Now this is what actually happens.

Thus we have, firstly, a glass rod with its twist; secondly, a mass of

metal with two twists on it—one near the glass rod, and of the image

kind; the other at the other end of the mass of metal, and related to

the original twist in the following way. It is the image of its image.

Now it will be found by using a mirror that the image of the image of a

twist is the twist itself.

Hence on the other end of the conductor there is a twist of the same

kind as that on the glass rod.

And it is obvious that the rod, with its twist, is connected with the

twist on the conductor nearest to it, and the twist on the other end of

the conductor will, by the same arbitrary process which we have assumed

as real, call up a twist, the image of itself, on any object near it.

If it is touched by the object with this contrary twist the two will run

together, and the conductor will, if it is left free, have only one

twist—the silk kind, which has come up opposite the rod.

If now the rod be moved away, the conductor will be twisted as a whole;

that is, all the particles of which it is composed will be slightly

twisted with a twist of the silk kind.

In this state it is said to be charged.

Thus the assumption which we have made, that there is some process in

nature in virtue of which, opposite to any twist, its image twist is

produced as a real thing—this assumption is in harmony with the laws of

induction.

Instead of working with a glass rod it is more convenient to use a metal

rod. Suppose we take a poker and attach a handle of sealing wax to its

middle. See A B below. This will be easily imagined, and its two ends,

the handle and the black end, will be easily retained in the mind.

If now electricity be communicated from the glass rod to the poker by

touching the two together, what happens is this: The particles of the

glass being twisted communicate their twist to the particles of the

poker. The twist on the poker is of the same kind as the twist on the

glass rod, and the amount of twisting which the glass particles had is

divided between the glass rod and the poker. The use of the sealing-wax

handle is to keep the twist from communicating itself to the body of the

person holding it, and, through him, to the earth. It is found that

certain bodies, “non-conductors,” will not communicate the twist and

convey it along; whereas metallic bodies, and conductors generally, will

communicate this twist at once to great distances.

We will suppose that a metallic body consists of particles so arranged

together that it easily acts as a set of minute threads or chains of

particles which will twist, each thread or chain twisting as a whole.

Thus the conception which should be formed of a metallic body conducting

electricity along it is this:—Conceive a bundle of very fine but very

rigid wires, each wire twisting separately but with the same kind of

twist as all the others, and each, as it twists, rotating amongst its

fellow threads. If we have a metal rod we can twist it between the

finger and thumb. This is not the kind of twist we suppose, but that

each separate string of particles is thus twisted, so that each set

twisting remains in the same part of the metal rod—but is turning round

in its fixed position. This is a body conveying an electric current. If

the current will not pass, the set of minute wires must be conceived as

held at the far end, and given a twist, starting from the point where

the electricity is communicated. Now if a conductor is thus charged and

left, it is found that it retains its charge; to be discharged it must

be touched with another conductor. Hence this twist of minute threads

differs from a twist of a wire in that the threads cannot untwist of

themselves unless other threads come into contact with them to which

they can impart the twist. That this should be the case may depend on

the fact that the twisting strings are strings of molecules, and the

ends of them would thus be connected with other molecules with something

of the same tenacity as that with which the strings themselves cohere

together, and are unable to unlock themselves from these insulating or

untwisting molecules.

Let us consider the state due to these twisting strings of particles.

Place two pennies lying on the table before you, and suppose them to be

the sections in which two strings of a conductor are cut across, so that

you are looking at two particles, represented by the pennies in the

interior of a conductor; the strings, of which the pennies are sections,

come up towards your eye. Now twist the two pennies each in the same

direction—say that of the hands of a watch. From the outer edges you can

take the motion off; the edges are moving in the same direction. But

where the two pennies meet you will see that the edge of each is going

in a contrary direction to the other. And if one penny tends to move an

object in one way the other tends to move it in the contrary direction.

Hence these motions tend to neutralize each other in the interior of a

conducting wire.

Having now formed a conception of the state of the particles in an

electrified poker, suppose another poker likewise held by an insulating

handle is brought near the first. Let the pokers be so arranged that the

handles both point one way, the black ends another way, and let the

second poker be in the same line as the first, with its handle towards

the black end of the first.

Now the first poker is charged, it contains electricity, its particles

are twisted. What effect will it have on the second poker?

It is found that the second poker undergoes a certain change, but when

it is removed to a distance from the first poker all trace of this

change disappears.

On the end nearest the first poker—on the handle—is found silk

electricity; on the end furthest from the first poker—on the black

end—is found glass electricity.

A B C D

--- ---

+ + - +

Let A B be one poker, the + representing the charge of glass

electricity. Let C D represent the other poker, the - representing the

induced silk electricity, the + the glass electricity in it.

Let A be the handle of the first, B its black end. Let C be the handle

of the second, and D its black end. To explain this let us bring in our

imaginary principle. Let us suppose that when a charged body is brought

near an uncharged body, but is separated from it by some medium through

which electricity cannot pass—let us suppose that by some agency the

twist in the charged body calls up an image twist in the body opposite

it. Thus, due to the twist in the first poker there will be an image

twist in the handle part of the second poker.

But the strings of particles in the second poker are not twisted as a

whole; they are twisted in such a way that if they are removed from the

first poker, the twist, whatever it be, disappears.

Now this would obviously be effected if we suppose the same thing to go

on in the second poker as took place between the first poker and the

second. Let the twist in the handle end C of the second poker be

accompanied by the production of an image twist in the black end D. And

let us take this as a fair account of our observations. If a body, which

as a whole does not undergo a twist, has one part of it twisted, then

there will be the image twist in the other part of it. I say that the

poker as a whole is not twisted, and all that this means is that if it

be removed from the electrical influence it is found to be not charged;

and the idea which we may form is this: the strings of particles are

twisted like the two strips of paper round the pencil; they are twisted

so that they will exactly unwind if left alone.

Now of course all these suppositions are merely provisional, and must be

dismissed unless seen to be mechanically possible; but for the present

we are trying to see if our assumption will fit in with facts. And our

assumption is that there is in nature a power which amongst the

molecules produces that as a real thing which in our larger mode of

existence only occurs as a simulacrum and appearance. Our looking-glass

images are not real, but we suppose that real images are produced

amongst the molecules.[3]

We have seen that if we make a certain supposition as to the calling up

of an image twist by a twist of molecular matter, then the main facts of

electricity are capable of an explanation, which, involving merely the

motion of ordinary matter, is far preferable to the idea of there being

a mysterious fluid, and more in harmony with our present ideas of

electricity.

And yet it is impossible to retain this supposition unless a clear

mechanical explanation can be given of how a real image of itself can be

called up by the twist which we suppose electricity to be.

We can by intelligent agency produce a twist which is the real image of

a given twist. But it would be absurd to suppose amongst the molecules

an agency which, acting with prescribed aim, gave in that domain those

real simulacra, those evident images, those phantoms with which we in

our larger world of masses are for ever mocked.

And yet it would be curious if such an hypothesis were to claim a

recognized position in our mental apparatus with which we think about

nature.

For in that molecular world, if we imagine it to ourselves, there would

be a curious state.

If we consider a twist and its image, they are but the simplest and most

rudimentary type of an organism. What holds good of a twist and its

image twist would hold good of a more complicated arrangement also. If a

bit of structure apparently very unlike a twist, and with manifold parts

and differences in it—if such a structure were to meet its image

structure, each of them would instantly unwind the other, and what was

before a complex and compound whole, opposite to an image of itself,

would at once be resolved into a string of formless particles. A flash,

a blaze, and all would be over.

To realize what this would mean we must conceive that in our world there

were to be for each man somewhere a counter-man, a presentment of

himself, a real counterfeit, outwardly fashioned like himself, but with

his right hand opposite his original’s right hand. Exactly like the

image of the man in a mirror.

And then when the man and his counterfeit met, a sudden whirl, a blaze,

a little steam, and the two human beings, having mutually unwound each

other, leave nothing but a residuum of formless particles.

CHAPTER II.

What physical explanation is possible of this production of a real

image?

First of all we may note that the production of a real image of any

disturbance is one of the commonest phenomena.

If a piece of indiarubber lying on the table be pressed downwards with

the finger it will move up when the finger is removed. The yielding and

the resuming its original form are movement and image movement.

If the disturbance is simply a displacement in one line, then, if the

medium in which this displacement is produced is not permanently

displaced, but on the whole maintains its equilibrium, there invariably

accompanies any displacement its image displacement.

Moreover, to take the simple example of a wave propagated through

water—the particles of the water on the whole move about a mean

position; they are not displaced permanently in any one direction; and,

taking the distance from the crest to the hollow of a wave, then from

the hollow to the next crest, is the real image of the first part. Thus

in the complete movement in the wave measured from crest to crest, there

is displacement and its real image.

Thus there seems some consistency about this supposition of an image,

about the production of a real image in nature.

But there are two observations which we can make.

_Firstly_, if it is true in these complicated cases it ought to be true

in simpler cases also. That is, if this supposition is in harmony with

electrical actions, it ought to fit in with other actions of a simpler

kind.

_Secondly_, a supposition of this kind has no permanent value; it is

rather a feeler, by which we trace out our way in the darkness, than any

actual vision itself. In default of an actual realization of what the

electrical relations are we can treat them by means of a supposition.

But we must be ready at any moment to give up the supposition if it does

not harmonize with the facts.

And in the first case does the idea of a real image hold good about the

simplest possible actions?

If we push our fist towards a glass the image is that of a fist moving

in the opposite direction.

Now, suppose a pressure exerted on a wall, as, for instance, a hard

stone hitting it. The wall undergoes a displacement, but not as a

whole—only that part of it where the stone hits. And this displacement

is followed by the image displacement, for the wall in the part where it

has been hit and pressed back moves forward, and by its reaction throws

the stone off.

Every case of action and reaction is a case of a motion and its image

motion.

If a bullet strikes the wall and goes with such velocity that it lodges

in it, then the motion of the ball and the image motion of the wall

destroy one another, and the result is a shattering of the wall in the

path of the bullet.

Now in the case of a simple displacement of this kind there is a rule by

which we can form the image displacement. Take a point on the wall, and

about this point as a centre turn the displacement half way round, so

that it does not come to be itself again, but is opposite to itself.

By this turning, the displacement becomes the image of itself; a

movement into the wall becomes a movement out from the wall; and these

follow one another if the wall is not injured. It should be noticed that

the displacement is moved round this point, using a direction which is

_not_ in the displacement itself. The displacement goes straight into

the wall. The turning motion, which we suppose, needs another direction

than this.

Now suppose, instead of a simple displacement like this, we take a

displacement involving two directions, as in the case of a wave

disturbance—it will be found that the conditions are just the same. If a

wave movement falls on a medium which it does not destroy or move as a

whole, the displacement calls up its image displacement. And the image

displacement can be found, as before, by twisting the displacement round

so as to become opposite to itself—by twisting it half-way round. But in

this case, too, a direction must be used which is not used in the

displacement itself.

[Illustration: Diagram II.]

Let us look at the wave disturbance more closely.

The horizontal central line in Diagram II. will represent the positions

which a number of particles occupy when at rest. That is, let us suppose

there to be a number of particles lying in a series forming this line.

We can think of the portions of an elastic cord. An indiarubber tube may

be taken as an illustration, and made to vibrate by a motion of the

hand.

If now one of the particles be deflected from its natural

position—suppose it is moved to the position M—then we should have one

particle at M out of its place, and all the others in their places.

But this does not happen. If the particle is pulled to M, the particles

near it follow after it, and are also disturbed from their places,

though not so much as the particle at M.

We should have a set of particles forming a shape like L M N, only much

longer; in fact, the particles all along the cord would be raised.

If the cord is struck suddenly we do have a set arranging themselves

like L M N, but only for a limited distance along the cord.

And here we notice a curious thing.

If a set of particles is forced to go like L M N, removed from their

position of repose, then at once a set of particles goes like N O N′.

A displacement is accompanied by another displacement which is the

opposite of it. And this displacement and opposite displacement travels

along the elastic cord.

But the point of view which is the most natural one to regard it from is

a little different from this. Let us consider a single point, P. When

this is disturbed it moves above its original position to M, and below

to the other end of the dotted line. Its complete movement is from one

of these extremes to the other. And if we take the complete disturbance

as exhibited in all its phases by different points, we ought to look at

the portion of the diagram M N O. For here at N we have a point not

displaced at all; at M, one displaced to its full extent upwards; at O,

one displaced to its full extent downwards. And intermediate particles

have intermediate displacements.

Now when a complex displacement of this kind is put into a cord, its

image at once springs up. The displacement represented by M N O at once

calls up the displacement represented by O N′ M′, and this condition of

displacement and image displacement continues repeating itself till the

cord comes to rest.

If the diagram be closely looked at, it will be seen that it exhibits

the image relationship twice over. For the movement of the particle P

from P to M has its image in the motion of another particle from its

place of repose to the position O. The disturbance itself, M N O,

consists of displacements and image displacements; and this disturbance,

with its image O N′ M′, makes the wave from crest to crest.

The “twist” which we consider in these pages is like the wave motion,

but with a third component added, so that in the complete motion there

is a displacement coming out from the plane of the paper, as well as the

displacements in the plane of the paper itself.

And just as the wave displacement produces a real image of itself in a

medium which it does not distort as a whole, so there is nothing

arbitrary in our assuming that an electric twist calls up the real image

of itself in an insulating medium—that is, a medium which it cannot

twist as a whole.

If L M N O is a wave motion, then L′ M′ N′ O is its image, as produced

by moving it round out of the plane of the paper—Diagram II. If the wave

disturbance is moved round in the plane of the paper, the original wave

L M N O becomes L′ M′ N′ O—Diagram III.—a shape which bears no

resemblance to the transmitted wave.

Consider O N M L to be a bent piece of wire lying on the paper; if it is

moved round O, keeping on the paper, it becomes O N′ M′ L′. To become

like O N′ M′ L′ in Diagram II. it must move up from the paper and down

again on the right.

Thus adopting this artificial aid to thought—that a displacement calls

up an image displacement—we get the rule that this displacement, the

image, can be got from the original displacement by moving the original

displacement half-way round, using as the plane in which the turning is

made that plane which is given us by taking these two directions—the

direction in which the wave is moving, and a direction at right angles

to the directions in which the displacements which form the wave take

place.

Thus, with the wave motion shown, if we take the direction towards the

top of the page to be the up direction, and that from left to right to

be the sideways direction, then out of the paper towards us is the

“near” direction. So, too, in this case we have to turn the wave

disturbance out of the plane of the paper, and each point of it, to

produce the image, must turn in a circle (going half-way round it) lying

in a plane which has the two directions near and sideways. The motions

of the particles themselves are in the plane of the paper. So to get the

image by turning we use a direction—the “near” direction, which is not

involved in the wave motion itself.

[Illustration: Diagram III.]

Hence we may state, as a tentative principle, that when a disturbance

takes place in a medium which will not be disturbed as a whole, then

such disturbance is accompanied by a real image of itself; and this real

image of itself is the configuration which would be obtained by twisting

the original disturbance round in a direction not contained in the

original disturbance.

Thus the disturbance O N′ M′ L′ is obtained by twisting the disturbance

L M N O round. The direction in which it is twisted is the direction

coming out from the plane of the paper.

Now if this plane disturbance is in nature accompanied by its real

image, why should not a twist such as takes place in the electric

current also be accompanied by its image twist when it impinges on a

medium which it cannot twist as a whole—that is, when it comes to an

insulator in its path?

The reason, obviously, is that we cannot conceive such an image produced

mechanically. And the reason of this can be exhibited thus.

When we had a plane disturbance like L M N O we only used up two

dimensions of space, and we have a third coming up from the plane; and

this direction enables us to imagine a turning which will alter A B into

its image.

But when we have a twist proceeding along an axis, as in the case of

electricity, we have no direction left over in space whereby we may

conceive the twist turned round.

Now when the displacement itself involves all these directions how will

our rule hold?

How shall we get the image displacement? We can find what it is by using

a looking-glass; but the same rule which served in previous cases ought

to work here also.

We want a direction which is neither up and down, right and left,

towards and away.

Now let us adopt a mathematical device, and suppose there is such a

direction, and let us call it the X direction, the unknown direction.

Then if we turn the twist round, using this X direction, we shall get

the image if our rule is correct. And as a matter of fact, by twisting a

figure round in this way, using a direction different from any of the

three mentioned above, we do get its image.

Hence the rule we have formed works consistently.

It will be found that if there was another direction so that the spiral

disturbance could be turned independently of the directions used up in

it, that just as a plane disturbance can be turned into its image

disturbance, so the spiral disturbance of electricity could be turned

into its image spiral by a simple turning.

In this argument we have not looked at the matter directly, but from the

outside. To see it immediately requires us to gain a familiarity with

the properties of space with four independent directions, and that would

take too long for the present paper. The same conclusion can be arrived

at mathematically; but in these papers as far as possible we avoid

symbolism. We want to gain hold of scientific facts in a warm and living

way, to unwrap them from conventionalities and formulæ.

Thus if we suppose that in the minute motions which go on about us there

is a possibility of moving in a four-dimensional way, then it is

perfectly legitimate to assume that in a medium which cannot be twisted,

but which is elastic, a twist calls up a real image twist.

And thus the assumptions which we have made as the basis of an

electrical theory are justified on the assumption of a four-dimensional

space, are untenable except on that supposition.

The matter is of course perfectly open. The only way is this, by

adopting the assumption of a higher space to predict what the actions of

the molecules will be, then if a number of predictions are verified the

evidence will become strong. And I feel sure that there are some very

curious things to be made out here. For my own part the evidence of the

reality of four-dimensional space—in the sense in which we say that our

space is real—does not rest on the consideration of the molecular

movements about which it is not easy to get clear ideas, but on the

study of the facts of space. I hardly think that any one who spent a few

years in becoming familiar with the facts of space, not by the means of

symbolism or reasoning but by pure observation, could doubt that there

are really four dimensions.

In noticing the simpler actions and their image actions we find that the

real image does not coexist with its original, but rather follows and

succeeds it. If we push against a board the board yields, and springs

back when we leave off pushing. If the original displacement is

permanent as a point pressed against an elastic surface and making the

surface yield, then the image of this displacement is potential; it is

not actually there, but comes into play as soon as the original

displacement is removed.

Now in the electrical actions we have assumed both the original twist

and the image twist as concurrently existing.

In certain cases there is no doubt that they are coexistent as when a

glass rod is rubbed by silk.

But if the case of the action of a charged poker on an uncharged one be

examined it will be found that there is nothing to prove that the image

twist comes into existence until the original one is removed.

When the charged poker is brought near the other, the remote end of the

second is affected with the same kind of electricity as is on the

charged poker.

The appearance is just the same as if a thin wall were exposed to a

pressure on one side, and the other side were to bulge out. The

displacement is transmitted through the conductor.

It is only when the original charged body is removed that the image

charge is found to be in existence on the second conductor. There are

some peculiarities, however, which make electrical displacements

different in their appearances from ordinary displacements.

No body can be made to move in any direction without imparting an equal

motion in an opposite direction to another body—_e.g._, the motion of a

cannon ball is equalled by the recoil of the cannon.

And so no twist can be given to the particles of a body without an image

twist being given to other particles.

Now the image displacement or rectilinear motion, in the case of a

rectilinear motion, in straightforward movement seems to remain in the

place where it was produced. The recoil of the gun carriage produces a

strain on its bearings and friction, which produce heat, which gradually

dissipates.

But the image displacement, in the case of electricity, seems to have a

marvellous facility for running through the earth and meeting the

original displacement. An indefinitely long line of action seems in

electricity to take the place of a simple point. Our ordinary mechanical

forces are located in centres, or points of action. In electricity the

line seems to take the place of the point. Where the ordinary engineer

deals with points the electrical engineer deals with lines.

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.

[Illustration: Diagram IV.]

[Illustration: Diagram V.]

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.

[Illustration: Diagram I, a block consisting of 27 cubes.]

[Illustration: Diagram II.]

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.

[Illustration: Diagram III. is a block.]

[Illustration: Diagram V.]

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.

[Illustration: Diagram IV. is its image block.]

[Illustration: Diagram VI.]

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.

[Illustration: Diagram VII.—Block III. built up with Cube V.]

[Illustration: Diagram VIII.—Block IV. built up with Cube V.—a

disadjustment.]

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