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.”

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.

Poincare, 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

Proceed still further. Behind such organic change—assumed to be four-dimensional—there is the determination of some will-to-live, which manifests itself to consciousness as thought and as desire. Into these the idea of space does not enter: we think of them as in time. But if there are developments of other dimensions of space, thought and emotion may themselves be discovered to have space relations; that is, they may find expression in the forms of higher spaces. Thus is opened up one of those rich vistas in which the subject of the fourth dimension abounds, but into which we can only glance in passing. If there are such higher-dimensional thought-forms, our normal consciousness, limited to a world of three dimensions, can apprehend only their three-dimensional aspects, and these not simultaneously, but successively—that is, in time. According to this view, any unified series of actions—for example, the life of an individual, or of a group—would represent the straining, so to speak, of a thought-form through our time, as the bodies subject to these actions would represent its straining through our space.

EVOLUTION AS SPACE-CONQUEST

Evolution is a struggle for, and a conquest of, space; for evolution, as the word implies, is a drawing out of what is inherent from latency into objective reality, or in other words into spatial—and temporal—extension.

This struggle for space, by means of which the birth and growth of organisms is achieved, is the very texture of life, the plot of every drama. Cells subdivide; micro-organisms war on one another; plants contend for soil, light, moisture; flowers cunningly suborn the bee to bring about their nuptials; animals wage deadly warfare in their rivalry to bring more hungry animals into a space-hungry world. Man is not exempt from this law of the jungle. Nations intrigue and fight for land—of which wealth is only the symbol—and a nation's puissance is measured by its power to push forward into the territory of its neighbor. The self-same impulse drives the individual. One measure of the difference between men in the matter of efficiency is the amount of space each can command: one has a house and grounds in some locality where every square inch has an appreciable value; another some fractional part of a lodging house in the slums. When this bloodless, but none the less deadly, contest for space becomes acute, as in the congested quarters of great cities, man's ingenuity is taxed to devise effective ways of augmenting his space-potency, and he expands in a vertical direction. This third-dimensional extension, typified in the tunnel and in the skyscraper, is but the latest phase of a conquest of space which began with the line of the pioneer's trail through an untracked wilderness.

DIMENSIONAL SEQUENCES

Not only does nature everywhere geometrize, but she does so in a particular way, in which we discover dimensional sequences. Consider the transformation of solid, liquid, gas, from one to another, under the influence of heat. A solid, set in free motion, can follow only a line—as is the case of a thrown ball. A liquid has the added power of lateral extension. Its tendency, when intercepted, is to spread out in the two dimensions of a plane—as in the case of a griddle cake; while a gas expands universally in all directions, as shown by a soap-bubble. It is a reasonable inference that the fourth state of matter, the corpuscular, is affiliated to some four-dimensional manner of extension, and that there may be states beyond this, involving even higher development of space.

Next glance at the vegetable kingdom. The seed, a point, generates a line system, in stem, branches, twigs, from which depend planes in the form of leaves and flowers, and from these come fruit, solids.

  “The point, the line, the surface and the sphere,
  In seed, stem, leaf and fruit appear.”

A similar sequence may be noted within the body: the line -network of the nerves conveys the message of sensation from the surface of the body to some center in the solid, of the brain—and thence to the Silent Thinker, “he who is without and within,” or in terms of our hypothesis, “he who dwells in higher space.”

MAN THE GEOMETER

When man essays the role of creator he cannot do otherwise than follow similar sequences: it is easy to discern dimensional progression in the products of man's ingenuity and skill. Consider, for example, the evolution of a building from its inception to its completion. It exists first of all in the mind of the architect, and there it is indubitably higher-spatial, for he can interpenetrate and examine every part, and he can consider it all at once, viewing it simultaneously from without and from within, just as one would be able to do in a space of four dimensions. He begins to give his idea physical embodiment by making with a pencil-point, lines on a plane (a piece of paper), the third dimension being represented by means of the other two. Next (if he is careful and wise) he makes a three-dimensional model. From the architect's drawings the engineer establishes his points, lays out his angles, and runs his lines upon the site itself. The mason follows, and with his footing courses makes ponderable and permanent the lines of the engineer. These lines become in due course walls—vertical planes. Floors and roofs—horizontal planes—follow, until some portion of three-dimensional space has been enclosed.

Substantially the same sequence holds, whatever the kind of building or the character of the construction—whether a steel-framed skyscraper or a wooden shanty. A line system, represented by columns and girders in the one case, and by studs and rafters in the other, becomes, by overlay or interposition, a system of planes, so assembled and correlated as to define a solid.

With nearly everything of man's creating—be it a bureau or a battleship—the process is as above described. First, a pattern to scale; next, an actual linear framework; then planes defining a solid. Consider almost any of the industries practiced throughout the ages: they may be conceived of thus in terms of dimensions; for example, those ancient ones of weaving and basket making. Lines (threads in the one case, rushes in the other) are wrought into planes to clothe a body or to contain a burden. Or think, if you choose, of the modern industry of book-making, wherein types are assembled, impressed upon sheets of paper, and these bound into volumes— points, lines, planes, solids. The book in turn becomes the unit of another dimensional order, in the library whose serried shelves form lines, which, combined into planes, define the lateral limits of the room.

HIGHER—AND HIGHEST—SPACE

These are truisms. What have they to do, it may be asked, with the idea of higher spaces? They have everything to do with it, for in achieving the enclosure of any portion of solid space the limit of known dimensions has been reached without having come to any end. More dimensions—higher spaces—are required to account for higher things. All of the products of man's ingenuity are inanimate except as he himself animates them. They remain as they were made, machines, not organisms. They have no inherent life of their own, no power of growth and renewal. In this they differ from animate creation because the highest achievement of the creative faculty in man in a mechanical way lacks the life principle possessed by the plant. And as the most perfect machine is inferior in this respect to the humblest flower that grows, so is the highest product of the vegetable kingdom inferior to man himself, the maker of the machine; for he can reflect upon his own and the world's becoming, while the plant can only become.

What is the reason for these differences of power and function? According to the Higher Space Hypothesis they are due to varying potencies of movement in the secret causeways and corridors of space. The higher functions of consciousness—volition, emotion, intellection—may be in some way correlated with the higher powers of numbers, and with the corresponding higher developments of space. Thus would the difference between physics and metaphysics become a difference of degree and not of kind. Evolution is to be conceived of as a continuous pushing back of the boundary between representation and reality, or as a conquest of space. We may conceive of space as of an infinite number of dimensions, and of consciousness as a moving—or rather as an expanding—point, embracing this infinity, involving worlds, powers, knowledges, felicities, within itself in everlasting progression.

III. PHYSICAL PHENOMENA

LOOKING FOR THE GREATER IN THE LESS

After the assured way in which the author has conducted the reader repeatedly up and down the dimensional ladder, it may be a surprise to learn that physical phenomena offer no irrefragable evidences of hyper-dimensionality. We could not think in higher space if consciousness were limited to three dimensions. The mathematical reality of higher space is never in question: the higher dimensions are as valid as the lower, but the hyper-dimensionality of matter is still unproven. Man's ant-like efforts to establish this as a truth have thus far been vain.

Lest this statement discourage the reader at the very outset, he should understand the reason for such failure. We are embedded in our own space, and if that space be embedded in higher space, how are we going to discover it? If space is curved, how are we going to measure its curvature? Our efforts to do so may be compared to measuring the distance between the tips of a bent bow by measuring along the bow instead of along the string.

Imagine a scientifically-minded threadworm to inhabit a page of Euclid's solid geometry: the evidences of three-dimensionality are there, in the very diagrams underneath his eyes; but you could not show him a solid—the flat page could not contain it, any more than our space can contain a form of four dimensions. You could only say to him, “These lines represent a solid.” He would have to depend on his faith for belief and not on that “knowledge gained by exact observation and correct thinking” in which alone the scientist finds a sure ground for understanding.

It is an axiom of science never to look outside three-space horizons for an understanding of phenomena when these can logically be accounted for within those horizons. Now because, on the Higher Space Hypothesis, each space is the container of all phenomena of its own order, the futility, for practical purposes, of going outside is at once apparent. The highly intelligent threadworm neither knows nor cares that the point of intersection of two lines in his diagram represents a point in a space to which he is a stranger. The point is there, on his page: it is what he calls a fact. “Why raise” (he says) “these puzzling and merely academic questions? Why attempt to turn the universe completely upside down?”

But though no proofs of hyper-dimensionality have been found in nature, there are equally no contradictions of it, and by using a method not inductive, but deductive, the Higher Space Hypothesis is plausibly confirmed. Nature affords a sufficient number of representations of four-dimensional forms and movements to justify their consideration.

SYMMETRY

Let us first flash the light of our hypothesis upon an all but universal characteristic of living forms, yet one of the most inexplicable—symmetry.

Animal life exhibits the phenomenon of the right-and left-handed symmetry of solids. This is exemplified in the human body, wherein the parts are symmetrical with relation to the axial plane. Another more elementary type of symmetry is characteristic of the vegetable kingdom. A leaf in its general contour is symmetrical: here the symmetry is about a line—the midrib. This type of symmetry is readily comprehensible, for it involves simply a revolution through 180 degrees. Write a word on a piece of paper and quickly fold it along the line of writing so that the wet ink repeats the pattern, and you have achieved the kind of symmetry represented in a leaf.

With the symmetry of solids, or symmetry with relation to an axial plane, no such simple movement as the foregoing suffices to produce or explain it, because symmetry about a plane implies four-dimensional movement. It is easy to see why this must be so. In order to achieve symmetry in any space—that is, in any given number of dimensions—there must be revolution in the next higher space: one more dimension is necessary. To make the (two-dimensional) ink figure symmetrical, it had to be folded over in the third dimension. The revolution took place about the figure's line of symmetry, and in a higher dimension. In three-dimensional symmetry (the symmetry of solids) revolution must occur about the figure's plane of symmetry, and in a higher—i.e., the fourth dimension. Such a movement we can reason about with mathematical definiteness: we see the result in the right-and left-handed symmetry of solids, but we cannot picture the movement ourselves because it involves a space of which our senses fail to give any account.

Now could it be shown that the two-dimensional symmetry observed in nature is the result of a three-dimensional movement, the right-and left-handed symmetry of solids would by analogy be the result of a four-dimensional movement. Such revolution (about a plane) would be easily achieved, natural and characteristic, in four space, just as the analogous movement (about a line) is easy, natural, and characteristic, in our space of three dimensions.

OTHER ALLIED PHENOMENA

In the mirror image of a solid we have a representation of what would result from a four-dimensional revolution, the surface of the mirror being the plane about which the movement takes place. If such a change of position were effected in the constituent parts of a body as a mirror image of it represents, the body would have undergone a revolution in the fourth dimension. Now two varieties of tartaric acid crystallize in forms bearing the relation to one another of object to mirror image. It would seem more reasonable to explain the existence of these two identical, but reversed, varieties of crystal, by assuming the revolution of a single variety in the fourth dimension, than by any other method.

There are two forms of sugar found in honey, dextrose and levulose. They are similar in chemical constitution, but the one is the reverse of the other when examined by polarized light—that is, they rotate the plane of polarization of a ray of light in opposite ways. If their atoms are conceived to have the power of motion in the fourth dimension, it would be easy to understand why they differ. Certain snails present the same characteristics as these two forms of sugar. Some are coiled to the right and others to the left; and it is remarkable that, like dextrose and levulose, their juices are optically the reverse of each other when studied by polarized light.

Revolution in the fourth dimension would also explain the change in a body from producing a right-handed, to producing a left-handed, polarization of light.

ISOMERISM

In chemistry the molecules of a compound are assumed to consist of the atoms of the elements contained in the compound. These atoms are supposed to be at certain distances from one another. It sometimes happens that two compound substances differ in their chemical or physical properties, or both, even though they have like chemical elements in the same proportion. This phenomenon is called isomerism, and the generally accepted explanation is that the atoms in isomeric molecules are differently arranged, or grouped, in space. It is difficult to imagine how atoms, alike in number, nature, and relative proportion, can be so grouped as somehow to produce compounds with different properties, particularly as in three-dimensional space four is the greatest number of points whose mutual distances, six in number, are all independent of each other. In four-dimensional space, however, the ten equal distances between any two of five points are geometrically independent, thus greatly augmenting the number and variety of possible arrangements of atoms.

This just escapes being the kind of proof demanded by science. If the independence of all the possible distances between the atoms of a molecule is absolutely required by theoretical chemical research, then science is really compelled, in dealing with molecules of more than four atoms, to make use of the idea of a space of more than three dimensions.

THE ORBITAL MOTION OF SPHERES: CELL SUB-DIVISION

There is in nature another representation of hyper-dimensionality which, though difficult to demonstrate, is too interesting and significant to be omitted here.

Imagine a helix, intersected, in its vertical dimension, by a moving plane. If necessary to assist the mind, suspend a spiral spring above a pail of water, then raise the pail until the coils, one after another, become immersed. The spring would represent the helix, and the surface of the water the moving plane. Concentrating attention upon this surface, you would see a point—the elliptical cross-section of the wire where it intersected the plane—moving round and round in a circle. Next conceive of the wire itself as a lesser helix of many convolutions, and repeat the experiment. The point of intersection would then continually return upon its own track in a series of minute loops forming those lesser loops, which, moving circle-wise, registered the involvement of the helix in the plane.

It is easy to go on imagining complicated structures of the nature of the spiral, and to suppose also that these structures are distinguishable from each other at every section. If we think of the intersection of these with the rising surface, as the atoms, or physical units, of a plane universe, we shall have a world of apparent motion, with bodies moving harmoniously amongst one another, each a cross-section of some part of an unchanging and unmoving three-dimensional entity.

Now augment the whole by an additional dimension—raise everything one space. The helix of many helices would become four-dimensional, and superficial space would change to solid space: each tiny circle of intersection would become a sphere of the same diameter, describing, instead of loops, helices. Here we would be among familiar forms, describing familiar motions: the forms, for example, of the earth and the moon and of their motion about the sun; of the atom, as we imagine it, the molecule and the cell. For is not the sphere, or ovoid, the unit form of nature; and is not the spiral vortex its characteristic motion, from that of the nebula in the sky to the electron in the atom? Thus, on the hypothesis that our space is traversing four-dimensional space, and that the forms of our space are cross-sections of four-dimensional forms, the unity and harmony of nature would be accounted for in a remarkably simple manner.

The above exercise of the imagination is a good preparation for the next demand upon it. Conceive a dichotomous tree—one that always divides into two branches—to pass through a plane. We should have, as a plane section, a circle of changing size, which would elongate and divide into two circles, each of which would do the same. This reminds us of the segmentation of cell life observed under the microscope, as though a four-dimensional figure were registering its passage through our space.

THE ELECTRIC CURRENT

Hinton conceived of an electric current as a four-dimensional vortex. He declared that on the Higher Space Hypothesis the revolution of the ether would yield the phenomenon of the electric current. The reader is referred to Hinton's book, The Fourth Dimension, for an extended development of this idea. What follows is a brief summary of his argument. First, he examines the characteristics of a vortex in a three-dimensional fluid. Then he conceives of what such a vortex would be in a four-dimensional medium of analogous properties. The whirl would be about a plane, and the contour of this plane would correspond to the ends of the axis line in the former vortex; and as before, the vortex would extend to the boundary. Every electric current forms a closed circuit: this is equivalent to the hyper-vortex having its ends in the boundary of the hyper-fluid. The vortex with a surface as its axis, therefore, affords a geometric image of a closed circuit.

Hinton supposes a conductor to be a body which has the property of serving as a terminal abutment to such a hyper-vortex as has been described. The conception that he forms of a closed current, therefore, is of a vortex sheet having its edge along the circuit of the conducting wire. The whole wire would then be like the centers on which a spindle turns in three-dimensional space, and any interruption of the continuity of the wire would produce a tension in place of a continuous revolution. The phenomena of electricity—polarity, induction, and the like—are of the nature of the stress and strain of a medium, but one possessing properties unlike those of ordinary matter. The phenomena can be explained in terms of higher space. If Hinton's hypothesis be the true explanation, the universality of electro-magnetic action would again point to the conclusion that our three-dimensional world is superficial—the surface, that is, of a four-dimensional universe.

THE GREATER UNIVERSE

This practically exhausts the list of accepted and accredited indications of hyper-dimensionality in our physical environment. But if the collective human consciousness is moving into the fourth dimension, such indications are bound to multiply out of all measure. It should be remembered that in Franklin's day electricity was manifest only in the friction of surfaces and in the thunderbolt. To-day all physical phenomena, in their last analysis, are considered to be electrical. The world is not different, but perception has evolved, and is evolving.

There is another field, in which some of our ablest minds are searching for evidences of the curvature of space, the field of astronomy and astro-physics. But into this the layman hesitates to enter because the experts themselves have found no common ground of understanding. The ether of space is a battlefield strewn with dead and dying hypotheses; gravitation, like multiplication, is vexation; the very nature of time, form and movement is under vivid discussion, in connection with what is known as the Theory of Relativity.

Notwithstanding these counter-currents of speculation, which should make the wise man speak smilingly of his wisdom, this summary remains incomplete without a reference to the pressure of higher space upon those adventurous minds that essay to deal with the profound problems of the greater universe, and a statement of the reasons for their feeling this pressure. These reasons are well suggested by Professor B.G. Harrison, in his Popular Astronomy. He says: “With the idea of a universe of finite dimensions there is the obvious difficulty of the beyond. The truth is that a universe of finite proportions is equally difficult to realize as one of infinite extent. Perhaps the nearest analogy to infinity that we can understand lies in our conception of a closed curve. It seems easier to imagine the endless movement of a sphere in a circular path than the case of one travelling in a straight line. Possibly this analogy may apply in some way to fourth-dimensional space, but the manner of its application is certainly not easy to understand. If we would imagine that all co-ordinates of time and space were curved, and eventually return to the same point, it might bring the ultimate comprehension one degree nearer.”

A HINT FROM ASTRONOMY

The physical evidence that our space is thus curved in higher space, some have considered astronomy to furnish in what is called the “negative parallax” of certain distant stars. This cannot be passed by, though it is too deeply involved with the probable error of the observers themselves to be considered more than an interesting fact in this connection. Every one knows that the difference of angle under which an object is seen from two standpoints is called its parallax. The parallax of the stars—and the consequent knowledge of their distance—is obtained by observing them from opposite points of the earth's orbit around the sun. When a star is within measurable distance, these angles are acute, and the lines from the star to the earth at opposite sides of its orbit converge, therefore. But when these lines, as sometimes happens, appear to be divergent, the result is called a negative parallax, and is explainable by higher space relationships. Obviously, the divergence of the lines would indicate that the object lies behind the observer instead of in front of him. This anomaly can be explained by the curvature of space in the fourth dimension. If space is so curved, the path of light itself is curved also, and a man—were his vision immeasurably keen, not to say telescopic—could see the back of his own head! It is not worth while to give this question of negative parallax too much importance, by reason of the probability of error, but in this connection it should be stated that there appears to be an undue number of negative parallaxes recorded.

GRAVITATION

Gravitation remains a puzzle to science. The tendency of modern physics is to explain all material phenomena in terms of electrons and the ether, but the attempt to account for gravitation in this way is attended with difficulties. In order to cope with these, it seems necessary to assume that our universe is only a portion of a greater universe. This assumption readily lends itself to the conception of our universe as a three-dimensional meeting place of two portions of a universe of four dimensions—that is, its conception as a “higher" surface. This is a fundamental postulate of higher space speculation.

One hypothesis advanced to explain gravitation assumes the existence of a constant hydrostatic pressure transmitted through the ether. A steady flow of ether into every electron in a gravitating system of bodies would give rise to forces of attraction between them, varying inversely as the square of the distance, according to Newton's law. But in order to avoid the conception of the continual destruction and creation of ether, it is necessary to assume a steady flow through every electron between our universe and the greater universe of which it is assumed to form a part Now because the electrons, in order to receive this flow, must lie on the boundary of this greater universe, the latter must be four-dimensional. Every electron, in other words, must be the starting point of a pathway into—and a terminal point out of—four-dimensional space. Here we have another familiar higher space concept.

THE ETHER OF SPACE

The ether of space, because it has at last found entrance, must be given a grudging hospitality in these pages, even though the mysterious stranger prove but a ghost. The Relativists would have it that with the acceptance of their point of view the ether may be eliminated; but if they take away the ether, they must give us something in its stead. In whatever way the science of the future disposes of this problem, it must take into account the fact of light transmission. On the theory that the ether is an elastic solid of amazing properties, in which the light waves vibrate transversely to their direction, it assists the mind to think of the ether as four-dimensional, because then a light wave would be a superficial disturbance of the medium—superficial, but three-dimensional, as must needs be the case with the surface of a four-dimensional solid.

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This search for evidences of hyper-dimensionality in the universe accessible to our senses is like looking, not for a needle in a haystack, but for a haystack in a needle—for the greater in the less. From the purely physical evidences, all that can with certainty be said is that the hypothesis is not inconsistent with the facts of science or its laws; that it is being verified and rendered more probable by the investigations of science; that it is applicable to the description or explanation of all the observed phenomena, and assigns a cause fully adequate to have produced them.

Now there is an order of phenomena that we call psychic. Because they are phenomenal they cannot occur outside of time and space altogether; because they are psychic they defy explanation in terms of the space and time of every-day life. Let us next examine these in the light of our hypothesis.

IV. TRANSCENDENTAL PHYSICS

ZOeLLNER

In the year 1877, Johann Friedrich Zoellner, professor of physics and astronomy at the University of Leipsic, undertook to prove that certain (so-called) psychic phenomena were susceptible of explanation on the hypothesis of a four-dimensional space. He used as illustrations the phenomena induced by the medium Henry Slade. By the irony of events, Slade was afterward arrested and imprisoned for fraud, in England. This fact so prejudiced the public mind against Zoellner that his name became a word of scorn, and the fourth dimension a synonym for what is fatuous and false. Zoellner died of it, but since his death public opinion has undergone a change. There is a great and growing interest in everything pertaining to the fourth dimension, and belief in that order of phenomena upon which Zoellner based his deductions is supported by evidence at once voluminous and impressive.

It is unnecessary to go into the question of the genuineness of the particular phenomena which Zoellner witnessed. His conclusions are alone important, since they apply equally to other manifestations, whose authenticity has never been successfully impeached. Zoellner's reasoning with regard to certain psychic phenomena is somewhat along the following lines.

APPARITIONS

The intrusion (as an apparition) of a person or thing into a completely enclosed portion of three-space; or contrariwise, the exit (as an evanishment) out of such a space.

Because we lack the sense of four-dimensional space, we must here have recourse to analogy, and assume three-dimensional space to be the unsensed higher region encompassing a world of two dimensions, To a hypothetical flat-man of a two-space, any portion of his plane surrounded by an unbroken line would constitute an enclosure. Were he confined within it, escape would be impossible by any means known to him. Had he the ability to move in the third dimension, however, he could rise, pass over the enclosing line without disturbing it, and descend on the other side. The moment he forsook the plane he would disappear from two-dimensional space. Such a disappearance would constitute an occult phenomenon in a world of two dimensions.

Correspondingly, an evanishment from any three-dimensional enclosure—such as a room with locked doors and windows—might be effected by means of a movement in the fourth dimension. Because a body would disappear from our perception the moment it forsook our space, such a disappearance would be a mystery; it would constitute an occult phenomenon. The thing would be no more mysterious, however, to a consciousness embracing four dimensions within its ken, than the transfer of an object from the inside to the outside of a plane figure without crossing its linear boundary is mysterious to us.

POSSESSION

The temporary possession of a person's body, or some member of that body, by an alien will, as exemplified in automatic writing and obsession.

It would doubtless amaze the scientifically orthodox to know how many people habitually and successfully practice the dubious art of automatic writing—not mediums, so-called, but people of refinement and intelligence. Although the messages received in this way may emanate from the subconscious mind of the performer, there is evidence to indicate that they come sometimes from an intelligence discarnate, or from a person remote from the recipient in space.

If such is indeed the case, if the will is extraneous, how does it possess itself of the nerves and muscles of the hand of the writer? The Higher Space Hypothesis is of assistance here. It is only necessary to remember that from the fourth dimension the interior of a solid is as much exposed as the interior of a plane figure is exposed from the region of the third dimension. A four-dimensional being would experience no difficulty, under suitable conditions, in possessing itself of any part of the bodily mechanism of another.

The same would hold true in cases of possession and obsession; for if the bastion of the hand can thus be captured, so also may the citadel of the brain. Certain familiar forms of hypnotism are not different from obsession, the hypnotizer using the brain and body of his subject as though they were his own. All unconsciously to himself, he has called into play four-dimensional mechanics. Many cases of so-called dual personality are more easily explicable as possession by an alien will than on the less credible hypothesis that the character, habits, and language of a person can change utterly in a moment of time.

CLAIRVOYANCE IN SPACE

Vision at a distance and the exercise of a superior power of sight.

Clairvoyance in space is of various kinds and degrees. Sometimes it consists in the perception of super-physical phenomena—the unfurling of a strange and wonderful land; and again it appears to be a higher power of ordinary vision, a kind of seeing to which the opacity of solids offers no impediment, or one involving spatial distances too great and too impeded for normal physical vision to be effective.

That clairvoyance which consists in the ability to perceive not alone the superficies of things as ordinary vision perceives them, but their interiors as well, is analogous to the power given by the X-ray, by means of which, on a fluorescent screen, a man may behold the beating of his own heart. But, if the reports of trained clairvoyants are to be believed, there is this difference: everything appears to them without the distortions due to perspective, objects being seen as though they were inside and not outside of the perceiving organ, or as though the observer were in the object perceived; or in all places at the same time.

Our analogy makes all this intelligible. To the flat-man, clairvoyance in space would consist in that power of perception which we exercise in reference to his plane. From the third dimension the boundaries of plane figures offer no impediment to the view of their interiors, and they themselves in no way impede our vision of surrounding objects. If we assume that clairvoyance in space is the perception of the things of our world from the region of the fourth dimension, the phenomena exactly conform to the demands of our analogy. It is no more difficult for a four-dimensional intelligence to understand the appearance or disappearance of a body in a completely closed room, or the withdrawal of an orange from its skin, without cutting or breaking that skin, than it is for us to see the possibility of taking up a pencil point from the center of a circle and putting it down outside. We are under no compulsion to draw a line across the circumference of the circle in order to enter or leave it. Moreover, the volume of our sensible universe embraced in the clairvoyant's field of view will increase in the same way that a balloonist's view increases in area as he rises above the surface of the earth. To account for clairvoyant vision at a distance, it is of course necessary to posit some perceptive organ other than the eye, but the fact that in trance the eyes are closed, itself demands this assumption.

CLAIRVOYANCE IN TIME

The perception of a past event as in process of occurring, or the prevision of something which comes to pass later.

No mechanistic explanation will serve to account for this order of clairvoyance since it is inextricably involved in the mystery of consciousness itself. Yet our already overworked analogy can perhaps cast a little light even here.

To the flat-man, the third dimension of objects passing through his plane translates itself to his experience into time. Were he capable of rising in the positive direction of the third dimension, he would have pre-vision, because he would be cognizant of that which had not yet intersected his plane: by sinking in the negative direction, he would have post-vision, because he could re-cognize that which had already passed.

Now there are excellent reasons, other than those based on analogy, that the fourth-dimensional aspect of things may manifest itself to our ordinary experience, not as spatial extension, but as temporal change. Then, if we conceive of clairvoyance as a transcending by consciousness of our three-dimensional space, prevision and post-vision would be logically possible as corresponding to the positive and negative of the fourth dimension. This may be made clearer by the aid of a homely illustration.

PISGAH SIGHTS OF LIFE'S PAGEANT

Suppose you are standing on a street corner, watching a procession pass. You see the pageant as a sequence of objects and individuals appearing into view near by and suddenly, and disappearing in the same manner. This would represent our ordinary waking consciousness of what goes on in the world round about. Now imagine that you walk up the street in a direction opposite to that in which the procession is moving. You then rapidly pass in review a portion of the procession which had not yet arrived at the point you were a few moments before. This would correspond to the seeing of something before it “happened,” and would represent the positive aspect of clairvoyance in time—prevision. Were you to start from your original position, and moving in the direction in which the procession was passing, overtake it at some lower street corner, you could witness the thing you had already seen. This would represent post-vision—clairvoyance of the past.

A higher type of clairvoyance would be represented by the sweep of vision possible from a balloon. From that place of vantage the procession would be seen, not as a sequence, but simultaneously, and could be traced from its formation to its dispersal. Past, present and future would be merged in one.

It is true that this explanation raises more questions than it answers: to account in this way for a marvel, a greater marvel must be imagined—that of transport out of one's own “space.” The whole subject bristles with difficulties, not the least of which is that even to conceive of such a thing as prevision all our old ideas about time must be recast. This is being done in the Principle of Relativity, a subject which may appropriately engage our attention next.