What Is the Fourth Dimension?
1884
[From Scientific Romances, Vol. 1 (1884).
Copy-text: pp 1-22,
Speculations on the Fourth Dimension,
Selected Writings of Charles H. Hinton, Copyright 1980
by Dover Publications, Inc., ISBN 0-486-23916-0,
LC 79-54399.]
Limitations of Our Existence
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
civilizations 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 civilization,
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 realize 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.
Constructing a Four-Square
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 x 2. This 2 x
2 is generally written 22, and named "2
squared."
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 x 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 x 2 x 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 x 2 x 2 x 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. Figure 1
represents a straight line; figure 2 represents a
square formed by the motion of that straight
line; figure 3 represents perspectively a cube
formed by the motion of that square ABCD upwards.
It would be well, instead of using figure 3, to
place a cube on the paper. Its base would be
ABCD, its upper surface EFGH.
The straight
line AB gives rise to the square
ABCD by a movement at right angles to itself. If
motion be confined to the straight line AB, 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 AB, it would have no
idea of any other movement than to and fro. The
square ABCD is formed from the straight line by a
movement in a direction entirely different from
the direction which exists in AB. This motion is
not expressible by means of any possible motion in
AB. A being which existed in AB, and whose
experience was limited to what could occur in AB,
would not be able to understand the instructions
we should give to make AB trace out the figure
ABCD.
In the
figure ABCD 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 figure 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 figure 2 into what
figure 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 ABCDEFGH.
All the
directions, manifold as they are, in
which a creature existing in figure 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 figure 3, something of the same kind as was
done to figure 1 to turn it into figure 2, or to
figure 2 to turn it into figure 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 figure 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 figure 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
figure 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
AB, and ends as CD; thus the line appears twice,
or it is doubled. The two other lines in the
square, AC, BD, 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 manv lines as there are points in the
previous figure. Now in the cube there are twelve
lines and eight points. Hence we get 2 x 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 realizing 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. Figure 3 may
be considered to be such a representation if the
square CDGH be considered to be the one that has
been pushed away from lying originally under the
square ABEF. 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.
Properties of Matter in the Fourth Dimension
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.
If the
line be supposed to be affixed to all
parts of the framework, and to loop over one
another, and support one another (ABCD framework,
X and Y two lines interlinked), it is obvious that
they could assume all sorts of figures, and that
the points on the plane would move in very
complicated paths. Figure 4 represents a section
of such a framework. Two lines XX and YY 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
I 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 realization 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 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. Organized 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.
Evidence of a Fourth Dimension
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 cognizance 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 centers of attraction whose influence
radiates out in every direction, so in this ampler
space there are centers 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 center of
attraction lying out of our space, and acting on
all the matter in it? The effect of such a center
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 center 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 center 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 center 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
center of attraction lies obliquely to the plane;
he would find that in this position a liquid would
show a tendency 40 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 center, 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 dire(Sion 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.
