Chapter 1

FOUR-DIMENSIONAL VISTAS

by

Claude Bragdon

New York

"_Perception has a destiny_." _Emerson_.

INTRODUCTION

There are two notable emancipations of the mind from the tyranny of mere appearances that have received scant attention save from mathematicians and theoretical physicists.

In 1823 Bolyai declared with regard to Euclid's so-called axiom of parallels, "I will draw two lines through a given point, both of which will be parallel to a given line." The drawing of these lines led to the concept of the curvature of space, and this to the idea of _higher_ space.

The recently developed Theory of Relativity has compelled the revision of the time concept as used in classical physics. One result of this has been to introduce the notion of _curved_ time.

These two ideas, of curved time and higher space, by their very nature are bound to profoundly modify human thought. They loosen the bonds within which advancing knowledge has increasingly labored, they lighten the dark abysses of consciousness, they reconcile the discoveries of Western workers with the inspirations of Eastern dreamers; but best of all, they open vistas, they offer "glimpses that may make us less forlorn."

CONTENTS

I. THE QUEST OF FREEDOM

The Undiscovered Country--Miracles--The Failure of Common Sense--The Function of Science--Mathematics--Intuition--Our Sense of Space--The Subjectivity of Space--The Need of an Enlarged Space-Concept.

II. THE DIMENSIONAL LADDER

Learning to Think in Terms of Spaces--From the Cosmos to the Corpuscle--And Beyond--Evolution as Space-Conquest--Dimensional Sequences--Man the Geometer--Higher, and Highest, Space.

III. PHYSICAL PHENOMENA

Looking for the Greater in the Less--Symmetry--Other Allied Phenomena--Isomerism--The Orbital Motion of Spheres: Cell-Subdivision-- The Electric Current--The Greater Universe--A Hint from Astronomy-- Gravitation--The Ether of Space.

IV. TRANSCENDENTAL PHYSICS

Zöllner--Apparitions--Possession--Clairvoyance in Space--Clairvoyance in Time--Pisgah Sights of Life's Pageant.

V. CURVED TIME

Time from the Standpoint of Experiment and of Conscious Experience-- Relativity--The Spoon-Man--The Orbital Movement of Time--Materiality the Mirror of Consciousness--Periodicity.

VI. SLEEP AND DREAMS

Sleep--Dreams--Time in Dreams--The Eastern Teaching in regard to Sleep and Dreams--Space in Dreams--The Phenomenon of Pause.

VII. THE NIGHT SIDE OF CONSCIOUSNESS

The Field of Psychic Research--Modifying the Past--Karma and Reincarnation--Colonel De Rochas' Experiments.

VIII. THE EASTERN TEACHING

Oriental Physics and Metaphysics--The Self-Recovered Memory of past Births--Release.

IX. THE MYSTICS

Hermes Trismegistus--The Page and the Press--The Ship and its Captain--Direct Vision--Plato's Shadow-Watchers--Swedenborg--Man, the Space-Eater--The Within and Without--Intuition and Reason--The Coil of Life.

X. GENIUS

Immanence--Timelessness--Beyond Good and Evil: Beauty--The Daemonic-- "A Dream and a Forgetting"--The Play of Brahm.

XI. THE GIFT OF FREEDOM

Concept and Conduct--Selflessness--Humility--Solidarity--Live Openly-- Non-Resistance to Evil--The Immanent Divine.

FOUR-DIMENSIONAL VISTAS

I THE QUEST OF FREEDOM

THE UNDISCOVERED COUNTRY

Expectancy of freedom is the dominant note of to-day. Amid the crash of armies and the clash of systems we await some liberating stroke which shall release us from the old dreary thralldoms. As Nietzsche says, "It would seem as though we had before us, as a reward for all our toils, a country still undiscovered, the horizons of which no one has yet seen, a beyond to every country and every refuge of the ideal that man has ever known, a world so overflowing with beauty, strangeness, doubt, terror and divinity, that both our curiosity and our lust of possession are frantic with eagerness."

Should a name be demanded for this home of freedom, there are those who would unhesitatingly call it _The Fourth Dimension of Space_. For such readers as may be ignorant of the amazing content of this seemingly meaningless phrase, any summary attempt at enlightenment will lead only to deeper mystification. To the question, where and what is the fourth dimension, the answer must be, it is here--in us, and all about us--in a direction toward which we can never point because at right angles to all the directions that we know. Our space cannot contain it, because it contains our space. No walls separate us from this demesne, not even the walls of our fleshly prison; yet we may not enter, even though we are already "there." It is the place of dreams, of living dead men: it is _At the Back of the North Wind_ and _Behind the Looking Glass_.

So might one go on, piling figure upon figure and paradox upon paradox, to little profit. The effective method is the ordered and deliberate one; therefore the author asks of his reader the endurance of his curiosity pending certain necessary preparations of the mind.

MIRACLES

Could one of our aviators have landed in ancient Athens, doubtless he would have been given a place in the Greek Pantheon, for the old idea of a demigod was a man with wings. Why, then, does a flying man so little amaze us? Because we know about engines, and the smell of gasoline has dulled our sense of the sublime. The living voice of a dead man leaves us unterrified if only we can be sure that it comes from a phonograph; but let that voice speak to us out of vacancy and we fall a prey to the same order of alarm that is felt by a savage at the report of a gun that he has never seen.

This illustration very well defines the nature of a miracle: it is a manifestation of power new to experience, and counter to the current thought of the time, Miracles are therefore always in order, they always happen. It is nothing that the sober facts of to-day are more marvellous than the fictions of Baron Munchausen, so long as we understand them: it is everything that phenomena are multiplying, that we are unable to understand. This increasing pressure upon consciousness _from a new direction_ has created a need to found belief on something firmer than a bottomless gullibility of mind. This book is aimed to meet that need by giving the mind the freedom of new spaces; but before it can even begin to do so, the reader must be brought to see the fallacy of attempting to measure the limits of the possible by that faculty known as common sense. And by common sense is meant, not the appeal to abstract reason, but to concrete experience.

THE FAILURE OF COMMON SENSE

Common sense had scarce had its laugh at Bell, and its shout of "I told you so!" at poor Langley, when lo! the telephone became the world's nervous system, and aeroplanes began to multiply like summer flies. To common sense the alchemist's dream of transmuting lead into gold seems preposterous, yet in a hundred laboratories radium is breaking down into helium, and the new chemistry bids fair to turn the time-honored jeer at the alchemists completely upside down. A wife whose mind was oriented in the new direction effectually silenced her husband's ridicule of what he called her credulity by reminding him that when wireless telegraphy was first suggested he had exclaimed, "Ah, that, you know, is one of the things that is not possible!" He was betrayed by his common sense.

The lessons such things teach us are summed up in the reply of Arago, the great savant, to the wife of Daguerre. She asked him if he thought her husband was losing his mind because he was trying to make permanent the image in a mirror. Arago is said to have answered, "He who, outside of pure mathematics, says a thing is impossible, speaks without reason."

Common sense neither leads nor lags, but is ever limited to the passing moment: the common knowledge of to-day was the mystery and enchantment of the day before yesterday, and will be the mere commonplace of the day after to-morrow. If common sense can so little anticipate the ordinary and orderly advancement of human knowledge, it is still less able to take that leap into the dark which is demanded of it now. The course of wisdom is therefore to place reliance upon reason and intuition, leaving to common sense the task of guiding the routine affairs of life, and guiding these alone.

THE FUNCTION OF SCIENCE

In enlisting the aid of reason in our quest for freedom, we shall be following in the footsteps of mathematicians and theoretical physicists. In their arduous and unflinching search after truth they have attained to a conception of the background of phenomena of far greater breadth and grandeur than that of the average religionist of to-day. As a mathematician once remarked to a neo-theosophist, "Your idea of the ether is a more material one than the materialist's own." Science has, however, imposed upon itself its own limitations, and in this connection these should be clearly understood.

Science is that knowledge which can be gained by exact observation and correct thinking. If science makes use of any methods but these it ceases to be itself. Science has therefore nothing to do with morals: it gives the suicide his pistol, the surgeon his life-saving lance, but neither admonishes nor judges them. It has nothing to do with emotion: it exposes the chemistry of a tear, the mechanism of laughter; but of sorrow and happiness it has naught to say. It has nothing to do with beauty: it traces the movements of the stars, and tells of their constitution; but the fact of their singing together, and that "such harmony is in immortal souls," it leaves to poet and philosopher. The timbre, loudness, pitch, of musical tones, is a concern of science; but for this a Beethoven symphony is no better than the latest ragtime air from the music halls. In brief, science deals only with _phenomena_, and its gift to man is power over his material environment.

MATHEMATICS

The gift of pure mathematics, on the other hand, is primarily to the mind and spirit: the fact that man uses it to get himself out of his physical predicaments is more or less by the way. Consider for a moment this paradox. Mathematics, the very thing common sense swears by and dotes on, contradicts common sense at every turn. Common sense balks at the idea of _less than nothing_; yet the _minus_ quantity, which in one sense is less than nothing in that something must be added to it to make it equal to nothing, is a concept without which algebra would have to come to a full stop. Again, the science of quaternions, or more generally, a vector analysis in which the progress of electrical science is essentially involved, embraces (explicitly or implicitly) the extensive use of _imaginary_ or _impossible_ quantities of the earlier algebraists. The very words "imaginary" and "impossible" are eloquent of the defeat of common sense in dealing with concepts with which it cannot practically dispense, for even the negative or imaginary solutions of imaginary quantities almost invariably have some physical significance. A similar statement might also be made with regard to _transcendental_ functions.

Mathematics, then, opens up ever new horizons, and its achievements during the past one hundred years give to thought the very freedom it seeks. But if science is dispassionate, mathematics is even more austere and impersonal. It cares not for teeming worlds and hearts insurgent, so long as in the pure clarity of space, relationships exist. Indeed, it requires neither time nor space, number nor quantity. As the mathematician approaches the limits already achieved by study, the colder and thinner becomes the air and the fewer the contacts with the affairs of every day. The Promethean fire of pure mathematics is perhaps the greatest of all in man's catalogue of gifts; but it is not most itself, but least so, when, immersed in the manifoldness of phenomenal life, it is made to serve purely utilitarian ends.

INTUITION

Common sense, immersed in the mere business of living, knows no more about life than a fish knows about water. The play of reason upon phenomena dissects life, and translates it in terms of inertia. The pure logic of mathematics ignores life and disdains its limitations, leading away into cold, free regions of its own. Now our desire for freedom is not to vibrate in a vacuum, but to live more abundantly. _Intuition_ deals with life directly, and introduces us into life's own domain: it is related to reason as flame is related to heat. All of the great discoveries in science, all of the great solutions in mathematics, have been the result of a _flash_ of intuition, after long brooding in the mind. _Intuition illumines_. Intuition is therefore the light which must guide us into that undiscovered country conceded by mathematics, questioned by science, denied by common sense--_The Fourth Dimension of Space_.

OUR SENSE OF SPACE

Space has been defined as "room to move about." Let us accord to this definition the utmost liberty of interpretation. Let us conceive of space not alone as room to move ponderable bodies in, but as room to think, to feel, to strike out in unimaginable directions, to overtake felicities and knowledges unguessed by experience and preposterous to common sense. Space is not measurable: we attribute dimensionality to space because such is the method of the mind; and that dimensionality we attribute to space is progressive because progression is a law of the mind. The so-called dimensions of space are to space itself as the steps that a climber cuts in the face of a cliff are to the cliff itself. They are not necessary to the cliff: they are necessary only to the climber. Dimensionality is the mind's method of mounting to the idea of the infinity of space. When we speak of the fourth dimension, what we mean is the fourth stage in the apprehension of that infinity. We might as legitimately speak of a fifth dimension, but the profitlessness of any discussion of a fifth and higher stages lies in the fact that they can be intelligently approached only through the fourth, which is still largely unintelligible. The case is like that of a man promised an increase of wages after he had worked a month, who asks for his second month's pay before he is entitled to the first.

THE SUBJECTIVITY OF SPACE

Without going deep into the doctrine of the ideality--that is, the purely subjective reality--of space, it is easy to show that we have arrived at our conception of a space of three dimensions by an intellectual process. The sphere of the senses is two-dimensional: except for the slight aid afforded by binocular vision, sight gives us moving pictures _on a plane_, and touch contacts _surfaces_ only. What circumstances, we may ask, have compelled our intellect to conceive of _solid_ space? This question has been answered as follows:

"If a child contemplates his hand, he is conscious of its existence in a double manner--in the first place by its tangibility, the second by its image on the retina of his eye. By repeated groping about and touching, the child knows by experience that his hand retains the same form and extension through all the variations of distance and position under which it is observed, notwithstanding that the form and extension of the image on the retina constantly change with the different position and distance of his hand in respect to his eye. The problem is thus set to the child's understanding: how to reconcile to his comprehension the apparently contradictory facts of the _invariableness_ of the object together with the _variableness_ of its appearance. This is only possible within a space of three dimensions, in which, owing to perspective distortions and changes, these variations of projection can be reconciled with the constancy of the form of a body."

Thus we have come to the idea of a three-dimensional space in order to overcome the apparent contradictoriness of facts of sensible experience. Should we observe in three-dimensional space contradictory facts our reason would be forced to reconcile these contradictions, also, and if they could be reconciled by the idea of a four-dimensional space our reason would accept this idea without cavil. Furthermore, if from our childhood, phenomena had been of daily occurrence requiring a space of four or more dimensions for an explanation conformable to reason, we should feel ourselves native to a space of four or more dimensions.

Poincaré, the great French mathematician and physicist, arrived at these same conclusions by another route. By a process of mathematical reasoning of a sort too technical to be appropriately given here, he discovers an order in which our categories range themselves naturally, and which corresponds with the points of space; and that this order presents itself in the form of what he calls a "three circuit distribution board." "Thus the characteristic property of space," he says, "that of having three dimensions, is only a property of our distribution board, _a property residing, so to speak, in human intelligence_." He concludes that a different association of ideas would result in a different distribution board, and that might be sufficient to endow space with a fourth dimension. He concedes that there may be thinking beings, living in our world, whose distribution board has four dimensions, and who do consequently think in hyperspace.

THE NEED OF AN ENLARGED SPACE-CONCEPT

It is the contrariety in phenomena already referred to, that is forcing advanced minds to entertain the idea of higher space. Mathematical physicists have found that experimental contradictions disappear if, instead of referring phenomena to a set of three space axes and one time axis of reference, they be referred to a set of four interchangeable axes involving four homogeneous co-ordinates. In other words, _time_ is made the fourth dimension. Psychic phenomena indicate that occasionally, in some individuals, the will is capable of producing physical movements for whose geometrico-mathematical definition a four-dimensional system of co-ordinates is necessary. This is only another step along the road which the human mind has always travelled: our conception of the cosmos grows more complete and more just at the same time that it recedes more and more beneath the surface of appearances.

Far from the Higher Space Hypothesis complicating thought, it simplifies by synthesis and co-ordination in a manner analogous to that by which plane geometry is simplified when solid geometry becomes a subject of study. By immersing the mind in the idea of many dimensions, we emancipate it from the idea of dimensionality. But the mind moves most readily, as has been said, in ordered sequence. Frankly submitting ourselves to this limitation, even while recognizing it as such, let us learn such lessons from it as we can, serving the illusions that master us until we have made them our slaves.

II THE DIMENSIONAL LADDER

LEARNING TO THINK IN TERMS OF SPACES

The Reader who is willing to consider the Higher Space Hypothesis seriously, who would discover, by its aid, new and profound truths closely related to life and conduct, should first of all endeavor to arouse in himself a new power of perception. This he will best accomplish by learning to discern dimensional sequences, not alone in geometry, but in the cosmos and in the natural world. By so doing he may erect for himself a veritable Jacob's ladder,

"Pitched between Heaven and Charing Cross."

He should accustom himself to ascend it, step by step, dimension by dimension. Then he will learn to trust Emerson's dictum, "Nature geometrizes," even in regions where the senses fail him, and the mind alone leads on. Much profitable amusement is to be gained by such exercises as follow. They are in the nature of a running up and down the scales in order to give strength and flexibility to a new set of mental fingers. Learning to think in terms of spaces contributes to our emancipation from the tyranny of space.

FROM THE COSMOS TO THE CORPUSCLE

By way of a beginning, proceed, by successive stages, from the contemplation of the greatest thing conceivable to the contemplation of the most minute, and note the space sequences revealed by this shifting of the point of view.

The greatest thing we can form any conception of is the starry firmament made familiar to the mind through the study of astronomy. No limit to this vastitude has ever been assigned. Since the beginning of recorded time, the earth, together with the other planets and the sun, has been speeding through interstellar space at the rate of 300,000,000 miles a year, without meeting or passing a single star. A ray of light, travelling with a velocity so great as to be scarcely measurable within the diameter of the earth's orbit, takes years to reach even the nearest star, centuries to reach those more distant. Viewed in relation to this universe of suns, our particular sun and all its satellites--of which the earth is one--shrinks to a point (a _physical_ point, so to speak--not geometrical one).

The mind recoils from these immensities: let us forsake them, then, for more familiar spaces, and consider the earth in its relation to the sun. Our planet appears as a _moving_ point, tracing out a _line_--a _one-space_--its path around the sun. Now let us remove ourselves in imagination only far enough from the earth for human beings thereon to appear as minute moving things, in the semblance, let us say, of insects infesting an apple. It is clear that from this point of view these beings have a freedom of movement in their "space" (the surface of the earth), of which the larger unit is not possessed; for while the earth itself can follow only a _line_, its inhabitants are free to move in the two dimensions of the surface of the earth.

Abandoning our last coign of vantage, let us descend in imagination and mingle familiarly among men. We now perceive that these creatures which from a distance appeared as though flat upon the earth's surface, are in reality erect at right angles to its plane, and that they are endowed with the power to move their members in _three dimensions_. Indeed, man's ability to traverse the surface of the earth is wholly dependent upon his power of three-dimensional movement. Observe that with each transfer of our attention from greater units to smaller, we appear to be dealing with a power of movement in an additional dimension.

Looking now in thought not _at_ the body of man, but _within_ it, we apprehend an ordered universe immensely vast in proportion to that physical ultimate we name the electron, as is the firmament immensely vast in proportion to a single star. It has been suggested that in the infinitely minute of organic bodies there is a power of movement in a _fourth_ dimension. If so, such four-dimensional movement may be the proximate cause of the phenomenon of _growth_--of those chemical changes and renewals whereby an organism is enabled to expand in three-dimensional space, just as by a three-dimensional power of movement (the act of walking) man is able to traverse his two-dimensional space--the surface of the earth.

--AND BEYOND