At the present time our actions are largely influenced by our theories.
We have abandoned the simple and instinctive mode of life of the earlier
civilisations for one regulated by the assumptions of our knowledge and
supplemented by all the devices of intelligence. In such a state it is
possible to conceive that a danger may arise, not only from a want of
knowledge and practical skill, but even from the very presence and
possession of them in any one department, if there is a lack of
information in other departments. If, for instance, with our present
knowledge of physical laws and mechanical skill, we were to build houses
without regard to the conditions laid down by physiology, we should
probably—to suit an apparent convenience—make them perfectly
draught-tight, and the best-constructed mansions would be full of
suffocating chambers. The knowledge of the construction of the body and
the conditions of its health prevent it from suffering injury by the
development of our powers over nature.
In no dissimilar way the mental balance is saved from the dangers
attending an attention concentrated on the laws of mechanical science by
a just consideration of the constitution of the knowing faculty, and the
conditions of knowledge. Whatever pursuit we are engaged in, we are
acting consciously or unconsciously upon some theory, some view of
things. And when the limits of daily routine are continually narrowed by
the ever-increasing complication of our civilisation, it becomes doubly
important that not one only but every kind of thought should be shared
in.
There are two ways of passing beyond the domain of practical certainty,
and of looking into the vast range of possibility. One is by asking,
“What is knowledge? What constitutes experience?” If we adopt this
course we are plunged into a sea of speculation. Were it not that the
highest faculties of the mind find therein so ample a range, we should
return to the solid ground of facts, with simply a feeling of relief at
escaping from so great a confusion and contradictoriness.
The other path which leads us beyond the horizon of actual experience is
that of questioning whatever seems arbitrary and irrationally limited in
the domain of knowledge. Such a questioning has often been successfully
applied in the search for new facts. For a long time four gases were
considered incapable of being reduced to the liquid state. It is but
lately that a physicist has succeeded in showing that there is no such
arbitrary distinction among gases. Recently again the question has been
raised, “Is there not a fourth state of matter?” Solid, liquid, and
gaseous states are known. Mr. Crookes attempts to demonstrate the
existence of a state differing from all of these. It is the object of
these pages to show that, by supposing away certain limitations of the
fundamental conditions of existence as we know it, a state of being can
be conceived with powers far transcending our own. When this is made
clear it will not be out of place to investigate what relations would
subsist between our mode of existence and that which will be seen to be
a possible one.
In the first place, what is the limitation that we must suppose away?
An observer standing in the corner of a room has three directions
naturally marked out for him; one is upwards along the line of meeting
of the two walls; another is forwards where the floor meets one of the
walls; a third is sideways where the floor meets the other wall. He can
proceed to any part of the floor of the room by moving first the right
distance along one wall, and then by turning at right angles and walking
parallel to the other wall. He walks in this case first of all in the
direction of one of the straight lines that meet in the corner of the
floor, afterwards in the direction of the other. By going more or less
in one direction or the other, he can reach any point on the floor, and
any movement, however circuitous, can be resolved into simple movements
in these two directions.
But by moving in these two directions he is unable to raise himself in
the room. If he wished to touch a point in the ceiling, he would have to
move in the direction of the line in which the two walls meet. There are
three directions then, each at right angles to both the other, and
entirely independent of one another. By moving in these three directions
or combinations of them, it is possible to arrive at any point in a
room. And if we suppose the straight lines which meet in the corner of
the room to be prolonged indefinitely, it would be possible by moving in
the direction of those three lines, to arrive at any point in space.
Thus in space there are three independent directions, and only three;
every other direction is compounded of these three. The question that
comes before us then is this. “Why should there be three and only three
directions?” Space, as we know it, is subject to a limitation.
In order to obtain an adequate conception of what this limitation is, it
is necessary to first imagine beings existing in a space more limited
than that in which we move. Thus we may conceive a being who has been
throughout all the range of his experience confined to a single straight
line. Such a being would know what it was to move to and fro, but no
more. The whole of space would be to him but the extension in both
directions of the straight line to an infinite distance. It is evident
that two such creatures could never pass one another. We can conceive
their coming out of the straight line and entering it again, but they
having moved always in one straight line, would have no conception of
any other direction of motion by which such a result could be effected.
The only shape which could exist in a one-dimensional existence of this
kind would be a finite straight line. There would be no difference in
the shapes of figures; all that could exist would simply be longer or
shorter straight lines.
Again, to go a step higher in the domain of a conceivable existence.
Suppose a being confined to a plane superficies, and throughout all the
range of its experience never to have moved up or down, but simply to
have kept to this one plane. Suppose, that is, some figure, such as a
circle or rectangle, to be endowed with the power of perception; such a
being if it moves in the plane superficies in which it is drawn, will
move in a multitude of directions; but, however varied they may seem to
be, these directions will all be compounded of two, at right angles to
each other. By no movement so long as the plane superficies remains
perfectly horizontal, will this being move in the direction we call up
and down. And it is important to notice that the plane would be
different, to a creature confined to it, from what it is to us. We think
of a plane habitually as having an upper and a lower side, because it is
only by the contact of solids that we realize a plane. But a creature
which had been confined to a plane during its whole existence would have
no idea of there being two sides to the plane he lived in. In a plane
there is simply length and breadth. If a creature in it be supposed to
know of an up or down he must already have gone out of the plane.
Is it possible, then, that a creature so circumstanced would arrive at
the notion of there being an up and down, a direction different from
those to which he had been accustomed, and having nothing in common with
them? Obviously nothing in the creature’s circumstances would tell him
of it. It could only be by a process of reasoning on his part that he
could arrive at such a conception. If he were to imagine a being
confined to a single straight line, he might realise that he himself
could move in two directions, while the creature in a straight line
could only move in one. Having made this reflection he might ask, “But
why is the number of directions limited to two? Why should there not be
three?”
A creature (if such existed), which moves in a plane would be much more
fortunately circumstanced than one which can only move in a straight
line. For, in a plane, there is a possibility of an infinite variety of
shapes, and the being we have supposed could come into contact with an
indefinite number of other beings. He would not be limited, as in the
case of the creature in a straight line, to one only on each side of
him.
It is obvious that it would be possible to play curious tricks with a
being confined to a plane. If, for instance, we suppose such a being to
be inside a square, the only way out that he could conceive would be
through one of the sides of the square. If the sides were impenetrable,
he would be a fast prisoner, and would have no way out.
What his case would be we may understand, if we reflect what a similar
case would be in our own existence. The creature is shut in in all the
directions he knows of. If a man is shut in in all the directions he
knows of, he must be surrounded by four walls, a roof and a floor. A
two-dimensional being inside a square would be exactly in the same
predicament that a man would be, if he were in a room with no opening on
any side. Now it would be possible to us to take up such a being from
the inside of the square, and to set him down outside it. A being to
whom this had happened would find himself outside the place he had been
confined in, and he would not have passed through any of the boundaries
by which he was shut in. The astonishment of such a being can only be
imagined by comparing it to that which a man would feel, if he were
suddenly to find himself outside a room in which he had been, without
having passed through the window, doors, chimney or any opening in the
walls, ceiling or floor.
Another curious thing that could be effected with a two-dimensional
being, is the following. Conceive two beings at a great distance from
one another on a plane surface. If the plane surface is bent so that
they are brought close to one another, they would have no conception of
their proximity, because to each the only possible movements would seem
to be movements in the surface. The two beings might be conceived as so
placed, by a proper bending of the plane, that they should be absolutely
in juxtaposition, and yet to all the reasoning faculties of either of
them a great distance could be proved to intervene. The bending might be
carried so far as to make one being suddenly appear in the plane by the
side of the other. If these beings were ignorant of the existence of a
third dimension, this result would be as marvellous to them, as it would
be for a human being who was at a great distance—it might be at the
other side of the world—to suddenly appear and really be by our side,
and during the whole time he not to have left the place in which he was.
CHAPTER II.
The foregoing examples make it clear that beings can be conceived as
living in a more limited space than ours. Is there a similar limitation
in the space we know?
At the very threshold of arithmetic an indication of such a limitation
meets us.
If there is a straight line before us two inches long, its length is
expressed by the number 2. Suppose a square to be described on the line,
the number of square inches in this figure is expressed by the number 4,
_i.e._, 2 × 2. This 2 × 2 is generally written 2², and named “2 square.”
Now, of course, the arithmetical process of multiplication is in no
sense identical with that process by which a square is generated from
the motion of a straight line, or a cube from the motion of a square.
But it has been observed that the units resulting in each case, though
different in kind, are the same in number.
If we touch two things twice over, the act of touching has been
performed four times. Arithmetically, 2 × 2 = 4. If a square is
generated by the motion of a line two inches in length, this square
contains four square inches.
So it has come to pass that the second and third powers of numbers are
called “square” and “cube.”
We have now a straight line two inches long. On this a square has been
constructed containing four square inches. If on the same line a cube be
constructed, the number of cubic inches in the figure so made is 8,
_i.e._, 2 × 2 × 2 or 2³. Here, corresponding to the numbers 2, 2², 2³,
we have a series of figures. Each figure contains more units than the
last, and in each the unit is of a different kind. In the first figure a
straight line is the unit, viz., one linear inch; it is said to be of
one dimension. In the second a square is the unit, viz., one square
inch. The square is a figure of two dimensions. In the third case a cube
is the unit, and the cube is of three dimensions. The straight line is
said to be of one dimension because it can be measured only in one way.
Its length can be taken, but it has no breadth or thickness. The square
is said to be of two dimensions because it has both length and breadth.
The cube is said to have three dimensions, because it can be measured in
three ways.
The question naturally occurs, looking at these numbers 2, 2², 2³, by
what figure shall we represent 2⁴, or 2 × 2 × 2 × 2. We know that in the
figure there must be sixteen units, or twice as many units as in the
cube. But the unit also itself must be different. And it must not differ
from a cube simply in shape. It must differ from a cube as a cube
differs from a square. No number of squares will make up a cube, because
each square has no thickness. In the same way, no number of cubes must
be able to make up this new unit. And here, instead of trying to find
something already known, to which the idea of a figure corresponding to
the fourth power can be affixed, let us simply reason out what the
properties of such a figure must be. In this attempt we have to rely,
not on a process of touching or vision, such as informs us of the
properties of bodies in the space we know, but on a process of thought.
Each fact concerning this unknown figure has to be reasoned out; and it
is only after a number of steps have been gone through, that any
consistent familiarity with its properties is obtained. Of all
applications of the reason, this exploration is perhaps the one which
requires, for the simplicity of the data involved, the greatest exercise
of the abstract imagination, and on this account is well worth patient
attention. The first steps are very simple. We must imagine a finite
straight line to generate a square by moving on the plane of the paper,
and this square in its turn to generate a cube by moving vertically
upwards. Fig. 1 represents a straight line; Fig. 2 represents a square
formed by the motion of that straight line; Fig. 3 represents
perspectively a cube formed by the motion of that square A B C D
upwards. It would be well, instead of using figure 3, to place a cube on
the paper. Its base would be A B C D, its upper surface E F G H.
[Illustration: Generating a line, square, and cube.]
The straight line A B gives rise to the square A B C D by a movement at
right angles to itself. If motion be confined to the straight line A B,
a backward and forward motion is the only one possible. No sideway
motion is admissible. And if we suppose a being to exist which could
only move in the straight line A B, it would have no idea of any other
movement than to and fro. The square A B C D is formed from the straight
line by a movement in a direction entirely different from the direction
which exists in A B. This motion is not expressible by means of any
possible motion in A B. A being which existed in A B, and whose
experience was limited to what could occur in A B, would not be able to
understand the instructions we should give to make A B trace out the
figure A B C D.
In the figure A B C D there is a possibility of moving in a variety of
directions, so long as all these directions are confined to one plane.
All directions in this plane can be considered as compounded of two,
from A to B, and from A to C. Out of the infinite variety of such
directions there is none which tends in a direction perpendicular to
Fig. 2; there is none which tends upwards from the plane of the paper.
Conceive a being to exist in the plane, and to move only in it. In all
the movements which he went through there would be none by which he
could conceive the alteration of Fig. 2 into what Fig. 3 represents in
perspective. For 2 to become 3 it must be supposed to move
perpendicularly to its own plane. The figure it traces out is the cube A
B C D E F G H.
All the directions, manifold as they are, in which a creature existing
in Fig. 3 could move, are compounded of three directions. From A to B,
from A to C, from A to E, and there are no other directions known to it.
But if we suppose something similar to be done to Fig. 3, something of
the same kind as was done to Fig. 1 to turn it into Fig. 2, or to Fig. 2
to turn it into Fig. 3, we must suppose the whole figure as it exists to
be moved in some direction entirely different from any direction within
it, and not made up of any combination of the directions in it. What is
this? It is the fourth direction.
We are as unable to imagine it as a creature living in the plane Fig. 2
would be to imagine a direction such that moving in it the square 2
would become the cube 3. The third dimension to such a creature would be
as unintelligible as the fourth is to us. And at this point we have to
give up the aid that is to be got from any presentable object, and we
have simply to investigate what the properties of the simplest figure in
four dimensions are, by pursuing further the analogy which we know to
exist between the process of formation of 2 from 1, and of 3 from 2, and
finally of 4 from 3. For the sake of convenience, let us call the figure
we are investigating—the simplest figure in four dimensions—a
four-square.
First of all we must notice, that if a cube be formed from a square by
the movement of the square in a new direction, each point of the
interior of the square traces out part of the cube. It is not only the
bounding lines that by their motion form the cube, but each portion of
the interior of the square generates a portion of the cube. So if a cube
were to move in the fourth dimension so as to generate a four-square,
every point in the interior of the cube would start _de novo_, and trace
out a portion of the new figure uninterfered with by the other points.
Or, to look at the matter in another light, a being in three dimensions,
looking down on a square, sees each part of it extended before him, and
can touch each part without having to pass through the surrounding
parts, for he can go from above, while the surrounding parts surround
the part he touches only in one plane.
So a being in four dimensions could look at and touch every point of a
solid figure. No one part would hide another, for he would look at each
part from a direction which is perfectly different from any in which it
is possible to pass from one part of the body to another. To pass from
one part of the body to another it is necessary to move in three
directions, but a creature in four dimensions would look at the solid
from a direction which is none of these three.
Let us obtain a few facts about the fourth figure, proceeding according
to the analogy that exists between 1, 2, 3, and 4. In the Fig. 1 there
are two points. In 2 there are four points—the four corners of the
square. In 3 there are eight points. In the next figure, proceeding
according to the same law, there would be sixteen points.
In the Fig. 1 there is one line. In the square there are four lines. In
the cube there are twelve lines. How many lines would there be in the
four-square? That is to say that there are three numbers—1, 4, and 12.
What is the fourth, going on accordingly to the same law?
To answer this question let us trace out in more detail how the figures
change into one another. The line, to become the square, moves; it
occupies first of all its original position, and last of all its final
position. It starts as A B, and ends as C D; thus the line appears
twice, or it is doubled. The two other lines in the square, A C, B D,
are formed by the motions of the points at the extremities of the moving
line. Thus, in passing from the straight line to the square the lines
double themselves, and each point traces out a line. If the same
procedure holds good in the case of the change of the square into the
cube, we ought in the cube to have double the number of lines as in the
square—that is eight—and every point in the square ought to become a
line. As there are four points in the square, we should have four lines
in the cube from them, that is, adding to the previous eight, there
should be twelve lines in the cube. This is obviously the case. Hence we
may with confidence, to deduce the number of lines in a four-square,
apply this rule. _Double the number of lines in the previous figure, and
add as many lines as there are points in the previous figure._ Now in
the cube there are twelve lines and eight points. Hence we get 2 × 12 +
8, or thirty-two lines in the four-square.
In the same way any other question about the four-square can be
answered. We must throw aside our realising power and answer in
accordance with the analogy to be worked out from the three figures we
know.
Thus, if we want to know how many plane surfaces the four-square has, we
must commence with the line, which has none; the square has one; the
cube has six. Here we get the three numbers, 0, 1, and 6. What is the
fourth?
Consider how the planes of the cube arise. The square at the beginning
of its motion determines one of the faces of the cube, at the end it is
the opposite face, during the motion each of the lines of the square
traces out one plane face of the cube. Thus we double the number of
planes in the previous figure, and every line in the previous figure
traces out a plane in the subsequent one.
Apply this rule to the formation of a square from a line. In the line
there is no plane surface, and since twice nothing is nothing, we get,
so far, no surface in the square; but in the straight line there is one
line, namely itself, and this by its motion traces out the plane surface
of the square. So in the square, as should be, the rule gives one
surface.
Applying this rule to the case of the cube, we get, doubling the
surfaces, 12; and adding a plane for each of the straight lines, of
which there are 12, we have another 12, or 24 plane surfaces in all.
Thus, just as by handling or looking at it, it is possible to describe a
figure in space, so by going through a process of calculation it is
within our power to describe all the properties of a figure in four
dimensions.
There is another characteristic so remarkable as to need a special
statement. In the case of a finite straight line, the boundaries are
points. If we deal with one dimension only, the figure 1, that of a
segment of a straight line, is cut out of and separated from the rest of
an imaginary infinitely long straight line by the two points at its
extremities. In this simple case the two points correspond to the
bounding surface of the cube. In the case of a two-dimensional figure an
infinite plane represents the whole of space. The square is separated
off by four straight lines, and it is impossible for an entry to be made
into the interior of the square, except by passing through the straight
lines. Now, in these cases, it is evident that the boundaries of the
figure are of one dimension less than the figure itself. Points bound
lines, lines bound plane figures, planes bound solid figures. Solids
then must bound four dimensional figures. The four-square will be
bounded in the following manner. First of all there is the cube which,
by its motion in the fourth direction, generates the figure. This, in
its initial position, forms the base of the four-square. In its final
position it forms the opposite end. During the motion each of the faces
of the cube give rise to another cube. The direction in which the cube
moves is such that of all the six sides none is in the least inclined in
that direction. It is at right angles to all of them. The base of the
cube, the top of the cube, and the four sides of the cube, each and all
of them form cubes. Thus the four-square is bounded by eight cubes.
Summing up, the four-square would have 16 points, 32 lines, 24 surfaces,
and it would be bounded by 8 cubes.
If a four-square were to rest in space it would seem to us like a cube.
To justify this conclusion we have but to think of how a cube would
appear to a two-dimensional being. To come within the scope of his
faculties at all, it must come into contact with the plane in which he
moves. If it is brought into as close a contact with this plane as
possible, it rests on it by one of its faces. This face is a square, and
the most a two-dimensional being could get acquainted with of a cube
would be a square.
Having thus seen how it is possible to describe the properties of the
simplest shape in four dimensions, it is evident that the mental
construction of more elaborate figures is simply a matter of time and
patience.
In the study of the form and development of the chick in the egg, it is
impossible to detect the features that are sought to be observed, except
by the use of the microscope. The specimens are accordingly hardened by
a peculiar treatment and cut into thin sections. The investigator going
over each of these sections, noticing all their peculiarities,
constructs in his mind the shape as it originally existed from the
record afforded by an indefinite number of slices. So, to form an idea
of a four-dimensional figure, a series of solid shapes bounded on every
side differing gradually from one another, proceeding, it may be, to the
most diverse forms, has to be mentally grasped and fused into a unitary
conception.
If, for instance, a small sphere were to appear, this to be replaced by
a larger one, and so on, and then, when the largest had appeared,
smaller and smaller ones to make their appearance, what would be
witnessed would be a series of sections of a four-dimensional sphere.
Each section in space being a sphere.
Again, just as solid figures can be represented on paper by perspective,
four-dimensional figures can be represented perspectively by solids. If
there are two squares, one lying over the other, and the underneath one
be pushed away, its sides remaining parallel with the one that was over
it, then if each point of the one be joined to the corresponding point
of the other, we have a fair representation on paper of a cube. Fig. 3
may be considered to be such a representation if the square C D G H be
considered to be the one that has been pushed away from lying originally
under the square A B E F. Each of the planes which bound the cube is
represented on the paper. The only thing that is wanting is the
three-dimensional content of the cube. So if two cubes be placed with
their sides parallel, but one somewhat diagonally with regard to the
other, and all their corresponding points be supposed joined, there will
be found a set of solid figures, each representing (though of course
distortedly) the bounding cubes of the four-dimensional figure, and
every plane and line in the four-dimensional figure will be found to be
represented in a kind of solid perspective. What is wanting is of course
the four-dimensional content.
CHAPTER III.
Having now passed in review some of the properties of four-dimensional
figures, it remains to ask what relations beings in four dimensions, if
they did exist, would have with us.
And in the first place, a being in four dimensions would have to us
exactly the appearance of a being in space. A being in a plane would
only know solid objects as two-dimensional figures—the shapes namely in
which they intersected his plane. So if there were four-dimensional
objects, we should only know them as solids—the solids, namely, in which
they intersect our space. Why, then, should not the four-dimensional
beings be ourselves, and our successive states the passing of them
through the three-dimensional space to which our consciousness is
confined?
Let us consider the question in more detail. And for the sake of
simplicity transfer the problem to the case of three and two dimensions
instead of four and three.
Suppose a thread to be passed through a thin sheet of wax placed
horizontally. It can be passed through in two ways. Either it can be
pulled through, or it can be held at both ends, and moved downwards as a
whole. Suppose a thread to be grasped at both ends, and the hands to be
moved downwards perpendicularly to the sheet of wax. If the thread
happens to be perpendicular to the sheet it simply passes through it,
but if the thread be held, stretched slantingwise to the sheet, and the
hands are moved perpendicularly downwards, the thread will, if it be
strong enough, make a slit in the sheet.
If now the sheet of wax were to have the faculty of closing up behind
the thread, what would appear in the sheet would be a moving hole.
Suppose that instead of a sheet and a thread, there were a straight line
and a plane. If the straight line were placed slantingwise in reference
to the plane and moved downwards, it would always cut the plane in a
point, but that point of section would move on. If the plane were of
such a nature as to close up behind the line, if it were of the nature
of a fluid, what would be observed would be a moving point. If now there
were a whole system of lines sloping in different directions, but all
connected together, and held absolutely still by one framework, and if
this framework with its system of lines were as a whole to pass slowly
through the fluid plane at right angles to it, there would then be the
appearance of a multitude of moving points in the plane, equal in number
to the number of straight lines in the system. The lines in the
framework will all be moving at the same rate—namely, at the rate of the
framework in which they are fixed. But the points in the plane will have
different velocities. They will move slower or faster, according as the
lines which give rise to them are more or less inclined to the plane. A
straight line perpendicular to the plane will, on passing through, give
rise to a stationary point. A straight line that slopes very much
inclined to the plane will give rise to a point moving with great
swiftness. The motions and paths of the points would be determined by
the arrangement of the lines in the system. It is obvious that if two
straight lines were placed lying across one another like the letter X,
and if this figure were to be stood upright and passed through the
plane, what would appear would be at first two points. These two points
would approach one another. When the part where the two strokes of the X
meet came into the plane, the two points would become one. As the upper
part of the figure passed through, the two points would recede from one
another.
[Illustration: Rectangle A B C D, crossed by two interlinked lines X
and Y two lines.]
If the line be supposed to be affixed to all parts of the framework, and
to loop over one another, and support one another,[1] it is obvious that
they could assume all sorts of figures, and that the points on the plane
would move in very complicated paths. The annexed figure represents a
section of such a framework. Two lines X X and Y Y are shown, but there
must be supposed to be a great number of others sloping backwards and
forwards as well as sideways.
Let us now assume that instead of lines, very thin threads were attached
to the framework: they on passing through the fluid plane would give
rise to very small spots. Let us call the spots atoms, and regard them
as constituting a material system in the plane. There are four
conditions which must be satisfied by these spots if they are to be
admitted as forming a material system such as ours. For the ultimate
properties of matter (if we eliminate attractive and repulsive forces,
which may be caused by the motions of the smallest particles), are—1,
Permanence; 2, Impenetrability; 3, Inertia; 4, Conservation of energy.
According to the first condition, or that of permanence, no one of these
spots must suddenly cease to exist. That is, the thread which by sharing
in the general motion of the system gives rise to the moving point, must
not break off before the rest of them. If all the lines suddenly ended
this would correspond to a ceasing of matter.
2. Impenetrability.—One spot must not pass through another. This
condition is obviously satisfied. If the threads do not coincide at any
point, the moving spots they give rise to cannot.
3. Inertia.—A spot must not cease to move or cease to remain at rest
without coming into collision with another point. This condition gives
the obvious condition with regard to the threads, that they, between the
points where they come into contact with one another, must be straight.
A thread which was curved would, passing through the plane, give rise to
a point which altered in velocity spontaneously. This the particles of
matter never do.
4. Conservation of energy.—The energy of a material system is never
lost; it is only transferred from one form to another, however it may
seem to cease. If we suppose each of the moving spots on the plane to be
the unit of mass, the principle of the conservation of energy demands
that when any two meet, the sum of the squares of their several
velocities before meeting shall be the same as the sum of the squares of
their velocities after meeting. Now we have seen that any statement
about the velocities of the spots in the plane is really a statement
about the inclinations of the threads to the plane. Thus the principle
of the conservation of energy gives a condition which must be satisfied
by the inclinations of the threads of the plane. Translating this
statement, we get in mathematical language the assertion that the sum of
the squares of the tangents of the angles the threads make with the
normal to the plane remains constant.
Hence, all complexities and changes of a material system made up of
similar atoms in a plane could result from the uniform motion as a whole
of a system of threads.
We can imagine these threads as weaving together to form connected
shapes, each complete in itself, and these shapes as they pass through
the fluid plane give rise to a series of moving points. Yet, inasmuch as
the threads are supposed to form consistent shapes, the motion of the
points would not be wholly random, but numbers of them would present the
semblance of moving figures. Suppose, for instance, a number of threads
to be so grouped as to form a cylinder for some distance, but after a
while to be pulled apart by other threads with which they interlink.
While the cylinder was passing through the plane, we should have in the
plane a number of points in a circle. When the part where the threads
deviated came to the plane, the circle would break up by the points
moving away. These moving figures in the plane are but the traces of the
shapes of threads as those shapes pass on. These moving figures may be
conceived to have a life and a consciousness of their own.
Or, if it be irrational to suppose them to have a consciousness when the
shapes of which they are momentary traces have none, we may well suppose
that the shapes of threads have consciousness, and that the moving
figures share this consciousness, only that in their case it is limited
to those parts of the shapes that simultaneously pass through the plane.
In the plane, then, we may conceive bodies with all the properties of a
material system, moving and changing, possessing consciousness. After a
while it may well be that one of them becomes so disassociated that it
appears no longer as a unit, and its consciousness as such may be lost.
But the threads of existence of such a figure are not broken, nor is the
shape which gave it origin altered in any way. It has simply passed on
to a distance from the plane. Thus nothing which existed in the
conscious life on the plane would cease. There would in such an
existence be no cause and effect, but simply the gradual realisation in
a superficies of an already existent whole. There would be no progress,
unless we were to suppose the threads as they pass to interweave
themselves in more complex shapes.
Can a representation, such as the preceding, be applied to the case of
the existence in space with which we have to do? Is it possible to
suppose that the movements and changes of material objects are the
intersections with a three-dimensional space of a four-dimensional
existence? Can our consciousness be supposed to deal with a spatial
profile of some higher actuality?
It is needless to say that all the considerations that have been brought
forward in regard to the possibility of the production of a system
satisfying the conditions of materiality by the passing of threads
through a fluid plane, holds good with regard to a four-dimensional
existence passing through a three-dimensional space. Each part of the
ampler existence which passed through our space would seem perfectly
limited to us. We should have no indication of the permanence of its
existence. Were such a thought adopted, we should have to imagine some
stupendous whole, wherein all that has ever come into being or will come
co-exists, which passing slowly on, leaves in this flickering
consciousness of ours, limited to a narrow space and a single moment, a
tumultuous record of changes and vicissitudes that are but to us. Change
and movement seem as if they were all that existed. But the appearance
of them would be due merely to the momentary passing through our
consciousness of ever existing realities.
In thinking of these matters it is hard to divest ourselves of the habit
of visual or tangible illustration. If we think of a man as existing in
four dimensions, it is hard to prevent ourselves from conceiving him as
prolonged in an already known dimension. The image we form resembles
somewhat those solemn Egyptian statues which in front represent well
enough some dignified sitting figure, but which are immersed to their
ears in a smooth mass of stone which fits their contour exactly.
No material image will serve. Organised beings seem to us so complete
that any addition to them would deface their beauty. Yet were we
creatures confined to a plane, the outline of a Corinthian column would
probably seem to be of a beauty unimprovable in its kind. We should be
unable to conceive any addition to it, simply for the reason that any
addition we could conceive would be of the nature of affixing an
unsightly extension to some part of the contour. Yet, moving as we do in
space of three dimensions, we see that the beauty of the stately column
far surpasses that of any single outline. So all that we can do is to
deny our faculty of judging of the ideal completeness of shapes in four
dimensions.
CHAPTER IV.
Let us now leave this supposition of framework and threads. Let us
investigate the conception of a four-dimensional existence in a simpler
and more natural manner—in the same way that a two-dimensional being
should think about us, not as infinite in the third dimension, but
limited in three dimensions as he is in two. A being existing in four
dimensions must then be thought to be as completely bounded in all four
directions as we are in three. All that we can say in regard to the
possibility of such beings is, that we have no experience of motion in
four directions. The powers of such beings and their experience would be
ampler, but there would be no fundamental difference in the laws of
force and motion.
Such a being would be able to make but a part of himself visible to us,
for a cube would be apprehended by a two-dimensional being as the square
in which it stood. Thus a four-dimensional being would suddenly appear
as a complete and finite body, and as suddenly disappear, leaving no
trace of himself, in space, in the same way that anything lying on a
flat surface, would, on being lifted, suddenly vanish out of the
cognisance of beings, whose consciousness was confined to the plane. The
object would not vanish by moving in any direction, but disappear
instantly as a whole. There would be no barrier, no confinement of our
devising that would not be perfectly open to him. He would come and go
at pleasure; he would be able to perform feats of the most surprising
kind. It would be possible by an infinite plane extending in all
directions to divide our space into two portions absolutely separated
from one another; but a four-dimensional being would slip round this
plane with the greatest ease.
To see this clearly, let us first take the analogous case in three
dimensions. Suppose a piece of paper to represent a plane. If it is
infinitely extended in every direction, it will represent an infinite
plane. It can be divided into two parts by an infinite straight line. A
being confined to this plane could not get from one part of it to the
other without passing through the line. But suppose another piece of
paper laid on the first and extended infinitely, it will represent
another infinite plane. If the being moves from the first plane by a
motion in the third dimension, it will move into this new plane. And in
it it finds no line. Let it move to such a position that when it goes
back to the first plane it will be on the other side of the line. Then
let it go back to the first plane. It has appeared now on the other side
of the line which divides the infinite plane into two parts.
Take now the case of four dimensions. Instead of bringing before the
mind a sheet of paper conceive a solid of three dimensions. If this
solid were to become infinite it would fill up the whole of
three-dimensional space. But it would not fill up the whole of
four-dimensional space. It would be to four-dimensional space what an
infinite plane is to three-dimensional space. There could be in
four-dimensional space an infinite number of such solids, just as in
three-dimensional space there could be an infinite number of infinite
planes.
Thus, lying alongside our space, there can be conceived a space also
infinite in all three directions. To pass from one to the other a
movement has to be made in the fourth dimension, just as to pass from
one infinite plane to another a motion has to be made in the third
dimension.
Conceive, then, corresponding to the first sheet of paper mentioned
above, a solid, and as the sheet of paper was supposed to be infinitely
extended in two dimensions, suppose the solid to be infinitely extended
in its three dimensions, so that it fills the whole of space as we know
it.
Now divide this infinite solid in two parts by an infinite plane, as the
infinite plane of paper was divided in two parts by an infinite line. A
being cannot pass from one part of this infinite solid to another, on
the other side of this infinite plane, without going through the
infinite plane, _so long as he keeps within the infinite solid_.
But suppose beside this infinite solid a second infinite solid, lying
next to it in the fourth dimension, as the second infinite plane of
paper was next to the first infinite plane in the third dimension. Let
now the being that wants to get on the other side of the dividing plane
move off in the fourth dimension, and enter the second infinite solid.
In this second solid there is no dividing plane. Let him now move, so
that coming back to the first infinite solid he shall be on the other
side of the infinite plane that divides it into two portions. If this is
done, he will now be on the other side of the infinite plane, without
having gone through it.
In a similar way a being, able to move in four dimensions, could get out
of a closed box without going through the sides, for he could move off
in the fourth dimension, and then move about, so that when he came back
he would be outside the box.
Is there anything in the world as we know it, which would indicate the
possibility of there being an existence in four dimensions? No definite
answer can be returned to this question. But it may be of some interest
to point out that there are certain facts which might be read by the
light of the fourth dimensional theory.
To make this clear, let us suppose that space is really four
dimensional, and that the three-dimensional space we know is, in this
ampler space, like a surface is in our space.
We should then be in this ampler space like beings confined to the
surface of a plane would be in ours. Let us suppose that just as in our
space there are centres of attraction whose influence radiates out in
every direction, so in this ampler space there are centres of attraction
whose influence radiates out in every direction. Is there anything to be
observed in nature which would correspond to the effect of a centre of
attraction lying out of our space, and acting on all the matter in it?
The effect of such a centre of attraction would not be to produce motion
in any known direction, because it does not lie off in any known
direction.
Let us pass to the corresponding case in three and two dimensions,
instead of four and three. Let us imagine a plane lying horizontally,
and in it some creatures whose experience was confined to it. If now
some water or other liquid were poured on to the plane, the creatures,
becoming aware of its presence, would find that it had a tendency to
spread out all over the plane. In fact it would not be to them as a
liquid is to us—it would rather correspond to a gas. For a gas, as we
know it, tends to expand in every direction, and gradually increase so
as to fill the whole of space. It exercises a pressure on the walls of
any vessel in which we confine it.
The liquid on the plane expands in all the dimensions which the
two-dimensional creatures on the plane know, and at the same time
becomes smaller in the third dimension, its absolute quantity remaining
unchanged. In like manner we might suppose that gases (which by
expansion become larger in the dimensions that we know) become smaller
in the fourth dimension.
The cause in this case would have to be sought for in an attractive
force, acting with regard to our space as the force of gravity acts with
regard to a horizontal plane.
Can we suppose that there is a centre of attraction somewhere off in the
fourth dimension, and that the gases, which we know are simply more
mobile liquids, expanding out in every direction under its influence.
This view receives a certain amount of support from the fact proved
experimentally that there is no absolute line of demarcation between a
liquid and a gas. The one can be made to pass into the other with no
moment intervening in which it can be said that now a change of state
has taken place.
We might then suppose that the matter we know extending in three
dimensions has also a small thickness in the fourth dimension; that
solids are rigid in the fourth as in the other three dimensions; that
liquids are too coherent to admit of their spreading out in space, and
becoming thinner in the fourth dimension, under the influence of an
attractive centre lying outside of our space; but that gases, owing to
the greater mobility of their particles, are subject to its action, and
spread out in space under its influence, in the same manner that
liquids, under the influence of gravity, spread out on a plane.
Then the density of a gas would be a measure of the relative thickness
of it in the fourth dimension: and the diminution of the density would
correspond to a diminution of the thickness in the fourth dimension.
Could this supposition be tested in any way?
Suppose a being confined to a plane; if the plane is moved far off from
the centre of attraction lying outside it, he would find that liquids
had less tendency to spread out than before.
Or suppose he moves to a distant part of the plane so that the line from
his position to the centre of attraction lies obliquely to the plane; he
would find that in this position a liquid would show a tendency to
spread out more in one direction than another.
Now our space considered as lying in four-dimensional space, as a plane
does in three-dimensional space, may be shifted. And the expansive force
of gases might be found to be different at different ages. Or, shifting
as we do our position in space during the course of the earth’s path
round the sun, there might arise a sufficient difference in our position
in space, with regard to the attractive centre, to make the expansive
force of gases different at different times of the year, or to cause
them to manifest a greater expansive force in one direction than in
another.
But although this supposition might be worked out at some length, it is
hard to suppose that it could afford any definite test of the physical
existence of a fourth dimension. No test has been discovered which is
decisive. And, indeed, before searching for tests, a theoretical point
of the utmost importance has to be settled. In discussing the
geometrical properties of straight lines and planes, we suppose them to
be respectively of one and two dimensions, and by so doing deny them any
real existence. A plane and a line are mere abstractions. Every portion
of matter is of three dimensions. If we consider beings on a plane not
as mere idealities, we must suppose them to be of some thickness. If
their experience is to be limited to a plane this thickness must be very
small compared to their other dimensions. Transferring our reasoning to
the case of four dimensions, we come to a curious result.
If a fourth dimension exists there are two possible alternatives.
One is, that there being four dimensions, we have a three-dimensional
existence only. The other is that we really have a four-dimensional
existence, but are not conscious of it. If we are in three dimensions
only, while there are really four dimensions, then we must be relatively
to those beings who exist in four dimensions, as lines and planes are in
relation to us. That is, we must be mere abstractions. In this case we
must exist only in the mind of the being that conceives us, and our
experience must be merely the thoughts of his mind—a result which has
apparently been arrived at, on independent grounds, by an idealist
philosopher.
The other alternative is that we have a four-dimensional existence. In
this case our proportions in it must be infinitely minute, or we should
be conscious of them. If such be the case, it would probably be in the
ultimate particles of matter, that we should discover the fourth
dimension, for in the ultimate particles the sizes in the three
dimensions are very minute, and the magnitudes in all four dimensions
would be comparable.
The preceding two alternative suppositions are based on the hypothesis
of the reality of four-dimensional existence, and must be conceived to
hold good only on that hypothesis.
It is somewhat curious to notice that we can thus conceive of an
existence relative to which that which we enjoy must exist as a mere
abstraction.
Apart from the interest of speculations of this kind they have
considerable value; for they enable us to express in intelligible terms
things of which we can form no image. They supply us, as it were, with
scaffolding, which the mind can make use of in building up its
conceptions. And the additional gain to our power of representation is
very great.
Many philosophical ideas and doctrines are almost unintelligible because
there is no physical illustration which will serve to express them. In
the imaginary physical existence which we have traced out, much that
philosophers have written finds adequate representation. Much of
Spinoza’s Ethics, for example, could be symbolized from the preceding
pages.
Thus we may discuss and draw perfectly legitimate conclusions with
regard to unimaginable things.
It is, of course, evident that these speculations present no point of
direct contact with fact. But this is no reason why they should be
abandoned. The course of knowledge is like the flow of some mighty
river, which, passing through the rich lowlands, gathers into itself the
contributions from every valley. Such a river may well be joined by a
mountain stream, which, passing with difficulty along the barren
highlands, flings itself into the greater river down some precipitous
descent, exhibiting at the moment of its union the spectacle of the
utmost beauty of which the river system is capable. And such a stream is
no inapt symbol of a line of mathematical thought, which, passing
through difficult and abstract regions, sacrifices for the sake of its
crystalline clearness the richness that comes to the more concrete
studies. Such a course may end fruitlessly, for it may never join the
main course of observation and experiment. But, if it gains its way to
the great stream of knowledge, it affords at the moment of its union the
spectacle of the greatest intellectual beauty, and adds somewhat of
force and mysterious capability to the onward current.
The Persian King.
_PART I._
CHAPTER I.
In Persia there was once a king. On one occasion when he was out hunting
he came to the narrow entrance of a valley. It was shut in on either
side by vast hills, seemingly the spurs from the distant mountains.
These great spurs spread out including a wide tract of land. Towards the
entrance where he stood they approached one another, and ended in abrupt
cliffs. Across the mouth of the valley stretched a deep ravine. The
king, followed by courtiers, galloped along, searching a spot where the
deep fissure might be shallower, so that descending into it he might
reach the valley by ascending on the opposite side.
But at every point the ravine stretched downwards dark and deep, from
cliff to cliff, shutting off all access to the valley.
At one point only was there a means of crossing. There were two masses
of rocks, jutting out one from either side like the abutments of a
natural bridge, and they seemed to meet in mid air.
The mass trembled and shook as the king spurred his horse over it, and
the dislodged stones reverberated from side to side of the chasm till
the noise of their falling was lost.
Before the first of his courtiers could follow him one of the great
piers or abutments gave way—the whole mass fell crashing down. The king
was alone in the valley.
“So ho,” he cried, “the kingdom of Persia is shrunk to this narrow
spot!” and without troubling himself for the moment how he should
return, he sped onward.
But when he had ridden far into the valley on his steed that could
outnumber ten leagues in an hour, and had returned to the entrance of
it, he saw no trace of a living soul on the opposite brink of the cleft.
No sign was left, save a few reeds bent down by the passage of the
mounted train, that any human being had stood on the opposite side for
ages.
The evening came on apace. Yet no one returned. Again he rode far into
the valley. For the most part it was covered with long grass, but here
and there a thick and tangled mass of vegetation attested to a great
luxuriance of soil, while the surface was intersected here and there
with rivulets of clear water, which finally lost their way in the dark
gorge over which he had just so rashly adventured. But on no side did
the steep cliffs offer any promise of escape.
When the night came on he stretched himself beneath one of the few trees
not far from the ravine, while his faithful horse stood tranquilly at
his head.
He did not awake till the moon had risen. But then suddenly he started
to his feet, and walking to the edge of the cleft, peered over to the
land from whence he had come. For he thought he heard sounds of some
kind that were not the natural ones of the rustling wind or the falling
water. Looking out he saw clearly opposite to him an old man in ragged
clothing, leaning against a rock, holding a long pipe in his hands, on
which he now and again played a few wild notes.
“Oho, peasant!” cried the king. “Run and tell the head man of your
village that the king bids him come directly, and will have him bring
with him the longest ropes and the strongest throwers under him.”
But the old man did not seem to give heed. Then the king cried,
“Hearken, old man, run quickly and tell your master that the king is
confined here, and will reward him beyond his dreams if he deliver him
quickly.”
Then the old man rose, and coming nearer to the edge of the ravine stood
opposite, still playing at intervals some notes on his long pipe. And
the king cried, “Canst thou hear? Dost thou dare to refuse to carry my
commands? For I am the king of Persia. Who art thou?”
Then the old man made answer, putting his pipe aside: “I am he who
appears only when a man has passed for ever beyond the ken of all that
have known him. I am Demiourgos, the maker of men.”
Then the king cried, “Mock me not, but obey my commands.”
The old man made answer, “I do not mock thee; and oh, my Lord, thou hast
moved the puppets I have made, and driven them so to dance on the
surface of the earth that I would willingly obey thee. But it is not
permitted me to pass between thee and the world of men thou hast known.”
Then the king was silent.
At length he said, “If thou art really what thou sayest, show me what
thou canst do; build me a palace.”
The old man lifted his pipe in both his trembling hands, and began to
blow.
It was a strange instrument, for it not only produced the shriller
sounds of the lute, and the piercing notes of the trumpet, but resounded
with the hollow booming of great organ pipes, and amongst all came ever
and again a sharp and sonorous clang as of some metal instrument
resounding when it was struck.
And then the king was as one who enjoys the delights of thought. For in
thought, delicate shades, impalpable nuances are ever passing. It is as
the blended strains of an invisible orchestra, but more subtle far, that
come and go in unexpected metres, and overwhelm you with their beauty
when all seemed silent. And lo, as the strains sound, outside—palpable,
large as the firmament, or real as the smallest thing you can take up
and know it is there—outside stands some existence revealed—to be known
and returned to for ever.
So the king, listening to this music, felt that something was rising
behind him. And turning, beheld course after course of a great building.
Almost as soon as he had looked it had risen completed, finished to the
last embossure on the windows, the tracery on the highest pinnacles. All
had happened while the old man was blowing on his pipe, and when he
ceased all was perfect.
And yet the appearance was very strange, for a finished and seemingly
habitable building rose out of waste unreclaimed soil, strewn with rocks
and barren. No dwellings were near the palace to wait on it, no roads
led to it or away from it.
“There should be houses around it, and roadways,” said the king; “make
them, and fields sown with corn, and all that is necessary for a state.”
Blowing on his pipe in regular recurrent cadences, the old man called up
houses close together, than scattered singly along roads which stretched
away into the distance, to be seen every here and there perfectly
clearly where they ascended a rising ground. And near at hand could be
distinguished fields of grain and pasture land.
Yet as the king turned to walk towards the new scene, the old man
laughed. “All this is a dream,” he cried; “so much I can do, but not at
once.” And breathing peals of music from his pipe, he said, “This can
be, but is not yet.”
“What,” asked the king, “is all a delusion?” and as he asked everything
sank down. There was no palace, no houses or fields, only the steep
precipice-locked valley, whither the king had ridden; and his horse
cowering behind him.
Then the king cried, “Thou art some moonstruck hermit, leading out a
life of folly alone. Get thee to the village thou knowest, and bring me
help.”
But the old man answered him saying, “Great king, I am bound to obey
thee, and all the creative might of my being I lay at thy feet; and lo,
in the midst of this valley I make for thee beings such as I can
produce. And all that thou hast seen is as nothing to what I can do for
thee. The depths of the starry heavens have no limit, nor what I do for
thee. Hast thou ever in thy life looked into the deep still ocean, and
lost thy sight in the unseen depths? Even so thou wilt find no end in
what I will give thee. Hast thou ever in thy life sought the depths of
thy love’s blue eyes, and found therein a world which stretched on
endlessly? Even so I bring all to thy feet. Now that all the gladness of
the world has departed from thee, behold, I am a more willing servant
than ever thou hast had.”
And again he played, and a hut rose up with a patch of cleared soil
around it, and a spring near by.
Then the king said, “Here will I dwell, and if I am to be cut off from
the rest of the world, I will lead a peaceful life in this valley.”
The sun was rising, the sounds had ceased, and the old man had
disappeared.