CHAPTER III.

THE ÆTHER.

There are some expressions which, being somewhat vaguely used, are apt to cause confusion in the mind of those who read or hear about higher space.

And perhaps the most mischievous is the expression, a curvature of space. Now of space as it is generally used, in its accepted significance, there can be no curvature. For space means a system of positions extending uniformly in the number of dimensions we choose to fix upon.

If we take the straight line as our space, we may call it 1 space; then the set of positions follow one on after the other without bending. If the line is bent it becomes a _line_, not a straight line. It should not be called 1 space, but a thing in 2 space. That is, it is a bent line in a plane.

A being who was on the line might not perceive the fact of this bending, and it might not affect the measurements he made. But if the line ran into itself again, and he found that he was moving on what we should call a circle, this would in no way affect his idea of space. He would recognize that what he called space, namely, his line, was not space, but a curved thing in 2 space.

Similarly, taking a plane—this is by definition not curved in any way, known or unknown, and it can only be conceived to be bent by ampler space being conceived, and its being imagined as having force applied to it so as to become a bent thing in this ampler space. In this case the term “plane” is not the correct name.

And so about our three-dimensional space; we cannot be robbed of that idea, although it might conceivably be proved that our earth and our whole universe were on a curved thing in 4 space.

We will then keep the term “space” for the ordinary conception; and call it 1, 2, 3, 4 space, according to the number of supposed independent directions.

A curved line or surface or solid we will call a 1, 2, or 3 thing, according to the number of dimensions in it.

A straight line is a 1 thing possible in 1 space. A circle is a 1 thing possible in 2 space. At any point of it a being in it is limited to motion in one direction, while the circle itself involves two dimensions. The surface of a sphere is a 2 thing possible in 3 space. The rind of an orange, or the orange itself, is a 3 thing possible in 3 space.

It will be observed that the surface of the sphere, although only a 2 thing, involves the conception of 3 space, and cannot be understood without the use of the idea of 3 space. It is a 2 thing because at any point of the surface a being can only move in two independent directions. A crooked line drawn on the surface of a sphere is a 1 thing in a 2 thing in 3 space.

Another very common misconception is occasioned by the use of a figure of this kind [Illustration: symbol] to represent a “knot” in 2 space.

It obviously corresponds in 2 space to an iron rod welded together at the crossing place of the loop, so that it is indistinguishable which is the one free end, which the other. At the crossing point the two lines represented by the two ink marks must be absolutely one and the same.

If one line be supposed to go over the other, by however small a distance, it would leave the plane. It would suddenly become invisible to the creature in the plane, and it would appear again at the other side of the line it crossed as if it came from nowhere.

It would be as extraordinary a sight as if we saw a pole going up to a brick wall, then beyond the brick wall the rest of the pole appearing—not going through the brick wall, nor coming round it—but somehow appearing; part of the same pole moving when it moved, obviously connected with it, and yet with no joining part which we could possibly discover.

Again, it sometimes appears to be thought that the fourth dimension is in some way different from the three which we know. But there is nothing mysterious at all about it. It is just an ordinary dimension tilted up in some way, which with our bodily organs we cannot point to. But if it is bent down it will be just like any ordinary dimension: a line which went up into the fourth dimension one inch will, when bent down, lie an inch in any known direction we like to point out. Only if this line in the fourth dimension be supposed to be connected rigidly with any rigid body, one of the directions in that rigid body must point away in the fourth dimension when the line that was in the fourth comes into a 3 space direction.

If the reader will refer back to the paper on the plane world he will find a description of the means by which a being there might know that he was in a limited world, and that his conception of space was not of what was really the whole of space, but of the limited portion of it to which he was confined by his manner of being.

The test by which such a being could discover his limitation was this. He found two things, each consisting of a multitude of parts—two triangles; and the relationship of the parts of the one was the same as the relationship of the parts of the other. For every point in the one there was a corresponding point in the other. For every pair of points in the one there was a corresponding pair of points in the other. In fact, considered as systems made up of mutually related parts, each was the same as the other.

Yet he could not make these two triangles coincide.

Now this impossibility of bringing together two things which he felt were really alike was the sign to him of his limitation; and by reflecting on the similar appearance which would present itself to a being limited to a straight line—by thinking of two systems of points which were really identical, and which he could make coincide, but which a line being could not make coincide, he would be led to conclude that he in his turn was subject to a limitation.

Now is there any object which we know which, considered as a whole consisting of parts, is exactly like another whole, the two having all their parts similarly arranged, so as to form in themselves two identical systems, and yet the one incapable of being made to coincide with the other, even in thought?

Let us look at our two hands.

They are (except for accidental variations) exactly alike. And yet they cannot be made to coincide.

And here, if we reflect on it, is the sign to us that we are limited in our notions of space—that we are really in a four-dimensional world.

Watching a ship as it recedes from the shore we see that it becomes hull down before it vanishes, and know that the earth is round. And no less certainly do our two hands, in their curious likeness and yet difference, afford to us a perpetual proof of our limitation, and indicate a larger world.

This sign really tells us more than the mere fact of our limitation: it tells us where to look for the possibility of four-dimensional movements. It tells us that movements of any degree of magnitude relative to us are not possible in the fourth dimension. It tells us to look for four-dimensional movements in the minute particles of matter, not in the movements of masses of about our own size.

The task before us is difficult. We have to make up from the outside what the appearances of a higher space existence are to us in our space, and then we have to look at the facts of nature and see if they correspond to these appearances.

Let us take a few isolated points and look at them patiently.

To a being standing on the rim of a plane world a straight line absolutely shuts out the prospect before him. If the straight line is infinite it cuts his world in two; he can never hope to get beyond it.

It is to him what an infinite plane would be to us, stretching impassably in front of us, cutting us off from all that lies on the other side.

But we know that a point can move round this line. It can revolve round it by going out of the plane, and coming down again into the plane on the other side of the line.

This movement would be inconceivable to a plane being; for he can only conceive it possible to get to the other side of the line by going to the end of it and coming back along the other side of the line.

Now take a piece of paper and put a dot right in the middle and suppose that it has no means of passing through the paper. We can only conceive the dot getting to the other side of the paper by passing round the edge and coming back again to the position underneath where it was.

But by a four-dimensional movement it can slip round the paper without going to the edge.

A set of words may help. In a plane a body rotates round a point—rotation takes place round a point. In space rotation is always round a line—the axis. In four-dimensional space rotation takes place round a plane.

To take a farther consideration of this point—a plane being can see one side or the opposite of a straight line. He can only see it in one direction or in the reverse direction. But we can look at a straight line from a direction at right angles to that in which a plane being looks at it. We can look at a straight line from points which go all round it.

Similarly, a being in four-dimensional space can look at a plane from a direction at right angles to that in which we look at it. If we try to think of this we shall imagine ourselves looking at the thin edge. But this is not what a four-dimensional being would mean. He would see the plane exactly as we see it, but it would be from a direction at right angles to that in which we look.

In working with four-dimensional models it is a curious sensation until we become used to it—that of looking at a plane at one time, and then looking at it again; and, although it seems just the same—as square in front of us as before—realizing that we are looking at it from a direction at right angles to that of our former view.

And in four dimensions a point which is quite close to a plane can revolve round it without passing through it, thus presenting to us the appearance of vibrating across the plane, but not passing through it.

The appearance is as wonderful to us as it would be for a plane being to see a point which was in front of a line quickly passing behind it without having gone round the end. Such a point would appear to the plane being to vibrate across his line without passing through it.

Now if we stand in front of a mirror we see the image of ourselves. If we were to go round the mirror and take behind it the position which our image seemed to occupy, we should not be able to make ourselves coincide with it. In the mirror opposite to our left hand is the image of our left hand; but if we passed round, our right hand would be in the place in which we imagined we saw the image of our left hand. And thus we cannot make ourselves coincide with our image. But by a rotation in four-dimensional space we could put ourselves so as exactly to coincide with our image. This can be seen by referring to the case of the straight line, Diagram IV.

Let A B C be a triangle, and G a line. If A B C moves round the end of the line, it can take up the position A′ B′ C′; but it cannot anyhow be made to take the position shown in Diagram V., A′ B′ C′.

But if we move the triangle A B C out of the plane round the line G as axis, it will, in the course of its twisting round this axis, come into the position A′ B′ C′. It will come into this position when it has twisted half-way round. The point A, for instance, twists round in a circle lying in a plane which contains the direction A to A′, and the direction at right angles to the paper. Twisting half-way round in this circle, it becomes A′, and so on for the other points. Now a being who did not know what a direction was which lay out of the plane would not be able to conceive this twisting and turning movement. It would be as impossible for him to conceive the triangle A B C turned into the triangle A′ B′ C′, as it would be for us to suppose ourselves turned into the looking-glass image of ourselves by a simple twisting.

Yet just as a thing inconceivable to the plane creature can be done, so we could be twisted round and turned into our image. But this only holds theoretically; our relation to the æther is such that we cannot be so turned, or any bodies of a magnitude appreciable to our senses.

If we consider the case of a being limited to a plane, we see that he would have two directions marked out for him at every point of the rim of matter on which he must be conceived as standing. This is up and down, and forwards and backwards—the up being away from the attracting mass on which he is.

Now, if he were to realize that he was in three-dimensional space, but confined to a plane surface in it, his first conclusion would be that there was a new direction starting from every point of matter, and that this new direction was not one of those which he knew. This new direction he could not represent in terms of the directions with which he was familiar, and he would have to invent new terms for it.

And so we, when we conceive that from every particle of matter there is a new direction not connected with any of those which we know, but independent of all the paths we can draw in space, and at right angles to them all—we also must invent a new name for this new direction. And let us suppose a force acting in a definite way in this new direction. Let there be a force like gravitation. If there is such a direction, there will probably be a force acting in it; for in every known direction we find forces of some kind or another acting. Let us call away from this force by the Greek word ana, and towards the centre of this force kata. Then from every point in addition to the directions up and down, right and left, away from and towards us, is the new direction ana and kata.

Now we must suppose something to prevent matter passing off in the direction kata. We must suppose something touching it at every point, and, like it, indefinitely extended in three dimensions.

But we need not suppose it—this unknown—to be infinitely extended in the new direction ana and kata. If matter is to move freely, it must be on the surface of this substratum. And when the word surface is used it does not mean surface in the sense that a table top is a surface; it is not a plane surface, but a solid space surface. If from every point of a material body a new direction goes off, the matter which fills up the space produced by the solid moving in this new direction will have the solid it started from as its surface, and will be to it as a solid cube is to the square which bounds it on the top.

Now this body which extends thus, bearing all solid portions of matter in contact with its surface by every point of them, may be thick in the kata direction or thin.

If it is thick, then the influence of any point streaming out in radiant lines will pass as in all space directions, so out also in this new direction.

And then if its influence spreads out in this new direction, its effect on any particle near it will diminish as the cube of the distance; for, besides filling all space, it will have also to fill space extended in this new direction.

But we know that the influence proceeding from a particle does not diminish as the cube of the distance, but as the square of the distance.

Hence the body which, touching all solid bodies by every point in them, and supports them extending itself in the kata direction—this body is not thick in this direction, but thin. It is so thin that over distances which we can measure the influence proceeding from a body is not lost by spreading in this new kind of depth.

Thus the supporting body resembles, as far as we know it, a portion of a vast bubble. But moving on the surface of this bubble we can pass up and down, near and far, right and left, without leaving the surface of the bubble. The direction in which it is thin is in a direction which we do not know, in which we cannot move. But although we cannot make any movements which we can observe with our eyes in this direction, still the thin film—thin though infinitely extended in any way which we can measure—this thin film vibrates and quivers in this new direction, and the effects of its trembling and quivering are visible in the results of molecular motion. It only affects matter by its movement in directions at right angles to any paths which we can point to or observe, and these movements are minute; but still they are incessant, all-pervading, and the cause of movements of matter. It is smooth—so smooth that it hinders not at all the gliding of our earth in its onward path. Hence it does not transmit a direct pull or push in any direction from one particle to another; but by the twistings and vibrations of the material particles it is affected, and conveys from one to another these movements. Yet to bear up all matter, and thus hold it on its vast solid surface, it must be extremely rigid and unshatterable; and hence it cannot be permanently altered or twisted by any force proceeding from matter; but receiving from matter any push or twist, it is impressed with it for some distance; then, reasserting itself, it produces an image displacement or twist, and this image it transfers to the particles of matter which it touches.

Sometimes, as when light comes from the sun, this displacement and image is repeated and repeated innumerable times before at last we, receiving it, become aware of the origin of the disturbance.

But the properties and powers of this solid sheet—this film quivering and trembling, yet infinite and solid—are too many to begin to enumerate. The æther is more solid than the vastest mountain chains, yet thinner than a leaf; undestroyed by the fiercest heat of any furnace, for the heat of the furnace is but its shaking and quivering; bearing all the heavenly bodies on it, and conveying their influence to all regions of what we call space.

And by some mysterious action it calls up magnetism from electricity; by its different movements it gives the different kinds of light their being.

Of itself untrammelled and unclogged by matter, it vibrates and shakes with the speed and rapidity of the vibrations of light. But when matter lies on it—when air, even in its rarest condition, lies on it—its proper movement is damped and some of its quick shakings that are light, slow down to the obscure vibrations of heat. Thus of itself it will not take up the vibration of a hot body, but selects only those orbs which are glowing with radiant light wherefrom to take its thrilling messages. But when matter lies on it, it takes obediently the less vivacious movements of terrestrial fires.

A being able to lay hold of the æther by any means would, unless he were instantly lost from amongst us by his staying still while the earth dashes on—he would be able to pass in any space direction in our world. He would not need to climb by stairs, nor to pass along resting on the ground.

And such a being, even as thin as ourselves, and as limited, if not even in physical powers, but merely in thought he became aware of his true relation to the æther, he would see all things differently.

From all shapes would fall that limitation of thought which makes us see them differently to what they are; and in largeness and liberty of possible movement his mind would travel where ours but creeps, and soar and extend where ours journeys and diverges.

It is impossible in contemplating the rudiments of four-dimensional existence to prevent a sense of largeness and liberty penetrating even through the profoundness of our ignorance.

Whether we shall find beings other than ourselves, when we have explored this larger space, cannot be said.

But there is a path which holds out a more distinct promise.

When the conditions of life on a plane are realized it becomes evident that much of that which is to us merely natural—obvious from the very conditions of our life—could only be attained by beings on the plane as the result of artificial contrivances and modifications of their natural tendencies. In their progress and development they would, as it were, represent on the plane the features of the normal and undeveloped life of three-dimensional beings, and they would attain, as a result of moral labour and energy, a position which children in our higher life are born to without trouble or thought.

And so we in our advancing civilization may to the eyes of some higher beings represent in our arrangements and institutions an approach to the simplest matters of fact in their existence. We are separated from such a view by our bodily conditions, but we are not to be prevented from taking it with our minds.

By building up the conception of higher space, by framing the mechanics of such a higher world, we may arrive at a fairly accurate knowledge of the conditions of life in it.

And then, with that element in our thought, with the reasoned-out characteristics present to our minds of what life on a higher physical basis would be, we may be able to judge amidst conflicting tendencies with more certainty and calmness.

In one of the following papers of this series an account will be given of some of the facts which we can discern about the machinery and appliances of four-dimensional beings.

But the work of real discernment belongs to those who will from childhood be brought up to the conception of higher space.

_APPENDIX I._

A supposition can be made with regard to the æther which renders clearer an idea often found in literature.

This idea is that of the freedom of the will. If the will is free, then it must affect the world so as to determine chains of actions about which the mechanical laws hold true. We know that these mechanical laws are invariably true. Hence, if the will is an independent cause, it must act so that its deeds produce to us the appearance of a set of events determined by our known laws of cause and effect. The idea of the freedom of the will is intimately connected with the assertion that apparent importance, command of power, greatness and estimation, are outside considerations, not affecting the real importance and value of any human agent. These ideas can easily be represented using the idea of the æther as here given.

For suppose the æther, instead of being perfectly smooth, to be corrugated, and to have all manner of definite marks and furrows. Then the earth, coming in its course round the sun on this corrugated surface, would behave exactly like the phonograph behaves.

In the case of the phonograph the indented metal sheet is moved past the metal point attached to the membrane. In the case of the earth it is the indented æther which remains still while the material earth slips along it. Corresponding to each of the marks in the æther there would be a movement of matter, and the consistency and laws of the movements of matter would depend on the predetermined disposition of the furrows and indentations of the solid surface along which it slips.

The sun, too, moving along the æther, would receive its extreme energy of vibration from the particular region along which it moved, and the furrows of the intervening distance give the phenomena actually observed of our relationship to the sun and other heavenly bodies.

Thus matter may be entirely passive, and the history of nations, stories of kings, down to the smallest details in the life of individuals, be phonographed out according to predetermined marks in the æther. In that case a man would, as to his material body, correspond to certain portions of matter; as to his actions and thoughts he would be a complicated set of furrows in the æther.

Now what the man is in himself may be left undetermined; but he would be more intimately connected with the æther than with the matter of his body. And we may suppose that the æther itself is capable of movement and alteration; that it moulds itself into new furrows and marks.

Thus the old woman smoking a pipe by the wayside years ago, and whom I somehow so often remember, is not much different from me—we are both corrugations of the same æther.

Now our consciousness is limited to our bodily surroundings. Yet it may be supposed that in an action of our wills we, whatever we are (and for the present let us suppose that we are a part of the æther), we may be altering these corrugations of the æther. A single act of our wills, when we really do act, may be a universal affair with quite infinite relations. Thus it may be the immediate presentation to us of an alteration proceeding from us of all that set of corrugations which represents our future life; it may be the whole disposition and lie of events, which are prepared for the earth to phonograph out, being differently disposed. And it evidently is quite independent of the particular furrows in which such alteration first occurs. That long strip of æther which is a very humble individual may, by an act of self-configuration, affect the neighbouring long strips and produce great changes. At any rate the intrinsic value of the will is quite independent of the kind of furrows along which any material human body is proceeding.

_APPENDIX II._

It is a good plan in fixing our attention to give definite names to the directions of space. Let U stand for up. Then the up direction we will call the U direction, or simply U.

Then sideways, from left to right, we will call V, so that moving in the V direction, or moving V, means moving to the right hand.

Then the away direction we will call W, so that a motion which goes away from us we call a W motion, and its direction we call W.

Then any other direction which we suppose independent of these we will call the X direction. Now the simple push or displacement takes place in direction V, or left to right. It is turned into its image by turning in the plane U V—_i.e._, the plane of the paper.

The wave motion takes up the directions U V, and it can be turned into its image by a turning in the plane W V—_i.e._, by turning out of the paper, as if the paper were folded over about the dotted line. Then finally the twisting motion takes up the directions U V W, and can be turned into its image by being turned in the plane V X. That is, if each point is turned half-way round in this plane it becomes the corresponding point in the image twist. Thus on the supposition of the preceding pages, if a positively electrified particle could be turned in 4 space, it would become a negatively electrified particle.

_APPENDIX III._

It remains now to examine if the supposition that the particles of a wire are twisting in strings fits in with observed facts of electricity.

And firstly, if the particles are twisting in this manner, it is only reasonable to suppose that they would take up a little more room than they did when not subject to this movement—that is, the wire would become a little thicker. But its volume remaining the same, if it becomes thicker it must compensate for this thickening by becoming shorter. And it is found that a wire through which an electric current is sent tends to become shorter when the current comes into it.

Again, suppose a wire through which a current has been sent suddenly isolated. It has a twist in it, and will keep this twist. But if it is connected up with any other wire forming a complete circuit through which it can untwist itself, it will probably do so, and in untwisting would very likely overshoot the mark and become twisted in the opposite direction. Thus it would make a series of twists, each less than the last before becoming quiescent. And it is observed that a wire if so isolated does produce a rapidly alternating series of very minute currents before it comes to rest; just as if it were untwisting itself and overshot the mark each way many times before the electrical state has altogether disappeared.

The question now comes before us, How is it that a wire gets twisted? Through what agency is a current of electricity urged through a wire, or a twist put into it?

This is often done by means of an electrical battery. We will take a simple instance.

Suppose a dish of sulphuric acid, and a bit of carbon and a piece of zinc put into it. Then the carbon and the zinc are connected outside the liquid by a wire. Along this wire electricity will pass. Now the twist put into the wire must come from somewhere. And it is found that the sulphuric acid, which is a very lively compound, and contains a great deal of energy, becomes quieted down, and is quite different after the battery has finished working. On examination afterwards it is found to consist of sulphate of zinc.

Sulphuric acid can be looked upon as consisting of two bodies—hydrogen and a sulphur and oxygen compound. This sulphur and oxygen compound is called SO₄. Now the SO₄ comes to the zinc, and with zinc forms quite a dead compound, with little energy in it, called zinc sulphate, or Zn SO₄. The hydrogen, on the other hand, comes off at the carbon in an energetic state.

Hence evidently the SO₄ has given up its energy, the hydrogen has not. So the twist in the wire probably comes from the SO₄ and thus the twist is started at the zinc end, and runs round the wire from zinc to carbon.

At the same time we may suppose that an image twist, starting also from the zinc, runs through the fluid of the battery and then along the wire, till meeting the twist the two mutually unwind each other.

Thus the battery will be as if one had a loop of thread, and at one point twisted it between one’s finger and thumb. Twist and image twist, starting from this point, unwind each other on the opposite part of the loop. And if the loop is not joined, but the threads are held, each will become twisted with increasing tension till they can twist no longer. The objects which hold the ends of the thread, and prevent them twisting, represent insulators.

It is found that when a strong current of electricity passes through some water which has had a little sulphuric acid added to it, two effects take place.

In the first place some of the current passes through as through a wire. In the next place a part of the current is used up in producing an effect on the water. It splits the water up into two parts, each of them containing very much more energy than the water. One part is called hydrogen, and comes off at the wire which comes from the zinc, which we will call the zinc wire. The other part of the water comes off at the wire coming from the carbon, or at the carbon wire, and is called oxygen.

Let us now suppose that the twist of the zinc wire calls up in the molecule of water next to it an image twist. If it could pass on its twist at once, the water would form an ordinary conductor; but the water is not a conductor. Hence we suppose the same relation to hold good between the end of the zinc wire and the water molecules as between the zinc wire and any other body to which the twist cannot be communicated.

Now in the part of the molecule nearest the zinc wire an image twist is called up. And hence the molecule, being unable to twist as a whole, in the end of it away from the zinc wire a twist is produced. Thus the water molecule is strained into image twist and twist. Now let us suppose that by a powerful current it is wrenched in two. It is separated into a part having an image twist “hydrogen,” which comes off at the zinc wire, and into a part with the twist “oxygen.”

But this part with the twist calls up an image twist in the molecule next to it, wrenches it in two. Thus the oxygen of the first molecule separates up the next molecule into hydrogen and oxygen. The oxygen has a twist, the hydrogen an image twist. These twists run each other out, and leave an oxygen part free.

This oxygen part does the same to the next molecule, and so this action is transmitted through the whole body of the water till the carbon wire is reached—when, the oxygen part finding no other molecule to wrench asunder, is left isolated, and comes off in the form of gas.

Thus we see that oxygen and hydrogen would be bodies having in them twist and image twist—that is, that they would have an active rotation each of them; but the rotation would be different in the two cases, and such that if put together they would run each other out: the light and heat produced by the union of the two being probably the exhibition of the effects of this running out.

If we adopt the supposition, which seems most in accordance with facts, that there are in water two different elements occurring in distinct particles, the one called oxygen, the other hydrogen; and if, moreover, we suppose that these particles are perpetually changing places, and that each oxygen particle is sometimes linked with this hydrogen particle, sometimes with that, then it is obvious that the oxygen and the hydrogen in the water are in such a state that, if collected together separately, they would form liquid oxygen and liquid hydrogen; and the effect of the electric twist is to give them those active image rotations, or strains, which make them take the gaseous form, and assume that peculiar relation to each other which exhibits itself so strikingly in combustion.

With regard to magnetism, the same phenomenon of a particular state or disturbance of matter and its image state or disturbance is very strikingly obvious.

For take the case of a magnet. By the influence of an electric current passing round it, it can be turned into a magnet with opposite poles. That is to say, the small particles of the iron have been so shifted that, whatever their disposition was in the first case, they have now the reverse disposition. If we suppose the small particles to be magnets like the whole magnet, and all to have their north poles pointing in one direction, then after the action of the current they have their north poles pointing in the opposite direction. But they have not turned in space, for, if they were to turn, each must turn about some axis. But if there was some axis then, with regard to this axis, the magnetic influence would have a definite relation; the turning of the particles would take place in a certain plane, and there would be a certain plane in the magnet which would have special properties.

But a magnet is perfectly symmetrical in all its properties round its axis. The magnet which has had its poles reversed is, as an arrangement, the image of itself in its first condition. In the solid mass of iron which forms the magnet, by the action of electricity, a particular arrangement and its real image are alternately produced.

There are some very important electrical phenomena which have been left out of consideration altogether—namely, the repulsions and attractions exercised by electrified bodies.

Adopting the conceptions here laid down with regard to electricity—that the two kinds are in the relation of twist and image twist—we find that certain conclusions force themselves upon us.

A positively electrified body attracts a negatively electrified body.

A positively electrified body repels a positively electrified body.

Or, as it is put in a shorter form, one kind of electricity attracts the opposite kind, and repels the same kind.

Now, if our theory is true, a twist ought to attract its image twist, and repel a twist like itself.

And as far as can be observed it is always a fact that a movement of any kind taking place in a medium does attract its image movement, and repel a movement like itself.

Some very instructive experiments have been made with bodies suspended in water, and caused to pulsate or twist. It would be found, on referring to the details of these experiments, that if two spheres are pulsating or throbbing, so that the movements of the one _are_ at any instant what the movements of the other would _seem like_, if looked at in a mirror, then these two spheres will attract each other. If the one is a real copy of the other, then they repel each other. And this law holds good not only for throbbing movements, but also for twisting movements.

If now we supposed that what held good for movements held good for tensions of the same nature as the movements, these results would be in exact accordance with our suppositions. If a twisting movement attracts its image twisting movement, will a twist attract its image twist by means of its effect on the medium in which it is, and on which it exerts tension? This point must be left undecided.

Casting Out the Self.

The words which I have chosen as the title of this paper are the expression for a process which has been asserted to be one that occurs alike in our mental and in our moral life. It has so happened that in certain of my own inquiries I have applied this process; and the details may be of interest. But I must warn the reader not to expect any wide views on life, or far-reaching thoughts, or any of the warmth of human affairs. What I think about is Space; and it is the application of the principle of casting out the self in attaining a knowledge of Space about which I have something to say.

And, firstly, as a bit of absolute human experience is never without value, but that which we make up is often so, I may as well cast the fear of ridicule aside and enable the reader to take in, in a few lines, the exact commencement of my inquiry.

The beginning of it was this. I gradually came to find that I had no knowledge worth calling by that name, and that I had never thoroughly understood anything which I had heard. I will not go into the matter further; simply this was what I found, and at a time when I had finished the years set apart for acquiring knowledge, and was far removed from contact with learned men. I could not take up my education again, but although I regretted my lost opportunities I determined to know something. With this view I tried to acquire knowledge in various ways, but in all of them knowledge was too impalpable for me to get hold of it. And I would earnestly urge all students to make haste in acquiring real knowledge while they are in the way with those that can impart it; and not rush on too quickly, thinking that they can get knowledge afterwards. For out in the world knowledge is hard to find.

At length I came to find that the only thing I could know was of this kind. If, for instance, there were several people in a room, I could not know them themselves, for they were too infinitely complicated for my mind to grasp; but I could know if they were at right or left hand of one another, close together, or far apart. And the same of, to take another instance, botanical specimens in a book. I could not grasp the specimens—each was too infinitely complicated, and each part too infinitely complex—but I could tell which specimen was next which.

Accordingly, being desirous to learn something thoroughly, and since, in the arrangement of any different objects, there was such a lot of ignorance introduced by the objects being different—each bringing in its own ignorance and feeling of bewilderment—I determined to learn an arrangement of a number of objects as much alike as possible.

Accordingly I took a number of cubes, which were as simple objects as I could get, arranged them in a large block, and proceeded to learn how they were placed with regard to each other. In order to learn them I gave each of them a name. The name meant the particular cube in the particular position.

Thus, taking any three names, I could say, about the three cubes denoted, how they were placed with regard to one another: one, say, would be straight above the first with four intervening, the third would touch the second on the right hand, or some similar arrangement.

Now in this way I got what I conceived to be knowledge. It was of no use or beauty apparently, but I had no reason to use it or to show it.

It is about this bit of knowledge that I want to speak now—a block of cubes, and the cubes are known each one where it is.

Sometimes I have been tempted to call this absolute knowledge, but have been reminded that I did not know the cube itself. Against this I have argued. But in argument we say many things which we do not understand, and my conclusion is, on the whole, that the objection is well founded. Still, if not knowledge absolute, the knowledge of this block approaches more nearly to knowledge absolute than any other with which I am acquainted, because each cube is the same as its neighbour, and instead of an arrangement of all sorts of diverse ignorances we have only one kind of ignorance—that of the cube. Each of the cubes was an inch each way, and I learnt a cubic yard of them. That is to say, when the name of any cube was said, I could tell at once those which it lay next to; and if a set of names were said, I could tell at once what shape composed of cubes was denoted. There were 216 primary names, and these, taken in pairs, were enough to name the cubic yard.

For the practical purpose of this paper, however, it will suffice if the reader will imagine a block of twenty-seven cubes, forming a larger cube, each cube being denoted by a name (see Diagram I. below). Then it is evident that two names mean a certain arrangement consisting of two cubes in definite places with regard to one another—three names denote three cubes, and so on. And I would ask the reader not to mind taking a little trouble at this point, and to look at the diagram for a little while. If there is anything about which we can form perfectly clear ideas, it is a little heap of cubes. And if the reader will simply look at them for a little space of time, he will realize clearly every word of what I have to say; for I am going to talk about nothing else than this little block of cubes.

Thus, looking at the cube with the figure 1 upon it, this numeral will serve for the name of the cube, and similarly the number written on every cube will serve for its name. So if I say cubes 1 and 2, I mean the two which lie next to each other, as shown in the diagram; and the numbers 1, 4, 7, denote three cubes standing above each other. If I say cubes 1 and 10, I mean the first cube and one behind it hidden by it in the diagram.

Now this is the bit of knowledge on which I propose to demonstrate the process of casting out the self. It is not a high form of knowledge, but it is a bit of knowledge with as little ignorance in it as we can have; and just as it is permitted a worm or reptile to live and breathe, so on this rudimentary form of knowledge we may be able to demonstrate the functions of the mind.

And first of all, when I had learnt the cubes, I found that I invariably associated some with the idea of being above others. When two names were said, I had the idea of a direction of up and down. But with regard to the cubes themselves, there was no absolute direction of up or down. I only conceive of an up and down in virtue of being on the earth’s surface, and because of the frequent experience of weight. Now this condition affecting myself I found was present in my knowledge of the cubes. When certain of the names were said, I conceived of a figure having an upper part and a lower part. Now, considered as a set of cubes related to one another and not to me, the block had nothing to do with up and down. As long ago as Ptolemy, men have known that there is no such thing as an absolute up and an absolute down. And yet I found that in my knowledge of the set of cubes there was firmly embedded this absolute up and this absolute down. Here, then, was an element arising from the particular conditions under which I was placed, and the next step after recognizing it was to cast it out. This was easily done. The block had to be turned upside down and learnt over again with the cubes all in their new positions. It was, I found, quite necessary to learn them all over again, for, if not, I found that I simply went over them mentally the way first learnt, and then about any particular one made the alteration required, by a rule. Unless they were learnt all over again the new knowledge of them was a mere external and simulated affair, and the up and down would be cast out in name, not in reality. It would be a curious kind of knowing, indeed, if one had to reflect what one knew and then, to get the facts, say the opposite.

It may seem as if, when the cubes were known in an upright position, they would be easily imagined in an inverted position. But practice shows that this is very far from being the case. It requires considerable mental effort to determine the alterations in position, and to get an immediate knowledge requires a considerable time.

It may seem as if it were a dubious way of getting rid of gravity, or up and down, just to reverse the action of it.

But this way is the only way, for we, I have found, cannot conceive it away; we have to conceive it acting every way, then, affecting each view impartially, it affects none more than another, and is practically eliminated.

The cube had not only to be turned upside down, but also laid on each of its sides and then learnt. There were a considerable number of positions, twenty-four in number, which had to be brought close to the mind, so that the lie of each cube, relative to its neighbours and the whole block, was a matter of immediate apprehension in each of the positions.

If a single cube be taken and moved about, it will be found that there are twenty-four positions in which it can be put by turning it, keeping one point fixed, and letting each turning be a twist of a right angle. The whole block had to be turned into each of these positions and learnt in each.

Thus the block of cubes seemed to be thoroughly known.

At any rate, up and down was cast out. And we can now attach a definite meaning to the expression “casting out the self.” One’s own particular relation to any object, or group of objects, presents itself to us as qualities affecting those objects—influencing our feeling with regard to them, and making us perceive something in them which is not really there.

Thus up and down is not really in the set of cubes.

Now these qualities or apparent facts of the objects can be got rid of one at a time. To cast out the self is to get rid of them altogether.

As soon as I had got rid of Up and Down out of the set of cubes I was struck by a curious fact.

If in building up the block of cubes one _goes_ to the left instead of to the right, keeping all other directions the same, a new cube is built up having a curious relation to the old cube. It is like the looking-glass image of the old cube. Every cube in the new block corresponds to every cube in the old block, but in the new figure it is as much to the left as before it was to the right. And any set of names in the block so put up gives a shape which is like the shape denoted by the same set of names in the old block, but which cannot be made to coincide with it, however turned about. It is the looking-glass image of the old shape. The one block was just like the other block, except that right was changed into left. Now, was it necessary to cast out right and left as had been done with up and down? or was right and left, as giving distinctions in the block and in shapes formed of cubes, to remain? It seemed as if right and left belonged more to me than to the set of cubes. And yet the right-handed set of cubes could not be made by moving about to coincide with the left-handed set of cubes. And this power of coincidence was the test which had convinced me of the self nature of “Up and Down.”

Let Diagram I. represent a small block of cubes. It is itself in the form of a cube, and it contains 27 cubes. For purposes of reference we will give a number to each cube, and the number will denote the cube where it is.

In the front slice are cubes numbered from 1 up to 9, in the second slice are cubes numbered from 10 to 18, and so on. Thus behind 1 is the cube 10. This cube and the cube 11 are hidden, but the cube 12 is shown in the perspective.

Now in this block of cubes there is a part which is known and a part which is unknown. The part which is known is how they come or the arrangement of them. The part that is unknown is the cube itself, repetitions of which in different positions forms the block.

The cube itself is unknown, because, being a piece of matter, it possesses endless qualities, each of which grows more incomprehensible the more we study it. It is also unknown in having in it a multitude of positions which are not known. The cube itself is, amongst other things, a vastly complicated arrangement of particles. Hence, _putting all together_, we are justified in calling the cube the unknown part; the arrangement, the known part.

The single cube thus is unknown in two ways. It is unknown in respect to the qualities of hardness, density, chemical composition, &c. It is also unknown as a shape. If it really consisted of a certain number of parts, each of which was clear and comprehensible in itself, then we should know it if we grasped in our minds the relationship of all these parts. But there are no definite parts of which a cube can be said to be made up. We can suppose it divided into a number of exactly similar parts, and suppose that all are like one of these parts. But this part itself remains, and the problem remains just the same about this part as about the whole cube.

Now there is a double perplexity: one about the nature of the matter, the other about the cube as to the arrangement of its parts. We will give up any question about the matter of which the cube is composed; to know anything about that is out of the question. But, supposing it to be of some kind of matter, it presents an inexhaustible number of positions. It can be divided again and again.

Let us look at the block again, and for the moment dismiss from our minds the question just raised as to the single cubes of which it is built up. Let us look on each of these cubes as a unit. Then two of the units, taken together, form a shape; three or five of them would form a more complicated shape, and so on.

We can also suppose the cubes away, and think merely of the places which they occupied. In this manner, by first thinking of the 27 cubes, and then simply by keeping the places of them in our minds, we get 27 positions, and in these positions we can suppose placed any small objects we choose. Each of these positions may be called a unit position, and we can form different arrangements of small objects by putting them in different ones of these positions. Now in all this we do not divide the cube up. We simply think of it as a whole—we think of it as a unit. Or if we take the room of the cube instead of the cube, and think of the place it occupies, which I call a position, we do not divide that position up. We take it, if I may use the expression, as a unit position. And _without asking any question as to the nature of these positions, whether they are complicated ideas or not_, we have a kind of knowledge of the whole block, in that it consists of this collection of 27 cubes, or of this set of 27 positions.

Thus in a rough and ready manner there is something which we can take. If we do not inquire about one of the cubes itself, we are all right; that being granted we can know the block.

But if we look into what each of these unit cubes, or what each of these unit positions is, we find quite an infinity opening before us. There is nothing definitely of which we can say that the whole unit cube is built up, and each of the positions has a perfectly endless number of positions in it, if we come to examine it closely. All that we can say is that our ignorance about each of the unit positions is of the same kind as our ignorance about every other, and, taking one as granted, we may as well take the 27 as granted; and so out of a lot of similar ignorances we get a kind of knowledge of the whole. And this knowledge is not a mere indefinite thing, but it can be worked at, improved, and made perfect after its kind. For suppose we limit ourselves to the 27 positions numbered in Diagram I. Two of these positions form one shape, three of them will form another shape, and so on. And in going over each of these arrangements we gradually get to know the whole set of them which form the block.

Having given up for the time any question as to the possible subdivisions of the cube, and looking on each cube as a unit position, we have 27 positions. These positions can be taken in different selections, and each selection is a shape. To know the block or set of positions means to form a clear idea of every shape, consisting of selections of positions, which can be formed out of the 27.

But each of the cubes, 27 of which form the whole block, can be divided up. Each of these cubes contains a great many positions. There must, for instance, be positions in each cube for every one of its molecules.

Thus it is evident that the cube supplies an inexhaustible number of positions to be learnt. I call the cube unknown in the sense that there are a great number of positions in it which are not clearly realized by the mind.

By a very simple device it is possible to penetrate a little into the unknown part. The whole set of cubes forms a cube. Let us consider the small cube to be a model of the whole cube. Let us consider it as consisting of 27 parts, each related to the other as the 27 first cubes were related amongst themselves. Thus the unknown part, the material cube, which is used to build up the whole, becomes reduced in size. Diagram II. represents such a cube.

This is the theory. The practical work consisted in learning the names denoting these smaller cubes in connection with their positions, so that, the names being said, the small cubes meant were present to the mind, and a set of names being said, the shape, consisting of a set of cubes in definite relations to each other, came vividly before one. A complete knowledge of the block of cubes would be a complete appreciation of all the possible shapes which selections of the cubes would form, and this I strove to attain. Here at length I found real knowledge, and after a time I was able to reduce the size of the unknown still further, and to obtain a solid mass of knowledge fairly well worked all through.

And now it all seemed satisfactory enough. There was real knowledge in knowledge of the arrangement; and the material cube, which must be assumed, could be made smaller and smaller, it could be turned into knowledge, thus affording a prospect of obtaining endless knowledge. Thus I found the real home of my mind, the only knowledge I had ever had, and I hoped always to continue to add to it, and always to reduce the unknown in size.

Presently, too, the forms of the outward world began to fall in with this knowledge; and as the mass of known cubes became larger in number, a group of them would fairly well represent a wall, a door, a house, a simple natural object such as a stone or a fruit.

Yet amidst all this delight I became conscious, dimly enough, of a self-element in the knowledge of blocks.

If, putting up the block of cubes, we go to the left instead of the right, but in all other respects build up in the same way, we obtain a block which has a curious relation to the first block.

The ordinary block is shown over again in Diagram III. Diagram IV. is the new block. The new block is like a looking-glass image of the old block. It is just the same, but that left and right is reversed.

Also, if we take selections of blocks we get figures which are just reversed. Thus 1, 4, 7, 8, in Block III., means a figure turned to the right; in Block IV. a figure turned to the left.

Again, consider the two figures formed by selecting the cubes 1, 4, 7, 8, 17, from Diagrams III. and IV. respectively. We get two figures which are just like one another as arrangements, but which we cannot turn into one another by twisting.

Considered as arrangements in themselves, these figures and these blocks seem to be identical, for the relationships of cube to cube which are present in the one are all present in the other. But considered as shapes they are not identical. For they will not coincide.

The whole matter becomes much more clear if we consider the relationship between the individual cube used and the block which it forms.

There are two starting-points, either of which we can adopt. We can start with the real material cube, or we can start with the act of arranging. When I speak of the real material cube I do not want to call attention to the kind of matter of which it is composed, or to the nature of matter, but to the fact that it is to be a real cube such as can be made, and which, if one edge or corner be marked, will retain that mark just where it is—a cube which is not a product of the imagination, but an object, with the properties of objects in general.

Let us start with the real material cube. Let us take the cube shown in Diagram V., which is the model on a small scale of the Block III. The numbers in it show the small cubes of which we suppose it to be built up after the pattern of Block III. The numbers also serve to show the distinction of positions—that is, we can refer to the right-hand corner or edge, &c., by saying the numbers of the small cube which lies there.

Now, using the cube of Diagram V. to build up the block in Diagram III. we get a perfectly orderly result, as shown in Diagram VII., and we can go to bigger and bigger blocks, or down to smaller and smaller ones without any hitch. But if we use the cube of Diagram V. to build up the block of Diagram IV., there is a disadjustment which can be discerned in Diagram VIII. Thus, when V. is used to build up III., the small cubes in V., 1, 4, 7, lie in same edge as the cubes 1, 4, 7, in the big Cube III. But when V. is used to build up IV., the small cubes 3, 6, 9, lie on the edge which is occupied by the cubes 1, 4, 7, in big Cube IV.

Thus, if the same material cube is used, there is a disadjustment, and the figure IV. cannot be considered the same as the figure III. even as an arrangement, for the same parts of the cubes do not lie in an analogous manner. A certain corner of Cube V. is marked with the figure 7; this corner would be on the outside in Block III., but in building up Block IV. it would lie on the inside.

It is somewhat difficult to express this fact, but if the real cubes are looked at it becomes perfectly obvious.

Imagine the whole Block III. to be built up of a number of cubes, every one of which is alike. If the sides of these cubes be distinguished by any markings—if, for instance, the left-hand side is blue and the other sides are each of some special colour, then on building up the whole block the left-hand side of the whole block will be blue.

If, now, the same cubes be taken, and the attempt be made to build up the looking-glass image of the block with them, it will be found that there will be a disadjustment. If the blue sides are made to go to the right, as they must, to form an image block, then some other sides will be in different places to what they should be in order to produce an image of the original block. Although considered as an arrangement of cubes the new block will be an image of the original block, still, looking at the individual cubes of which it is composed, it will be seen that the new block is not an exact image of the old block.

If, however, we take the other starting-point, and, not assuming any fixed fundamental cube, look only at the act of arrangement, the two Blocks, III. and IV., are found to be identical in every internal relationship.

For, taking the act of arrangement as the basis, if, when we have built up the Block IV., we look upon each of the cubes as an arrangement of the same kind as the whole, then the cube 1 in Diag. IV. is represented in Diag. VI. And it is evident that if Diag. IV. is built up out of cubes like Diag. VI., the small cubes, 1, 4, 7, lie in the same edge as the cubes 1, 4, 7, in Diag. IV. Thus it will be found for every relationship in Diag. III. there is an exactly similar relationship in Diag. IV.

In this case if, for the sake of material illustration, we use marked cubes, it seems that we must not suppose each particular cube to have a fixed marking of its own, but that we must suppose the markings to spring up on the sides of the cubes in accordance with the places into which they are put.

There is another manner of regarding the matter which may help to bring out the point at issue.

If we suppose that we are putting up the cubes in one room while another person is putting up cubes in an adjoining room; if we can tell him what we are doing, using the words right and left, he will be able to put up a block exactly like ours. But if we do not allow ourselves to use the words right and left, but speak to the other person as if he were simply an intelligence without having the same kind of bodily organization as ourselves, we should find that, supposing he could put up the block of cubes, it would be a mere matter of chance whether he had put up the block as we had put it, or whether he had put it up in an image way. And the same with regard to any shape. We could tell him that the cubes should be put together, and we could tell him the relationship which they should have with regard to one another; but the figure he put up would just as likely be an image of our shape as not.

And we could go on for ever building more and more complicated shapes and telling him to do the same, and no hitch or difficulty would come. But at the end all his shapes might be ours just reversed, as if seen in a mirror.

And if, having put up the block, we coloured the sides of the cube we used as the fundamental cube, and told him how we had coloured it: if he coloured his and brought it to us, and we compared them, his would just as likely be the image of our cube, and not able to be turned into it. So that although, as arrangements, the structures we had put up were alike, still neither of us could use the other’s fundamental cube; and if we exchanged the fundamental cubes there would be an inconsistency in each of our arrangements.

Now, are these blocks of cubes really the same? Are III. and IV. really the same in themselves, as all relationships in the one are to be found in the other? If so, the feeling on my part that they are different, and the inconceivability of their coinciding, must be due to some self-element which is mixed up with my apprehension of the cube.

The Block IV. is like the Block III. in its known part—in its arrangement. It is unlike Block III. in its unknown part—the cube which must ultimately be supposed as the fundamental cube, by using which over and over again the whole is built up.

Now, the properties of the unknown part—the little cube of matter which of some size or another, we must assume, are so mysterious that one does not feel any argument very safe which rests on it.

Moreover, there is a very obvious consideration which reduces the importance of the part played by the material cube very considerably.

It is possible to consider the Cube V., which is used to build up III., as the total of 27 cubes.

But each of these cubes—the small cubes in Diag. V.—can be considered to be made up of 27 still smaller cubes.

By going on in this way we can get our fundamental cube very small indeed. The difference between the Cubes III. and IV., in respect to this fundamental cube, will still remain. But omitting this difference they will be, considered as arrangements, identical.

To state the matter over again. We start with a real cube, one inch each way, and build up the block in Diagram III. with it. If we try to build up the block in Diagram IV. with this same inch cube, we find that there is a disadjustment.

But we are not obliged to have our fundamental cube one inch in size. We can take it as small as we like, and build up the block, using a greater number of such cubes. We can take it the twenty-seventh of the twenty-seventh of an inch cube; or, in fact, as small as ever we like. And if we take a very small cube as the fundamental one with which we build up the Block III., then, using this same fundamental cube to build up Block IV., we should find a disadjustment, although this disadjustment would only come in when we come down to the very minute cube, and studied its relationship to the whole Block IV.

Thus, apparently, the Block IV. could never be built up consistently, using as its fundamental cube the fundamental cube out of Block III. But in saying this we have really made an assumption.

It is obvious that Cubes V. and VI., just like Cubes III. and IV., considered as shapes made up of matter, are very different, and could not be shifted one on to the other.

But all our laws and feelings about movements and possibilities are founded on the observation of objects having a certain degree of magnitude.

But the fundamental cube, which we must assume, may be supposed to be of a degree of magnitude less than any known degree.

In cubes of a certain size V. and VI. are different, and cannot be made to coincide.

But we are absolutely unable to say anything about cubes beyond a certain degree of smallness. With cubes of a certain degree of minuteness, V. and VI. might be able to be made coincide.

Thus, for instance, we feel as if we could divide a piece of matter on and on for ever. But chemists tell us that, after a certain number of divisions, the next division would split it up into two different kinds of matter. Since all our reasoning is founded on the behaviour of objects of known size, we can tell nothing at all by inference about the behaviour of very small objects.

It is obvious that, from our customary experience, we can assert absolutely nothing at all about the extremely minute or the extremely large. All reasoning which is founded on the likeness between the extremely small and the ordinary objects of our observation is absolutely valueless as telling us any truth.

Of course, by saying this we have not got rid of the argument for the difference of III. and IV. But we have put the thing from the observation of which that argument is drawn out of the region of known things. We have put it into the hazy land of the extremely minute. Its argument is good, but it depends on its being of a certain size. We suppose it less than that size, and we can consider the subject without regard to its argument.

The question then before me was, Is “Right and Left” to be cast out? And connected with this was the consideration of whether it was possible for extremely minute cubes to be “pulled through,” that is, to be treated somehow which would turn one like V. into one like VI.

Now, if “right and left” was a self-element, it could be cast out; if it was a permanent distinction in the cubes themselves, it could not be cast out. The thing to do was evidently to try. The method was to learn the cubes over again, in a set of new positions. For every one of the ways in which they were learnt before, there was an inverted or pulled through way to be learnt.

While I was engaged in this attempt another inquiry suddenly coincided with this, and explained it all.

Much has been said about the fourth dimension of space and the inconceivability of it to us. Now, if there are beings who live in a four-dimensional world, they must feel as habituated to it as we do to ours, and the conceptions which seem so impossible to us must be every-day matters to them. It would be impossible for us to try to enter at once into the serious thoughts of these denizens of higher space. But amongst them there would probably be some with whose occupations we might become familiar, and with whose ideas we might gain some acquaintance. Amongst these beings there must be children, and just as children on the earth gain their familiarity with space by means of bricks and blocks and toys, so these higher children must have their own simple objects wherewith they grow into familiarity with their complex world.

Now it is easy to make a set of simple objects such as these higher children would use. And it seemed a practical thing to do with regard to the conceivability or inconceivability of the fourth dimension to give the matter a fair trial, by going through those processes and those experiences which must be gone through by the beings in higher space to gain their acquaintance with it.

When I say that it is easy to make a set of objects, such as the higher children use, I do not mean to say that they can be made completely in every part at once. But we can make the ends and sides of them, and we can look at the ends and sides of them as they appear to us in space, and we can make up exactly what sides come into space when the simple objects are twisted and moved.

Just as a being living on a plane could tell about all the faces and edges of a cube or other simple solid figure by looking at what he could see when the cube was laid on his plane, and when it was twisted and laid down again; so we can tell all about the sides, faces, and edges of a higher solid.

And the project seems less uninviting if we reflect on how complicated a matter the formation of our own conceptions of a solid are. What a lot of faces and edges a cube has! And, moreover, it must be remembered that we never touch or see a solid; we only see the surface and touch the surface. If we cut away the surface that we first saw or touched, we come on another surface, and so on.

Now, of course, the surfaces of a solid are given to us by nature in their right connection and relation. Each of the edges of the cube, for instance, can be noticed and remarked without any difficulty, and they are all on the same bit of space, to be looked at one at the same time as another.

But the sides, faces, and edges of a higher solid cannot be in our space all at once. They must come separately, be looked at one by one.

Thus a being in a plane could not see the lower side and the front of a cube at once. He would first have to look at the lower side as the cube rested on his plane, then if the cube were turned over he would see the front, and the lower side would be gone. If he got the set of right appearances which a cube would present to him when, turning about in a systematic way, it came at intervals into his plane, and if, moreover, he fixed his mind on these appearances, he might at last, if it was in him, rise to the conception of a cube as we know it.

Now, the parts by which a higher solid comes into our space are solids, and what we have to form is a set of solids coming and going in a systematic way, as the higher figure is moved about in a systematic way.

This afforded a welcome exercise, for conceiving the solid shapes, and how they went and came, increased my familiarity with the set of cubes.

Moreover, in trying to get the piece of ignorance—the necessary real cube—as small as possible, I had got the block which I knew to a somewhat fine state of division, and could, by picking out a particular set of cubes from the whole number, obtain a mental model of any shape I wanted. The whole block of cubes formed a kind of solid paper in which one could mentally put down any solid shape one wanted. And just as it is a great convenience to have a piece of paper for drawing figures one wants to think about, so it was a great convenience to have this solid paper.

The subject, however, abounds in abysses for stupidity to fall into, and I had to clamber out of each of them; so it took me several years before I got quite on the right tack. Then it was easy enough: any one in a few weeks could learn to conceive four-dimensional figures. Not only is it easy, but there are abundant traces that we do it continually without being aware of it. I am sure if the loveliness of the work while one is doing it, and the simplicity and self-evident nature of the results when obtained, were generally known, it would be a favourite amusement.

Now one of the first things that presented itself to my attention when I began to move the four-dimensional figures about was a fact which bore curious reference to my difficulty about the fundamental cube. If the reader remembers, it seemed to me as if the cube out of which the whole block of known cubes was built ought to be able to be inverted. That is to say, it seemed to me that there was a self-element present in my knowledge of the cubes. But in order to cast out that self-element the fundamental cube which lay at the basis of the whole block would have to be able to be inverted, or pulled through.

Now I found that when I took a four-dimensional figure which came into space by a cube—that is, a figure which rested on space by a cube, or one of whose sides was a cube—when I took a figure of that sort up in the fourth dimension and twisted it round and brought it down again, this cube would sometimes be inverted or pulled through—although I had done nothing to it, but had simply twisted the whole figure round without disturbing the arrangement of its parts.

Thus evidently to a higher child it would be no more difficult to invert or pull through a cube or a figure than it would be to me to twist one round.

Hence it was obvious that right and left was really a self-element in my block of cubes. I being in our space was under a certain limitation, and that limitation made me feel as if a right-handed arrangement was different from a left-handed arrangement.

A being who was not limited as I was would see that they were one and the same. Hence, in knowing the set of blocks it was necessary to cast out “right and left,” and the names had to be learnt over again in new positions.

Thus it is evident that there are three expressions which may be considered in reference to a knowledge of a block of cubes as almost identical: “Casting out the self”—“Seeing as a higher child”—and thirdly, “Acquiring an intuitive knowledge of four-dimensional space.”

Thus, taking the simplest and most obvious facts—the arrangement of a few cubes—we found that there was a known part and an unknown part; the known part corresponding to our act of putting, the unknown part the cube which, of some size or another, must be taken as given in the external world. Then there was obviously a self-element present in the Up and Down felt as in the cubes. This being removed, Right and Left had also to go. So, to get the knowledge of this simple set of objects clear of self-elements, two universe transforming thoughts have to be used; and when these thoughts are thus incorporated the cubes become different.

It will be obvious to the reader that in these pages I have merely touched the surface of the subject. But the deeper matters which are contained in the knowledge of a block of cubes are difficult to express, and are so mixed up with the practical work, as far as I conceive them at present, that it is best to consider in some detail the applications to the world about us of those truths of which we have already got a clear apprehension from the block of cubes.

Instead, then, of going on, let us conclude the present paper by going back, and taking a simple instance of the general truth that progress in the knowledge of a block of cubes is casting out the self.

Let the reader turn to Diagram I. and make out the shape which the following numbers denote—namely, 1, 4, 5. If the following numbers be said, 18, 27, 26, it will be found that they denote the same shape, but in a different position. Now if the block of cubes be well known, these two sets of names, 1, 4, 5, and 18, 27, 26, ought to convey instantly to the mind the same idea. However quickly they are realized, it ought to be evident that they are the same shape.

And a good deal of the practical work in learning a block of cubes consists in gaining this faculty of immediate apprehension. But when it is gained it is seen to consist much more of getting rid of an imperfection than in being any real advance. For if the two shapes are identical we need not ask ourselves how it is we see them as the same, but we have to ask ourselves what is the reason why we do not recognize their identity; and the answer evidently is that, if we do not recognize their identity, it is due to the particular relationship of each shape to ourselves. One is down on our left hand, another is up on our right, and they are turned relatively to us different ways. Now these differences, which are merely relative to us, we impress upon the shapes, and really feel the shapes to be different. The practice consists in getting rid of the influence of these self-elements, so that two shapes, however complicated, being alike, when their names are said, we feel them to be alike without calculation or reflection. Thus the power of seeing likeness and analogy in this domain is merely another name for the power of casting out the self-elements from our mental presentation of any objects with which we come into contact.

Footnotes

[1]A B C D framework, X and Y two lines interlinked.

[2]_See_ Appendix.

[3]For details, see Appendix III.

Transcriber’s Notes

--Retained publication information from the printed edition: this eBook is public-domain in the country of publication.

--Silently corrected a few palpable typos; left intentionally nonstandard usage unchanged.

--In the text versions only, text in italics is delimited by _underscores_.

--In the text versions only, superscript text is preceded by ^caret.

--In the text versions only, subscript text is preceded by _underscore.

End of Project Gutenberg's Scientific Romances (First Series), by C. H. Hinton