phe- nomena. Let us try to imagine what an enormous mental evolution he would have to undergo in order to understand that these five separate phenomena on his plane are the finger-tips of the hand of a large, active and intelligent being — man. To make out, step by step, how the plane being would attain to an understanding of our world, lying in the region of the to him mysterious third dimension — i. е., partly in the past, partly in the future — would be interesting in the highest degree. First of all, in order to understand the world of three dimensions, he must cease to be two dimensional — he must become three dimensional himself or, in other words, he must feel an interest in the life of three-dimensional space. After having felt the interest of this life, he will by so doing transcend his plane, and will never be in a position thereafter to return to it. Entering more and more within the circle of ideas and concepts which were entirely incomprehen- sible to him before, he will have already become, not two-dimen- sional, but three-dimensional. But all along the plane being will have been essentially three-dimensional, that is, he will have had the third dimension, without his being conscious of it himself. To become three-dimensional he must be three-dimensional. Then as the end of ends he can address himself to the self-liberation from the illusion of the two-dimensionality of himself and the world, and to the apprehension of the three-dimensional world. CHAPTER VII. The imDOSsibility of the mathematical definition of dimensions. Why ТЬе К not mathematics sense dimensions? The entire cond^onahty of the representation of dimensions by powers The possibility of Representing all powers on a line. Kant and b°bachevAjN The difference between non-Euclidian geometry andmetageometry Where shall we find the explanation of the three-dimensionality of the world, if Kant's ideas are true? Are not the conditions of the three-dimensionality of the world confined to our receptive appa- ratus, to our psyche? 'OW that we have studied those "relations which our space itself bears within it" we shall return to the questions: But what in reality do the dimensions of space represent? —and why are there three of them? _ The fact that it is impossible to define three-di- mensionality mathematically must appear most strange. We are little conscious of this, and it seems to us a paradox, because we speak of the dimensions of space, but it remains a fact that mathematics does not sense the dimensions of space. _ The question arises, how can such a fine instrument of analysis as mathematics not feel dimensions, if they represent some real properties of space. . Speaking of mathematics, it is necessary to recognize nrst of all, as a fundamental premise, that correspondent to each mathematical expression is always the relation of some realities. If there is no such a thing, if it be not true— then there is no mathematics. This is its principal substance, its principal con- tents. To express the correlations of magnitudes, such is the problem of mathematics. But these correlations shall be between something. Instead of algebraical a, b and с it must be possible to substitute some reality. This is the ABC of all mathematics; a, b and c— these are credit bills, they can be good ones only if behind them there is a real something, and they can be counter- feited if behind them there is no reality whatever. 63 64 TERTIUM ORGANUM "Dimensions" play here a very strange role. If we designate them by the algebraic symbols a, b and c, they have the character of counterfeit credit bills. For this a, b and с it is impossible to substitute any real magnitudes which are capable of expressing the correlations of dimensions. Usually dimensions are represented by powers: the first, the second, the third; that is, if a line is called a, then a square, the sides of which are equal to this line, is called a 2 , and a cube, the face of which is equal to this square, is called a 3 . This among other things gave Hinton the foundation on which he constructed his theory of lesser acts, four-dimensional solids — a 4 . But this is "belles lettres" of the purest sort. First of all, because the representation of "dimensions" by powers is entirely condi- tional. It is possible to represent all powers on a line. For exam- ple, take the segment of a line equal to five millimeters; then a segment equal to twenty-five millimeters will be the square of it, i. е., a 2 ; and a segment of one hundred and twenty-five milli- meters will be the cube — a 3 . How shall we understand that mathematics does not feel dimen- sions — that it is impossible to express mathematically the differ- ence between dimensions?