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Title: The Fourth Dimension
Author: C. Howard Hinton
Release Date: January 12, 2022 [eBook #67153]
Language: English
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Transcriber’s Notes
Obvious typographical errors have been silently corrected. All other
spelling and punctuation remains unchanged.
Italics are represented thus _italic_, bold thus =bold= and
superscripts thus y^{en}.
It should be noted that much of the text is a discussion centred on the
many illustrations which have not been included.
THE FOURTH DIMENSION
SOME OPINIONS OF THE PRESS
“_Mr. C. H. Hinton discusses the subject of the higher dimensionality
of space, his aim being to avoid mathematical subtleties and
technicalities, and thus enable his argument to be followed by readers
who are not sufficiently conversant with mathematics to follow these
processes of reasoning._”—NOTTS GUARDIAN.
“_The fourth dimension is a subject which has had a great fascination
for many teachers, and though one cannot pretend to have quite grasped
Mr. Hinton’s conceptions and arguments, yet it must be admitted that
he reveals the elusive idea in quite a fascinating light. Quite
apart from the main thesis of the book many chapters are of great
independent interest. Altogether an interesting, clever and ingenious
book._”—DUNDEE COURIER.
“_The book will well repay the study of men who like to exercise their
wits upon the problems of abstract thought._”—SCOTSMAN.
“_Professor Hinton has done well to attempt a treatise of moderate
size, which shall at once be clear in method and free from
technicalities of the schools._”—PALL MALL GAZETTE.
“_A very interesting book he has made of it._”—PUBLISHERS’ CIRCULAR.
“_Mr. Hinton tries to explain the theory of the fourth dimension so
that the ordinary reasoning mind can get a grasp of what metaphysical
mathematicians mean by it. If he is not altogether successful it is not
from want of clearness on his part, but because the whole theory comes
as such an absolute shock to all one’s preconceived ideas._”—BRISTOL
TIMES.
“_Mr. Hinton’s enthusiasm is only the result of an exhaustive study,
which has enabled him to set his subject before the reader with far
more than the amount of lucidity to which it is accustomed._”—PALL MALL
GAZETTE.
“_The book throughout is a very solid piece of reasoning in the domain
of higher mathematics._”—GLASGOW HERALD.
“_Those who wish to grasp the meaning of this somewhat difficult
subject would do well to read_ The Fourth Dimension. _No mathematical
knowledge is demanded of the reader, and any one, who is not afraid of
a little hard thinking, should be able to follow the argument._”—LIGHT.
“_A splendidly clear re-statement of the old problem of the fourth
dimension. All who are interested in this subject will find the
work not only fascinating, but lucid, it being written in a style
easily understandable. The illustrations make still more clear
the letterpress, and the whole is most admirably adapted to the
requirements of the novice or the student._”—TWO WORLDS.
“_Those in search of mental gymnastics will find abundance of exercise
in Mr. C. H. Hinton’s_ Fourth Dimension.”—WESTMINSTER REVIEW.
FIRST EDITION, _April 1904_; SECOND EDITION, _May 1906_.
Views of the Tessaract.
No. 1. No. 2. No. 3.
No. 4. No. 5. No. 6.
No. 7. No. 8. No. 9.
No. 10. No. 11. No. 12.
THE
FOURTH DIMENSION
BY
C. HOWARD HINTON, M.A.
AUTHOR OF “SCIENTIFIC ROMANCES”
“A NEW ERA OF THOUGHT,” ETC., ETC.
[Illustration: Colophon]
LONDON
SWAN SONNENSCHEIN & CO., LIMITED
25 HIGH STREET, BLOOMSBURY
1906
PRINTED BY
HAZELL, WATSON AND VINEY, LD.,
LONDON AND AYLESBURY.
PREFACE
I have endeavoured to present the subject of the higher dimensionality
of space in a clear manner, devoid of mathematical subtleties and
technicalities. In order to engage the interest of the reader, I have
in the earlier chapters dwelt on the perspective the hypothesis of a
fourth dimension opens, and have treated of the many connections there
are between this hypothesis and the ordinary topics of our thoughts.
A lack of mathematical knowledge will prove of no disadvantage to the
reader, for I have used no mathematical processes of reasoning. I have
taken the view that the space which we ordinarily think of, the space
of real things (which I would call permeable matter), is different from
the space treated of by mathematics. Mathematics will tell us a great
deal about space, just as the atomic theory will tell us a great deal
about the chemical combinations of bodies. But after all, a theory is
not precisely equivalent to the subject with regard to which it is
held. There is an opening, therefore, from the side of our ordinary
space perceptions for a simple, altogether rational, mechanical, and
observational way of treating this subject of higher space, and of
this opportunity I have availed myself.
The details introduced in the earlier chapters, especially in
Chapters VIII., IX., X., may perhaps be found wearisome. They are of
no essential importance in the main line of argument, and if left
till Chapters XI. and XII. have been read, will be found to afford
interesting and obvious illustrations of the properties discussed in
the later chapters.
My thanks are due to the friends who have assisted me in designing and
preparing the modifications of my previous models, and in no small
degree to the publisher of this volume, Mr. Sonnenschein, to whose
unique appreciation of the line of thought of this, as of my former
essays, their publication is owing. By the provision of a coloured
plate, in addition to the other illustrations, he has added greatly to
the convenience of the reader.
C. HOWARD HINTON.
CONTENTS
CHAP. PAGE
I. FOUR-DIMENSIONAL SPACE 1
II. THE ANALOGY OF A PLANE WORLD 6
III. THE SIGNIFICANCE OF A FOUR-DIMENSIONAL
EXISTENCE 15
IV. THE FIRST CHAPTER IN THE HISTORY OF FOUR
SPACE 23
V. THE SECOND CHAPTER IN THE HISTORY OF
FOUR SPACE 41
Lobatchewsky, Bolyai, and Gauss
Metageometry
VI. THE HIGHER WORLD 61
VII. THE EVIDENCE FOR A FOURTH DIMENSION 76
VIII. THE USE OF FOUR DIMENSIONS IN THOUGHT 85
IX. APPLICATION TO KANT’S THEORY OF EXPERIENCE 107
X. A FOUR-DIMENSIONAL FIGURE 122
XI. NOMENCLATURE AND ANALOGIES 136
XII. THE SIMPLEST FOUR-DIMENSIONAL SOLID 157
XIII. REMARKS ON THE FIGURES 178
XIV. A RECAPITULATION AND EXTENSION OF THE
PHYSICAL ARGUMENT 203
APPENDIX I.—THE MODELS 231
" II.—A LANGUAGE OF SPACE 248
THE FOURTH DIMENSION
CHAPTER I
FOUR-DIMENSIONAL SPACE
There is nothing more indefinite, and at the same time more real, than
that which we indicate when we speak of the “higher.” In our social
life we see it evidenced in a greater complexity of relations. But this
complexity is not all. There is, at the same time, a contact with, an
apprehension of, something more fundamental, more real.
With the greater development of man there comes a consciousness of
something more than all the forms in which it shows itself. There is
a readiness to give up all the visible and tangible for the sake of
those principles and values of which the visible and tangible are the
representation. The physical life of civilised man and of a mere savage
are practically the same, but the civilised man has discovered a depth
in his existence, which makes him feel that that which appears all to
the savage is a mere externality and appurtenage to his true being.
Now, this higher—how shall we apprehend it? It is generally embraced
by our religious faculties, by our idealising tendency. But the higher
existence has two sides. It has a being as well as qualities. And in
trying to realise it through our emotions we are always taking the
subjective view. Our attention is always fixed on what we feel, what
we think. Is there any way of apprehending the higher after the purely
objective method of a natural science? I think that there is.
Plato, in a wonderful allegory, speaks of some men living in such a
condition that they were practically reduced to be the denizens of
a shadow world. They were chained, and perceived but the shadows of
themselves and all real objects projected on a wall, towards which
their faces were turned. All movements to them were but movements
on the surface, all shapes but the shapes of outlines with no
substantiality.
Plato uses this illustration to portray the relation between true
being and the illusions of the sense world. He says that just as a man
liberated from his chains could learn and discover that the world was
solid and real, and could go back and tell his bound companions of this
greater higher reality, so the philosopher who has been liberated, who
has gone into the thought of the ideal world, into the world of ideas
greater and more real than the things of sense, can come and tell his
fellow men of that which is more true than the visible sun—more noble
than Athens, the visible state.
Now, I take Plato’s suggestion; but literally, not metaphorically.
He imagines a world which is lower than this world, in that shadow
figures and shadow motions are its constituents; and to it he contrasts
the real world. As the real world is to this shadow world, so is the
higher world to our world. I accept his analogy. As our world in three
dimensions is to a shadow or plane world, so is the higher world to our
three-dimensional world. That is, the higher world is four-dimensional;
the higher being is, so far as its existence is concerned apart from
its qualities, to be sought through the conception of an actual
existence spatially higher than that which we realise with our senses.
Here you will observe I necessarily leave out all that gives its
charm and interest to Plato’s writings. All those conceptions of the
beautiful and good which live immortally in his pages.
All that I keep from his great storehouse of wealth is this one thing
simply—a world spatially higher than this world, a world which can only
be approached through the stocks and stones of it, a world which must
be apprehended laboriously, patiently, through the material things of
it, the shapes, the movements, the figures of it.
We must learn to realise the shapes of objects in this world of the
higher man; we must become familiar with the movements that objects
make in his world, so that we can learn something about his daily
experience, his thoughts of material objects, his machinery.
The means for the prosecution of this enquiry are given in the
conception of space itself.
It often happens that that which we consider to be unique and unrelated
gives us, within itself, those relations by means of which we are able
to see it as related to others, determining and determined by them.
Thus, on the earth is given that phenomenon of weight by means of which
Newton brought the earth into its true relation to the sun and other
planets. Our terrestrial globe was determined in regard to other bodies
of the solar system by means of a relation which subsisted on the earth
itself.
And so space itself bears within it relations of which we can
determine it as related to other space. For within space are given the
conceptions of point and line, line and plane, which really involve the
relation of space to a higher space.
Where one segment of a straight line leaves off and another begins is
a point, and the straight line itself can be generated by the motion of
the point.
One portion of a plane is bounded from another by a straight line, and
the plane itself can be generated by the straight line moving in a
direction not contained in itself.
Again, two portions of solid space are limited with regard to each
other by a plane; and the plane, moving in a direction not contained in
itself, can generate solid space.
Thus, going on, we may say that space is that which limits two portions
of higher space from each other, and that our space will generate the
higher space by moving in a direction not contained in itself.
Another indication of the nature of four-dimensional space can be
gained by considering the problem of the arrangement of objects.
If I have a number of swords of varying degrees of brightness, I can
represent them in respect of this quality by points arranged along a
straight line.
If I place a sword at A, fig. 1, and regard it as having a certain
brightness, then the other swords can be arranged in a series along the
line, as at A, B, C, etc., according to their degrees of brightness.
[Illustration: Fig. 1.]
If now I take account of another quality, say length, they can be
arranged in a plane. Starting from A, B, C, I can find points to
represent different degrees of length along such lines as AF, BD, CE,
drawn from A and B and C. Points on these lines represent different
degrees of length with the same degree of brightness. Thus the whole
plane is occupied by points representing all conceivable varieties of
brightness and length.
[Illustration: Fig. 2.]
Bringing in a third quality, say sharpness, I can draw, as in fig. 3,
any number of upright lines. Let distances along these upright lines
represent degrees of sharpness, thus the points F and G will represent
swords of certain definite degrees of the three qualities mentioned,
and the whole of space will serve to represent all conceivable degrees
of these three qualities.
[Illustration: Fig. 3.]
If now I bring in a fourth quality, such as weight, and try to find a
means of representing it as I did the other three qualities, I find
a difficulty. Every point in space is taken up by some conceivable
combination of the three qualities already taken.
To represent four qualities in the same way as that in which I have
represented three, I should need another dimension of space.
Thus we may indicate the nature of four-dimensional space by saying
that it is a kind of space which would give positions representative
of four qualities, as three-dimensional space gives positions
representative of three qualities.
CHAPTER II
THE ANALOGY OF A PLANE WORLD
At the risk of some prolixity I will go fully into the experience of
a hypothetical creature confined to motion on a plane surface. By so
doing I shall obtain an analogy which will serve in our subsequent
enquiries, because the change in our conception, which we make in
passing from the shapes and motions in two dimensions to those in
three, affords a pattern by which we can pass on still further to the
conception of an existence in four-dimensional space.
A piece of paper on a smooth table affords a ready image of a
two-dimensional existence. If we suppose the being represented by
the piece of paper to have no knowledge of the thickness by which
he projects above the surface of the table, it is obvious that he
can have no knowledge of objects of a similar description, except by
the contact with their edges. His body and the objects in his world
have a thickness of which however, he has no consciousness. Since
the direction stretching up from the table is unknown to him he will
think of the objects of his world as extending in two dimensions only.
Figures are to him completely bounded by their lines, just as solid
objects are to us by their surfaces. He cannot conceive of approaching
the centre of a circle, except by breaking through the circumference,
for the circumference encloses the centre in the directions in which
motion is possible to him. The plane surface over which he slips and
with which he is always in contact will be unknown to him; there are no
differences by which he can recognise its existence.
But for the purposes of our analogy this representation is deficient.
A being as thus described has nothing about him to push off from, the
surface over which he slips affords no means by which he can move in
one direction rather than another. Placed on a surface over which he
slips freely, he is in a condition analogous to that in which we should
be if we were suspended free in space. There is nothing which he can
push off from in any direction known to him.
Let us therefore modify our representation. Let us suppose a vertical
plane against which particles of thin matter slip, never leaving the
surface. Let these particles possess an attractive force and cohere
together into a disk; this disk will represent the globe of a plane
being. He must be conceived as existing on the rim.
[Illustration: Fig. 4.]
Let 1 represent this vertical disk of flat matter and 2 the plane being
on it, standing upon its rim as we stand on the surface of our earth.
The direction of the attractive force of his matter will give the
creature a knowledge of up and down, determining for him one direction
in his plane space. Also, since he can move along the surface of his
earth, he will have the sense of a direction parallel to its surface,
which we may call forwards and backwards.
He will have no sense of right and left—that is, of the direction which
we recognise as extending out from the plane to our right and left.
The distinction of right and left is the one that we must suppose to
be absent, in order to project ourselves into the condition of a plane
being.
Let the reader imagine himself, as he looks along the plane, fig. 4,
to become more and more identified with the thin body on it, till he
finally looks along parallel to the surface of the plane earth, and up
and down, losing the sense of the direction which stretches right and
left. This direction will be an unknown dimension to him.
Our space conceptions are so intimately connected with those which
we derive from the existence of gravitation that it is difficult to
realise the condition of a plane being, without picturing him as in
material surroundings with a definite direction of up and down. Hence
the necessity of our somewhat elaborate scheme of representation,
which, when its import has been grasped, can be dispensed with for the
simpler one of a thin object slipping over a smooth surface, which lies
in front of us.
It is obvious that we must suppose some means by which the plane being
is kept in contact with the surface on which he slips. The simplest
supposition to make is that there is a transverse gravity, which keeps
him to the plane. This gravity must be thought of as different to the
attraction exercised by his matter, and as unperceived by him.
At this stage of our enquiry I do not wish to enter into the question
of how a plane being could arrive at a knowledge of the third
dimension, but simply to investigate his plane consciousness.
It is obvious that the existence of a plane being must be very limited.
A straight line standing up from the surface of his earth affords a bar
to his progress. An object like a wheel which rotates round an axis
would be unknown to him, for there is no conceivable way in which he
can get to the centre without going through the circumference. He would
have spinning disks, but could not get to the centre of them. The plane
being can represent the motion from any one point of his space to any
other, by means of two straight lines drawn at right angles to each
other.
Let AX and AY be two such axes. He can accomplish the translation from
A to B by going along AX to C, and then from C along CB parallel to AY.
The same result can of course be obtained by moving to D along AY and
then parallel to AX from D to B, or of course by any diagonal movement
compounded by these axial movements.
[Illustration: Fig. 5.]
By means of movements parallel to these two axes he can proceed (except
for material obstacles) from any one point of his space to any other.
If now we suppose a third line drawn out from A at right angles to the
plane it is evident that no motion in either of the two dimensions he
knows will carry him in the least degree in the direction represented
by AZ.
[Illustration: Fig. 6.]
The lines AZ and AX determine a plane. If he could be taken off his
plane, and transferred to the plane AXZ, he would be in a world exactly
like his own. From every line in his world there goes off a space world
exactly like his own.
[Illustration: Fig. 7.]
From every point in his world a line can be drawn parallel to AZ in
the direction unknown to him. If we suppose the square in fig. 7 to be
a geometrical square from every point of it, inside as well as on the
contour, a straight line can be drawn parallel to AZ. The assemblage
of these lines constitute a solid figure, of which the square in the
plane is the base. If we consider the square to represent an object
in the plane being’s world then we must attribute to it a very small
thickness, for every real thing must possess all three dimensions.
This thickness he does not perceive, but thinks of this real object as
a geometrical square. He thinks of it as possessing area only, and no
degree of solidity. The edges which project from the plane to a very
small extent he thinks of as having merely length and no breadth—as
being, in fact, geometrical lines.
With the first step in the apprehension of a third dimension there
would come to a plane being the conviction that he had previously
formed a wrong conception of the nature of his material objects. He
had conceived them as geometrical figures of two dimensions only. If a
third dimension exists, such figures are incapable of real existence.
Thus he would admit that all his real objects had a certain, though
very small thickness in the unknown dimension, and that the conditions
of his existence demanded the supposition of an extended sheet of
matter, from contact with which in their motion his objects never
diverge.
Analogous conceptions must be formed by us on the supposition of a
four-dimensional existence. We must suppose a direction in which we can
never point extending from every point of our space. We must draw a
distinction between a geometrical cube and a cube of real matter. The
cube of real matter we must suppose to have an extension in an unknown
direction, real, but so small as to be imperceptible by us. From every
point of a cube, interior as well as exterior, we must imagine that it
is possible to draw a line in the unknown direction. The assemblage of
these lines would constitute a higher solid. The lines going off in
the unknown direction from the face of a cube would constitute a cube
starting from that face. Of this cube all that we should see in our
space would be the face.
Again, just as the plane being can represent any motion in his space by
two axes, so we can represent any motion in our three-dimensional space
by means of three axes. There is no point in our space to which we
cannot move by some combination of movements on the directions marked
out by these axes.
On the assumption of a fourth dimension we have to suppose a fourth
axis, which we will call AW. It must be supposed to be at right angles
to each and every one of the three axes AX, AY, AZ. Just as the two
axes, AX, AZ, determine a plane which is similar to the original plane
on which we supposed the plane being to exist, but which runs off from
it, and only meets it in a line; so in our space if we take any three
axes such as AX, AY, and AW, they determine a space like our space
world. This space runs off from our space, and if we were transferred
to it we should find ourselves in a space exactly similar to our own.
We must give up any attempt to picture this space in its relation
to ours, just as a plane being would have to give up any attempt to
picture a plane at right angles to his plane.
Such a space and ours run in different directions from the plane of AX
and AY. They meet in this plane but have nothing else in common, just
as the plane space of AX and AY and that of AX and AZ run in different
directions and have but the line AX in common.
Omitting all discussion of the manner on which a plane being might be
conceived to form a theory of a three-dimensional existence, let us
examine how, with the means at his disposal, he could represent the
properties of three-dimensional objects.
There are two ways in which the plane being can think of one of our
solid bodies. He can think of the cube, fig. 8, as composed of a number
of sections parallel to his plane, each lying in the third dimension
a little further off from his plane than the preceding one. These
sections he can represent as a series of plane figures lying in his
plane, but in so representing them he destroys the coherence of them
in the higher figure. The set of squares, A, B, C, D, represents the
section parallel to the plane of the cube shown in figure, but they are
not in their proper relative positions.
[Illustration: Fig. 8.]
The plane being can trace out a movement in the third dimension by
assuming discontinuous leaps from one section to another. Thus,
a motion along the edge of the cube from left to right would be
represented in the set of sections in the plane as the succession of
the corners of the sections A, B, C, D. A point moving from A through
BCD in our space must be represented in the plane as appearing in A,
then in B, and so on, without passing through the intervening plane
space.
In these sections the plane being leaves out, of course, the extension
in the third dimension; the distance between any two sections is not
represented. In order to realise this distance the conception of motion
can be employed.
[Illustration: Fig. 9.]
Let fig. 9 represent a cube passing transverse to the plane. It will
appear to the plane being as a square object, but the matter of which
this object is composed will be continually altering. One material
particle takes the place of another, but it does not come from anywhere
or go anywhere in the space which the plane being knows.
The analogous manner of representing a higher solid in our case, is to
conceive it as composed of a number of sections, each lying a little
further off in the unknown direction than the preceding.
[Illustration: Fig. 10.]
We can represent these sections as a number of solids. Thus the cubes
A, B, C, D, may be considered as the sections at different intervals in
the unknown dimension of a higher cube. Arranged thus their coherence
in the higher figure is destroyed, they are mere representations.
A motion in the fourth dimension from A through B, C, etc., would be
continuous, but we can only represent it as the occupation of the
positions A, B, C, etc., in succession. We can exhibit the results of
the motion at different stages, but no more.
In this representation we have left out the distance between one
section and another; we have considered the higher body merely as a
series of sections, and so left out its contents. The only way to
exhibit its contents is to call in the aid of the conception of motion.
[Illustration: Fig. 11.]
If a higher cube passes transverse to our space, it will appear as
a cube isolated in space, the part that has not come into our space
and the part that has passed through will not be visible. The gradual
passing through our space would appear as the change of the matter
of the cube before us. One material particle in it is succeeded by
another, neither coming nor going in any direction we can point to. In
this manner, by the duration of the figure, we can exhibit the higher
dimensionality of it; a cube of our matter, under the circumstances
supposed, namely, that it has a motion transverse to our space, would
instantly disappear. A higher cube would last till it had passed
transverse to our space by its whole distance of extension in the
fourth dimension.
As the plane being can think of the cube as consisting of sections,
each like a figure he knows, extending away from his plane, so we can
think of a higher solid as composed of sections, each like a solid
which we know, but extending away from our space.
Thus, taking a higher cube, we can look on it as starting from a cube
in our space and extending in the unknown dimension.
[Illustration: Fig. 12.]
Take the face A and conceive it to exist as simply a face, a square
with no thickness. From this face the cube in our space extends by the
occupation of space which we can see.
But from this face there extends equally a cube in the unknown
dimension. We can think of the higher cube, then, by taking the set
of sections A, B, C, D, etc., and considering that from each of them
there runs a cube. These cubes have nothing in common with each other,
and of each of them in its actual position all that we can have in our
space is an isolated square. It is obvious that we can take our series
of sections in any manner we please. We can take them parallel, for
instance, to any one of the three isolated faces shown in the figure.
Corresponding to the three series of sections at right angles to each
other, which we can make of the cube in space, we must conceive of the
higher cube, as composed of cubes starting from squares parallel to the
faces of the cube, and of these cubes all that exist in our space are
the isolated squares from which they start.
CHAPTER III
THE SIGNIFICANCE OF A FOUR-DIMENSIONAL EXISTENCE
Having now obtained the conception of a four-dimensional space, and
having formed the analogy which, without any further geometrical
difficulties, enables us to enquire into its properties, I will refer
the reader, whose interest is principally in the mechanical aspect,
to Chapters VI. and VII. In the present chapter I will deal with
the general significance of the enquiry, and in the next with the
historical origin of the idea.
First, with regard to the question of whether there is any evidence
that we are really in four-dimensional space, I will go back to the
analogy of the plane world.
A being in a plane world could not have any experience of
three-dimensional shapes, but he could have an experience of
three-dimensional movements.
We have seen that his matter must be supposed to have an extension,
though a very small one, in the third dimension. And thus, in the
small particles of his matter, three-dimensional movements may well
be conceived to take place. Of these movements he would only perceive
the resultants. Since all movements of an observable size in the plane
world are two-dimensional, he would only perceive the resultants in
two dimensions of the small three-dimensional movements. Thus, there
would be phenomena which he could not explain by his theory of
mechanics—motions would take place which he could not explain by his
theory of motion. Hence, to determine if we are in a four-dimensional
world, we must examine the phenomena of motion in our space. If
movements occur which are not explicable on the suppositions of our
three-dimensional mechanics, we should have an indication of a possible
four-dimensional motion, and if, moreover, it could be shown that such
movements would be a consequence of a four-dimensional motion in the
minute particles of bodies or of the ether, we should have a strong
presumption in favour of the reality of the fourth dimension.
By proceeding in the direction of finer and finer subdivision, we come
to forms of matter possessing properties different to those of the
larger masses. It is probable that at some stage in this process we
should come to a form of matter of such minute subdivision that its
particles possess a freedom of movement in four dimensions. This form
of matter I speak of as four-dimensional ether, and attribute to it
properties approximating to those of a perfect liquid.
Deferring the detailed discussion of this form of matter to Chapter
VI., we will now examine the means by which a plane being would come to
the conclusion that three-dimensional movements existed in his world,
and point out the analogy by which we can conclude the existence of
four-dimensional movements in our world. Since the dimensions of the
matter in his world are small in the third direction, the phenomena in
which he would detect the motion would be those of the small particles
of matter.
Suppose that there is a ring in his plane. We can imagine currents
flowing round the ring in either of two opposite directions. These
would produce unlike effects, and give rise to two different fields
of influence. If the ring with a current in it in one direction be
taken up and turned over, and put down again on the plane, it would be
identical with the ring having a current in the opposite direction. An
operation of this kind would be impossible to the plane being. Hence
he would have in his space two irreconcilable objects, namely, the
two fields of influence due to the two rings with currents in them in
opposite directions. By irreconcilable objects in the plane I mean
objects which cannot be thought of as transformed one into the other by
any movement in the plane.
Instead of currents flowing in the rings we can imagine a different
kind of current. Imagine a number of small rings strung on the original
ring. A current round these secondary rings would give two varieties
of effect, or two different fields of influence, according to its
direction. These two varieties of current could be turned one into
the other by taking one of the rings up, turning it over, and putting
it down again in the plane. This operation is impossible to the plane
being, hence in this case also there would be two irreconcilable fields
in the plane. Now, if the plane being found two such irreconcilable
fields and could prove that they could not be accounted for by currents
in the rings, he would have to admit the existence of currents round
the rings—that is, in rings strung on the primary ring. Thus he would
come to admit the existence of a three-dimensional motion, for such a
disposition of currents is in three dimensions.
Now in our space there are two fields of different properties, which
can be produced by an electric current flowing in a closed circuit or
ring. These two fields can be changed one into the other by reversing
the currents, but they cannot be changed one into the other by any
turning about of the rings in our space; for the disposition of the
field with regard to the ring itself is different when we turn the
ring, over and when we reverse the direction of the current in the ring.
As hypotheses to explain the differences of these two fields and their
effects we can suppose the following kinds of space motions:—First, a
current along the conductor; second, a current round the conductor—that
is, of rings of currents strung on the conductor as an axis. Neither of
these suppositions accounts for facts of observation.
Hence we have to make the supposition of a four-dimensional motion.
We find that a four-dimensional rotation of the nature explained in a
subsequent chapter, has the following characteristics:—First, it would
give us two fields of influence, the one of which could be turned into
the other by taking the circuit up into the fourth dimension, turning
it over, and putting it down in our space again, precisely as the two
kinds of fields in the plane could be turned one into the other by a
reversal of the current in our space. Second, it involves a phenomenon
precisely identical with that most remarkable and mysterious feature of
an electric current, namely that it is a field of action, the rim of
which necessarily abuts on a continuous boundary formed by a conductor.
Hence, on the assumption of a four-dimensional movement in the region
of the minute particles of matter, we should expect to find a motion
analogous to electricity.
Now, a phenomenon of such universal occurrence as electricity cannot be
due to matter and motion in any very complex relation, but ought to be
seen as a simple and natural consequence of their properties. I infer
that the difficulty in its theory is due to the attempt to explain a
four-dimensional phenomenon by a three-dimensional geometry.
In view of this piece of evidence we cannot disregard that afforded
by the existence of symmetry. In this connection I will allude to the
simple way of producing the images of insects, sometimes practised by
children. They put a few blots of ink in a straight line on a piece of
paper, fold the paper along the blots, and on opening it the lifelike
presentment of an insect is obtained. If we were to find a multitude
of these figures, we should conclude that they had originated from a
process of folding over; the chances against this kind of reduplication
of parts is too great to admit of the assumption that they had been
formed in any other way.
The production of the symmetrical forms of organised beings, though not
of course due to a turning over of bodies of any appreciable size in
four-dimensional space, can well be imagined as due to a disposition in
that manner of the smallest living particles from which they are built
up. Thus, not only electricity, but life, and the processes by which we
think and feel, must be attributed to that region of magnitude in which
four-dimensional movements take place.
I do not mean, however, that life can be explained as a
four-dimensional movement. It seems to me that the whole bias of
thought, which tends to explain the phenomena of life and volition, as
due to matter and motion in some peculiar relation, is adopted rather
in the interests of the explicability of things than with any regard to
probability.
Of course, if we could show that life were a phenomenon of motion, we
should be able to explain a great deal that is at present obscure. But
there are two great difficulties in the way. It would be necessary to
show that in a germ capable of developing into a living being, there
were modifications of structure capable of determining in the developed
germ all the characteristics of its form, and not only this, but of
determining those of all the descendants of such a form in an infinite
series. Such a complexity of mechanical relations, undeniable though
it be, cannot surely be the best way of grouping the phenomena and
giving a practical account of them. And another difficulty is this,
that no amount of mechanical adaptation would give that element of
consciousness which we possess, and which is shared in to a modified
degree by the animal world.
In those complex structures which men build up and direct, such as a
ship or a railway train (and which, if seen by an observer of such a
size that the men guiding them were invisible, would seem to present
some of the phenomena of life) the appearance of animation is not due
to any diffusion of life in the material parts of the structure, but to
the presence of a living being.
The old hypothesis of a soul, a living organism within the visible one,
appears to me much more rational than the attempt to explain life as a
form of motion. And when we consider the region of extreme minuteness
characterised by four-dimensional motion the difficulty of conceiving
such an organism alongside the bodily one disappears. Lord Kelvin
supposes that matter is formed from the ether. We may very well suppose
that the living organisms directing the material ones are co-ordinate
with them, not composed of matter, but consisting of etherial bodies,
and as such capable of motion through the ether, and able to originate
material living bodies throughout the mineral.
Hypotheses such as these find no immediate ground for proof or disproof
in the physical world. Let us, therefore, turn to a different field,
and, assuming that the human soul is a four-dimensional being, capable
in itself of four dimensional movements, but in its experiences through
the senses limited to three dimensions, ask if the history of thought,
of these productivities which characterise man, correspond to our
assumption. Let us pass in review those steps by which man, presumably
a four-dimensional being, despite his bodily environment, has come to
recognise the fact of four-dimensional existence.
Deferring this enquiry to another chapter, I will here recapitulate the
argument in order to show that our purpose is entirely practical and
independent of any philosophical or metaphysical considerations.
If two shots are fired at a target, and the second bullet hits it
at a different place to the first, we suppose that there was some
difference in the conditions under which the second shot was fired
from those affecting the first shot. The force of the powder, the
direction of aim, the strength of the wind, or some condition must
have been different in the second case, if the course of the bullet
was not exactly the same as in the first case. Corresponding to every
difference in a result there must be some difference in the antecedent
material conditions. By tracing out this chain of relations we explain
nature.
But there is also another mode of explanation which we apply. If we ask
what was the cause that a certain ship was built, or that a certain
structure was erected, we might proceed to investigate the changes in
the brain cells of the men who designed the works. Every variation in
one ship or building from another ship or building is accompanied by
a variation in the processes that go on in the brain matter of the
designers. But practically this would be a very long task.
A more effective mode of explaining the production of the ship or
building would be to enquire into the motives, plans, and aims of the
men who constructed them. We obtain a cumulative and consistent body of
knowledge much more easily and effectively in the latter way.
Sometimes we apply the one, sometimes the other mode of explanation.
But it must be observed that the method of explanation founded on
aim, purpose, volition, always presupposes a mechanical system on
which the volition and aim works. The conception of man as willing and
acting from motives involves that of a number of uniform processes of
nature which he can modify, and of which he can make application. In
the mechanical conditions of the three-dimensional world, the only
volitional agency which we can demonstrate is the human agency. But
when we consider the four-dimensional world the conclusion remains
perfectly open.
The method of explanation founded on purpose and aim does not, surely,
suddenly begin with man and end with him. There is as much behind the
exhibition of will and motive which we see in man as there is behind
the phenomena of movement; they are co-ordinate, neither to be resolved
into the other. And the commencement of the investigation of that will
and motive which lies behind the will and motive manifested in the
three-dimensional mechanical field is in the conception of a soul—a
four-dimensional organism, which expresses its higher physical being
in the symmetry of the body, and gives the aims and motives of human
existence.
Our primary task is to form a systematic knowledge of the phenomena
of a four-dimensional world and find those points in which this
knowledge must be called in to complete our mechanical explanation of
the universe. But a subsidiary contribution towards the verification
of the hypothesis may be made by passing in review the history of
human thought, and enquiring if it presents such features as would be
naturally expected on this assumption.
CHAPTER IV
THE FIRST CHAPTER IN THE HISTORY OF FOUR SPACE
Parmenides, and the Asiatic thinkers with whom he is in close
affinity, propound a theory of existence which is in close accord
with a conception of a possible relation between a higher and a lower
dimensional space. This theory, prior and in marked contrast to the
main stream of thought, which we shall afterwards describe, forms a
closed circle by itself. It is one which in all ages has had a strong
attraction for pure intellect, and is the natural mode of thought for
those who refrain from projecting their own volition into nature under
the guise of causality.
According to Parmenides of the school of Elea the all is one, unmoving
and unchanging. The permanent amid the transient—that foothold for
thought, that solid ground for feeling on the discovery of which
depends all our life—is no phantom; it is the image amidst deception of
true being, the eternal, the unmoved, the one. Thus says Parmenides.
But how explain the shifting scene, these mutations of things!
“Illusion,” answers Parmenides. Distinguishing between truth and
error, he tells of the true doctrine of the one—the false opinion of a
changing world. He is no less memorable for the manner of his advocacy
than for the cause he advocates. It is as if from his firm foothold
of being he could play with the thoughts under the burden of which
others laboured, for from him springs that fluency of supposition and
hypothesis which forms the texture of Plato’s dialectic.
Can the mind conceive a more delightful intellectual picture than that
of Parmenides, pointing to the one, the true, the unchanging, and yet
on the other hand ready to discuss all manner of false opinion, forming
a cosmogony too, false “but mine own” after the fashion of the time?
In support of the true opinion he proceeded by the negative way of
showing the self-contradictions in the ideas of change and motion.
It is doubtful if his criticism, save in minor points, has ever been
successfully refuted. To express his doctrine in the ponderous modern
way we must make the statement that motion is phenomenal, not real.
Let us represent his doctrine.
[Illustration: Fig. 13.]
Imagine a sheet of still water into which a slanting stick is being
lowered with a motion vertically downwards. Let 1, 2, 3 (Fig. 13),
be three consecutive positions of the stick. A, B, C, will be three
consecutive positions of the meeting of the stick, with the surface of
the water. As the stick passes down, the meeting will move from A on to
B and C.
Suppose now all the water to be removed except a film. At the meeting
of the film and the stick there will be an interruption of the film.
If we suppose the film to have a property, like that of a soap bubble,
of closing up round any penetrating object, then as the stick goes
vertically downwards the interruption in the film will move on.
[Illustration: Fig. 14.]
If we pass a spiral through the film the intersection will give a point
moving in a circle shown by the dotted lines in the figure. Suppose
now the spiral to be still and the film to move vertically upwards,
the whole spiral will be represented in the film of the consecutive
positions of the point of intersection. In the film the permanent
existence of the spiral is experienced as a time series—the record
of traversing the spiral is a point moving in a circle. If now we
suppose a consciousness connected with the film in such a way that the
intersection of the spiral with the film gives rise to a conscious
experience, we see that we shall have in the film a point moving in a
circle, conscious of its motion, knowing nothing of that real spiral
the record of the successive intersections of which by the film is the
motion of the point.
It is easy to imagine complicated structures of the nature of the
spiral, structures consisting of filaments, and to suppose also that
these structures are distinguishable from each other at every section.
If we consider the intersections of these filaments with the film as it
passes to be the atoms constituting a filmar universe, we shall have in
the film a world of apparent motion; we shall have bodies corresponding
to the filamentary structure, and the positions of these structures
with regard to one another will give rise to bodies in the film moving
amongst one another. This mutual motion is apparent merely. The reality
is of permanent structures stationary, and all the relative motions
accounted for by one steady movement of the film as a whole.
Thus we can imagine a plane world, in which all the variety of motion
is the phenomenon of structures consisting of filamentary atoms
traversed by a plane of consciousness. Passing to four dimensions and
our space, we can conceive that all things and movements in our world
are the reading off of a permanent reality by a space of consciousness.
Each atom at every moment is not what it was, but a new part of that
endless line which is itself. And all this system successively revealed
in the time which is but the succession of consciousness, separate
as it is in parts, in its entirety is one vast unity. Representing
Parmenides’ doctrine thus, we gain a firmer hold on it than if we
merely let his words rest, grand and massive, in our minds. And we have
gained the means also of representing phases of that Eastern thought
to which Parmenides was no stranger. Modifying his uncompromising
doctrine, let us suppose, to go back to the plane of consciousness
and the structure of filamentary atoms, that these structures are
themselves moving—are acting, living. Then, in the transverse motion
of the film, there would be two phenomena of motion, one due to the
reading off in the film of the permanent existences as they are in
themselves, and another phenomenon of motion due to the modification of
the record of the things themselves, by their proper motion during the
process of traversing them.
Thus a conscious being in the plane would have, as it were, a
two-fold experience. In the complete traversing of the structure, the
intersection of which with the film gives his conscious all, the main
and principal movements and actions which he went through would be the
record of his higher self as it existed unmoved and unacting. Slight
modifications and deviations from these movements and actions would
represent the activity and self-determination of the complete being, of
his higher self.
It is admissible to suppose that the consciousness in the plane has
a share in that volition by which the complete existence determines
itself. Thus the motive and will, the initiative and life, of the
higher being, would be represented in the case of the being in the
film by an initiative and a will capable, not of determining any great
things or important movements in his existence, but only of small and
relatively insignificant activities. In all the main features of his
life his experience would be representative of one state of the higher
being whose existence determines his as the film passes on. But in his
minute and apparently unimportant actions he would share in that will
and determination by which the whole of the being he really is acts and
lives.
An alteration of the higher being would correspond to a different life
history for him. Let us now make the supposition that film after film
traverses these higher structures, that the life of the real being is
read off again and again in successive waves of consciousness. There
would be a succession of lives in the different advancing planes of
consciousness, each differing from the preceding, and differing in
virtue of that will and activity which in the preceding had not been
devoted to the greater and apparently most significant things in life,
but the minute and apparently unimportant. In all great things the
being of the film shares in the existence of his higher self as it is
at any one time. In the small things he shares in that volition by
which the higher being alters and changes, acts and lives.
Thus we gain the conception of a life changing and developing as a
whole, a life in which our separation and cessation and fugitiveness
are merely apparent, but which in its events and course alters,
changes, develops; and the power of altering and changing this whole
lies in the will and power the limited being has of directing, guiding,
altering himself in the minute things of his existence.
Transferring our conceptions to those of an existence in a higher
dimensionality traversed by a space of consciousness, we have an
illustration of a thought which has found frequent and varied
expression. When, however, we ask ourselves what degree of truth
there lies in it, we must admit that, as far as we can see, it is
merely symbolical. The true path in the investigation of a higher
dimensionality lies in another direction.
The significance of the Parmenidean doctrine lies in this that here, as
again and again, we find that those conceptions which man introduces of
himself, which he does not derive from the mere record of his outward
experience, have a striking and significant correspondence to the
conception of a physical existence in a world of a higher space. How
close we come to Parmenides’ thought by this manner of representation
it is impossible to say. What I want to point out is the adequateness
of the illustration, not only to give a static model of his doctrine,
but one capable as it were, of a plastic modification into a
correspondence into kindred forms of thought. Either one of two things
must be true—that four-dimensional conceptions give a wonderful power
of representing the thought of the East, or that the thinkers of the
East must have been looking at and regarding four-dimensional existence.
Coming now to the main stream of thought we must dwell in some detail
on Pythagoras, not because of his direct relation to the subject, but
because of his relation to investigators who came later.
Pythagoras invented the two-way counting. Let us represent the
single-way counting by the posits _aa_, _ab_, _ac_, _ad_, using these
pairs of letters instead of the numbers 1, 2, 3, 4. I put an _a_ in
each case first for a reason which will immediately appear.
We have a sequence and order. There is no conception of distance
necessarily involved. The difference between the posits is one of
order not of distance—only when identified with a number of equal
material things in juxtaposition does the notion of distance arise.
Now, besides the simple series I can have, starting from _aa_, _ba_,
_ca_, _da_, from _ab_, _bb_, _cb_, _db_, and so on, and forming a
scheme:
_da_ _db_ _dc_ _dd_
_ca_ _cb_ _cc_ _cd_
_ba_ _bb_ _bc_ _bd_
_aa_ _ab_ _ac_ _ad_
This complex or manifold gives a two-way order. I can represent it by
a set of points, if I am on my guard against assuming any relation of
distance.
[Illustration: Fig. 15.]
Pythagoras studied this two-fold way of counting in reference to
material bodies, and discovered that most remarkable property of the
combination of number and matter that bears his name.
The Pythagorean property of an extended material system can be
exhibited in a manner which will be of use to us afterwards, and which
therefore I will employ now instead of using the kind of figure which
he himself employed.
Consider a two-fold field of points arranged in regular rows. Such a
field will be presupposed in the following argument.
[Illustration: Fig. 16. 1 and 2]
It is evident that in fig. 16 four of the points determine a square,
which square we may take as the unit of measurement for areas. But we
can also measure areas in another way.
Fig. 16 (1) shows four points determining a square.
But four squares also meet in a point, fig. 16 (2).
Hence a point at the corner of a square belongs equally to four
squares.
Thus we may say that the point value of the square shown is one point,
for if we take the square in fig. 16 (1) it has four points, but each
of these belong equally to four other squares. Hence one fourth of each
of them belongs to the square (1) in fig. 16. Thus the point value of
the square is one point.
The result of counting the points is the same as that arrived at by
reckoning the square units enclosed.
Hence, if we wish to measure the area of any square we can take the
number of points it encloses, count these as one each, and take
one-fourth of the number of points at its corners.
[Illustration: Fig. 17.]
Now draw a diagonal square as shown in fig. 17. It contains one point
and the four corners count for one point more; hence its point value is
2. The value is the measure of its area—the size of this square is two
of the unit squares.
Looking now at the sides of this figure we see that there is a unit
square on each of them—the two squares contain no points, but have four
corner points each, which gives the point value of each as one point.
Hence we see that the square on the diagonal is equal to the squares
on the two sides; or as it is generally expressed, the square on the
hypothenuse is equal to the sum of the squares on the sides.
[Illustration: Fig. 18.]
Noticing this fact we can proceed to ask if it is always true. Drawing
the square shown in fig. 18, we can count the number of its points.
There are five altogether. There are four points inside the square on
the diagonal, and hence, with the four points at its corners the point
value is 5—that is, the area is 5. Now the squares on the sides are
respectively of the area 4 and 1. Hence in this case also the square
on the diagonal is equal to the sum of the square on the sides. This
property of matter is one of the first great discoveries of applied
mathematics. We shall prove afterwards that it is not a property of
space. For the present it is enough to remark that the positions in
which the points are arranged is entirely experimental. It is by means
of equal pieces of some material, or the same piece of material moved
from one place to another, that the points are arranged.
Pythagoras next enquired what the relation must be so that a square
drawn slanting-wise should be equal to one straight-wise. He found that
a square whose side is five can be placed either rectangularly along
the lines of points, or in a slanting position. And this square is
equivalent to two squares of sides 4 and 3.
Here he came upon a numerical relation embodied in a property of
matter. Numbers immanent in the objects produced the equality so
satisfactory for intellectual apprehension. And he found that numbers
when immanent in sound—when the strings of a musical instrument were
given certain definite proportions of length—were no less captivating
to the ear than the equality of squares was to the reason. What wonder
then that he ascribed an active power to number!
We must remember that, sharing like ourselves the search for the
permanent in changing phenomena, the Greeks had not that conception of
the permanent in matter that we have. To them material things were not
permanent. In fire solid things would vanish; absolutely disappear.
Rock and earth had a more stable existence, but they too grew and
decayed. The permanence of matter, the conservation of energy, were
unknown to them. And that distinction which we draw so readily between
the fleeting and permanent causes of sensation, between a sound and
a material object, for instance, had not the same meaning to them
which it has for us. Let us but imagine for a moment that material
things are fleeting, disappearing, and we shall enter with a far better
appreciation into that search for the permanent which, with the Greeks,
as with us, is the primary intellectual demand.
What is that which amid a thousand forms is ever the same, which we can
recognise under all its vicissitudes, of which the diverse phenomena
are the appearances?
To think that this is number is not so very wide of the mark. With
an intellectual apprehension which far outran the evidences for its
application, the atomists asserted that there were everlasting material
particles, which, by their union, produced all the varying forms and
states of bodies. But in view of the observed facts of nature as
then known, Aristotle, with perfect reason, refused to accept this
hypothesis.
He expressly states that there is a change of quality, and that the
change due to motion is only one of the possible modes of change.
With no permanent material world about us, with the fleeting, the
unpermanent, all around we should, I think, be ready to follow
Pythagoras in his identification of number with that principle which
subsists amidst all changes, which in multitudinous forms we apprehend
immanent in the changing and disappearing substance of things.
And from the numerical idealism of Pythagoras there is but a step to
the more rich and full idealism of Plato. That which is apprehended by
the sense of touch we put as primary and real, and the other senses we
say are merely concerned with appearances. But Plato took them all as
valid, as giving qualities of existence. That the qualities were not
permanent in the world as given to the senses forced him to attribute
to them a different kind of permanence. He formed the conception of a
world of ideas, in which all that really is, all that affects us and
gives the rich and wonderful wealth of our experience, is not fleeting
and transitory, but eternal. And of this real and eternal we see in the
things about us the fleeting and transient images.
And this world of ideas was no exclusive one, wherein was no place
for the innermost convictions of the soul and its most authoritative
assertions. Therein existed justice, beauty—the one, the good, all
that the soul demanded to be. The world of ideas, Plato’s wonderful
creation preserved for man, for his deliberate investigation and their
sure development, all that the rude incomprehensible changes of a harsh
experience scatters and destroys.
Plato believed in the reality of ideas. He meets us fairly and
squarely. Divide a line into two parts, he says; one to represent
the real objects in the world, the other to represent the transitory
appearances, such as the image in still water, the glitter of the sun
on a bright surface, the shadows on the clouds.
A B
——————————————————————————————|————————————————————————————————-
Real things: Appearances:
_e.g._, the sun. _e.g._, the reflection of the sun.
Take another line and divide it into two parts, one representing
our ideas, the ordinary occupants of our minds, such as whiteness,
equality, and the other representing our true knowledge, which is of
eternal principles, such as beauty, goodness.
A^1 B^1
——————————————————————————————|————————————————————————————————-
Eternal principles, Appearances in the mind,
as beauty as whiteness, equality
Then as A is to B, so is A^1 to B^1
That is, the soul can proceed, going away from real things to a region
of perfect certainty, where it beholds what is, not the scattered
reflections; beholds the sun, not the glitter on the sands; true being,
not chance opinion.
Now, this is to us, as it was to Aristotle, absolutely inconceivable
from a scientific point of view. We can understand that a being is
known in the fulness of his relations; it is in his relations to his
circumstances that a man’s character is known; it is in his acts under
his conditions that his character exists. We cannot grasp or conceive
any principle of individuation apart from the fulness of the relations
to the surroundings.
But suppose now that Plato is talking about the higher man—the
four-dimensional being that is limited in our external experience to a
three-dimensional world. Do not his words begin to have a meaning? Such
a being would have a consciousness of motion which is not as the motion
he can see with the eyes of the body. He, in his own being, knows a
reality to which the outward matter of this too solid earth is flimsy
superficiality. He too knows a mode of being, the fulness of relations,
in which can only be represented in the limited world of sense, as the
painter unsubstantially portrays the depths of woodland, plains, and
air. Thinking of such a being in man, was not Plato’s line well divided?
It is noteworthy that, if Plato omitted his doctrine of the independent
origin of ideas, he would present exactly the four-dimensional
argument; a real thing as we think it is an idea. A plane being’s idea
of a square object is the idea of an abstraction, namely, a geometrical
square. Similarly our idea of a solid thing is an abstraction, for
in our idea there is not the four-dimensional thickness which is
necessary, however slight, to give reality. The argument would then
run, as a shadow is to a solid object, so is the solid object to the
reality. Thus A and B´ would be identified.
In the allegory which I have already alluded to, Plato in almost as
many words shows forth the relation between existence in a superficies
and in solid space. And he uses this relation to point to the
conditions of a higher being.
He imagines a number of men prisoners, chained so that they look at
the wall of a cavern in which they are confined, with their backs to
the road and the light. Over the road pass men and women, figures and
processions, but of all this pageant all that the prisoners behold
is the shadow of it on the wall whereon they gaze. Their own shadows
and the shadows of the things in the world are all that they see, and
identifying themselves with their shadows related as shadows to a world
of shadows, they live in a kind of dream.
Plato imagines one of their number to pass out from amongst them
into the real space world, and then returning to tell them of their
condition.
Here he presents most plainly the relation between existence in a plane
world and existence in a three-dimensional world. And he uses this
illustration as a type of the manner in which we are to proceed to a
higher state from the three-dimensional life we know.
It must have hung upon the weight of a shadow which path he
took!—whether the one we shall follow toward the higher solid and the
four-dimensional existence, or the one which makes ideas the higher
realities, and the direct perception of them the contact with the truer
world.
Passing on to Aristotle, we will touch on the points which most
immediately concern our enquiry.
Just as a scientific man of the present day in reviewing the
speculations of the ancient world would treat them with a curiosity
half amused but wholly respectful, asking of each and all wherein lay
their relation to fact, so Aristotle, in discussing the philosophy
of Greece as he found it, asks, above all other things: “Does this
represent the world? In this system is there an adequate presentation
of what is?”
He finds them all defective, some for the very reasons which we esteem
them most highly, as when he criticises the Atomic theory for its
reduction of all change to motion. But in the lofty march of his reason
he never loses sight of the whole; and that wherein our views differ
from his lies not so much in a superiority of our point of view, as
in the fact which he himself enunciates—that it is impossible for one
principle to be valid in all branches of enquiry. The conceptions
of one method of investigation are not those of another; and our
divergence lies in our exclusive attention to the conceptions useful
in one way of apprehending nature rather than in any possibility we
find in our theories of giving a view of the whole transcending that of
Aristotle.
He takes account of everything; he does not separate matter and the
manifestation of matter; he fires all together in a conception of a
vast world process in which everything takes part—the motion of a grain
of dust, the unfolding of a leaf, the ordered motion of the spheres in
heaven—all are parts of one whole which he will not separate into dead
matter and adventitious modifications.
And just as our theories, as representative of actuality, fall before
his unequalled grasp of fact, so the doctrine of ideas fell. It is
not an adequate account of existence, as Plato himself shows in his
“Parmenides”; it only explains things by putting their doubles beside
them.
For his own part Aristotle invented a great marching definition which,
with a kind of power of its own, cleaves its way through phenomena
to limiting conceptions on either hand, towards whose existence all
experience points.
In Aristotle’s definition of matter and form as the constituent of
reality, as in Plato’s mystical vision of the kingdom of ideas, the
existence of the higher dimensionality is implicitly involved.
Substance according to Aristotle is relative, not absolute. In
everything that is there is the matter of which it is composed, the
form which it exhibits; but these are indissolubly connected, and
neither can be thought without the other.
The blocks of stone out of which a house is built are the material for
the builder; but, as regards the quarrymen, they are the matter of the
rocks with the form he has imposed on them. Words are the final product
of the grammarian, but the mere matter of the orator or poet. The atom
is, with us, that out of which chemical substances are built up, but
looked at from another point of view is the result of complex processes.
Nowhere do we find finality. The matter in one sphere is the matter,
plus form, of another sphere of thought. Making an obvious application
to geometry, plane figures exist as the limitation of different
portions of the plane by one another. In the bounding lines the
separated matter of the plane shows its determination into form. And
as the plane is the matter relatively to determinations in the plane,
so the plane itself exists in virtue of the determination of space. A
plane is that wherein formless space has form superimposed on it, and
gives an actuality of real relations. We cannot refuse to carry this
process of reasoning a step farther back, and say that space itself is
that which gives form to higher space. As a line is the determination
of a plane, and a plane of a solid, so solid space itself is the
determination of a higher space.
As a line by itself is inconceivable without that plane which it
separates, so the plane is inconceivable without the solids which
it limits on either hand. And so space itself cannot be positively
defined. It is the negation of the possibility of movement in more than
three dimensions. The conception of space demands that of a higher
space. As a surface is thin and unsubstantial without the substance of
which it is the surface, so matter itself is thin without the higher
matter.
Just as Aristotle invented that algebraical method of representing
unknown quantities by mere symbols, not by lines necessarily
determinate in length as was the habit of the Greek geometers, and so
struck out the path towards those objectifications of thought which,
like independent machines for reasoning, supply the mathematician
with his analytical weapons, so in the formulation of the doctrine
of matter and form, of potentiality and actuality, of the relativity
of substance, he produced another kind of objectification of mind—a
definition which had a vital force and an activity of its own.
In none of his writings, as far as we know, did he carry it to its
legitimate conclusion on the side of matter, but in the direction of
the formal qualities he was led to his limiting conception of that
existence of pure form which lies beyond all known determination
of matter. The unmoved mover of all things is Aristotle’s highest
principle. Towards it, to partake of its perfection all things move.
The universe, according to Aristotle, is an active process—he does
not adopt the illogical conception that it was once set in motion
and has kept on ever since. There is room for activity, will,
self-determination, in Aristotle’s system, and for the contingent and
accidental as well. We do not follow him, because we are accustomed to
find in nature infinite series, and do not feel obliged to pass on to a
belief in the ultimate limits to which they seem to point.
But apart from the pushing to the limit, as a relative principle
this doctrine of Aristotle’s as to the relativity of substance is
irrefragible in its logic. He was the first to show the necessity
of that path of thought which when followed leads to a belief in a
four-dimensional space.
Antagonistic as he was to Plato in his conception of the practical
relation of reason to the world of phenomena, yet in one point he
coincided with him. And in this he showed the candour of his intellect.
He was more anxious to lose nothing than to explain everything. And
that wherein so many have detected an inconsistency, an inability to
free himself from the school of Plato, appears to us in connection with
our enquiry as an instance of the acuteness of his observation. For
beyond all knowledge given by the senses Aristotle held that there is
an active intelligence, a mind not the passive recipient of impressions
from without, but an active and originative being, capable of grasping
knowledge at first hand. In the active soul Aristotle recognised
something in man not produced by his physical surroundings, something
which creates, whose activity is a knowledge underived from sense.
This, he says, is the immortal and undying being in man.
Thus we see that Aristotle was not far from the recognition of the
four-dimensional existence, both without and within man, and the
process of adequately realising the higher dimensional figures to which
we shall come subsequently is a simple reduction to practice of his
hypothesis of a soul.
The next step in the unfolding of the drama of the recognition of
the soul as connected with our scientific conception of the world,
and, at the same time, the recognition of that higher of which a
three-dimensional world presents the superficial appearance, took place
many centuries later. If we pass over the intervening time without a
word it is because the soul was occupied with the assertion of itself
in other ways than that of knowledge. When it took up the task in
earnest of knowing this material world in which it found itself, and of
directing the course of inanimate nature, from that most objective aim
came, reflected back as from a mirror, its knowledge of itself.
CHAPTER V
THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE
LOBATCHEWSKY, BOLYAI, AND GAUSS
Before entering on a description of the work of Lobatchewsky and Bolyai
it will not be out of place to give a brief account of them, the
materials for which are to be found in an article by Franz Schmidt in
the forty-second volume of the _Mathematische Annalen_, and in Engel’s
edition of Lobatchewsky.
Lobatchewsky was a man of the most complete and wonderful talents.
As a youth he was full of vivacity, carrying his exuberance so far
as to fall into serious trouble for hazing a professor, and other
freaks. Saved by the good offices of the mathematician Bartels, who
appreciated his ability, he managed to restrain himself within the
bounds of prudence. Appointed professor at his own University, Kasan,
he entered on his duties under the regime of a pietistic reactionary,
who surrounded himself with sycophants and hypocrites. Esteeming
probably the interests of his pupils as higher than any attempt at a
vain resistance, he made himself the tyrant’s right-hand man, doing an
incredible amount of teaching and performing the most varied official
duties. Amidst all his activities he found time to make important
contributions to science. His theory of parallels is most closely
connected with his name, but a study of his writings shows that he was
a man capable of carrying on mathematics in its main lines of advance,
and of a judgment equal to discerning what these lines were. Appointed
rector of his University, he died at an advanced age, surrounded by
friends, honoured, with the results of his beneficent activity all
around him. To him no subject came amiss, from the foundations of
geometry to the improvement of the stoves by which the peasants warmed
their houses.
He was born in 1793. His scientific work was unnoticed till, in 1867,
Houel, the French mathematician, drew attention to its importance.
Johann Bolyai de Bolyai was born in Klausenburg, a town in
Transylvania, December 15th, 1802.
His father, Wolfgang Bolyai, a professor in the Reformed College of
Maros Vasarhely, retained the ardour in mathematical studies which had
made him a chosen companion of Gauss in their early student days at
Göttingen.
He found an eager pupil in Johann. He relates that the boy sprang
before him like a devil. As soon as he had enunciated a problem the
child would give the solution and command him to go on further. As a
thirteen-year-old boy his father sometimes sent him to fill his place
when incapacitated from taking his classes. The pupils listened to him
with more attention than to his father for they found him clearer to
understand.
In a letter to Gauss Wolfgang Bolyai writes:—
“My boy is strongly built. He has learned to recognise many
constellations, and the ordinary figures of geometry. He makes apt
applications of his notions, drawing for instance the positions of the
stars with their constellations. Last winter in the country, seeing
Jupiter he asked: ‘How is it that we can see him from here as well as
from the town? He must be far off.’ And as to three different places
to which he had been he asked me to tell him about them in one word. I
did not know what he meant, and then he asked me if one was in a line
with the other and all in a row, or if they were in a triangle.
“He enjoys cutting paper figures with a pair of scissors, and without
my ever having told him about triangles remarked that a right-angled
triangle which he had cut out was half of an oblong. I exercise his
body with care, he can dig well in the earth with his little hands.
The blossom can fall and no fruit left. When he is fifteen I want to
send him to you to be your pupil.”
In Johann’s autobiography he says:—
“My father called my attention to the imperfections and gaps in the
theory of parallels. He told me he had gained more satisfactory
results than his predecessors, but had obtained no perfect and
satisfying conclusion. None of his assumptions had the necessary
degree of geometrical certainty, although they sufficed to prove the
eleventh axiom and appeared acceptable on first sight.
“He begged of me, anxious not without a reason, to hold myself aloof
and to shun all investigation on this subject, if I did not wish to
live all my life in vain.”
Johann, in the failure of his father to obtain any response from Gauss,
in answer to a letter in which he asked the great mathematician to make
of his son “an apostle of truth in a far land,” entered the Engineering
School at Vienna. He writes from Temesvar, where he was appointed
sub-lieutenant September, 1823:—
“Temesvar, November 3rd, 1823.
“DEAR GOOD FATHER,
“I have so overwhelmingly much to write about my discovery that I know
no other way of checking myself than taking a quarter of a sheet only
to write on. I want an answer to my four-sheet letter.
“I am unbroken in my determination to publish a work on Parallels, as
soon as I have put my material in order and have the means.
“At present I have not made any discovery, but the way I have followed
almost certainly promises me the attainment of my object if any
possibility of it exists.
“I have not got my object yet, but I have produced such stupendous
things that I was overwhelmed myself, and it would be an eternal shame
if they were lost. When you see them you will find that it is so. Now
I can only say that I have made a new world out of nothing. Everything
that I have sent you before is a house of cards in comparison with a
tower. I am convinced that it will be no less to my honour than if I
had already discovered it.”
The discovery of which Johann here speaks was published as an appendix
to Wolfgang Bolyai’s _Tentamen_.
Sending the book to Gauss, Wolfgang writes, after an interruption of
eighteen years in his correspondence:—
“My son is first lieutenant of Engineers and will soon be captain.
He is a fine youth, a good violin player, a skilful fencer, and
brave, but has had many duels, and is wild even for a soldier. Yet he
is distinguished—light in darkness and darkness in light. He is an
impassioned mathematician with extraordinary capacities.... He will
think more of your judgment on his work than that of all Europe.”
Wolfgang received no answer from Gauss to this letter, but sending a
second copy of the book received the following reply:—
“You have rejoiced me, my unforgotten friend, by your letters. I
delayed answering the first because I wanted to wait for the arrival
of the promised little book.
“Now something about your son’s work.
“If I begin with saying that ‘I ought not to praise it,’ you will be
staggered for a moment. But I cannot say anything else. To praise it
is to praise myself, for the path your son has broken in upon and the
results to which he has been led are almost exactly the same as my own
reflections, some of which date from thirty to thirty-five years ago.
“In fact I am astonished to the uttermost. My intention was to let
nothing be known in my lifetime about my own work, of which, for the
rest, but little is committed to writing. Most people have but little
perception of the problem, and I have found very few who took any
interest in the views I expressed to them. To be able to do that one
must first of all have had a real live feeling of what is wanting, and
as to that most men are completely in the dark.
“Still it was my intention to commit everything to writing in the
course of time, so that at least it should not perish with me.
“I am deeply surprised that this task can be spared me, and I am most
of all pleased in this that it is the son of my old friend who has in
so remarkable a manner preceded me.”
The impression which we receive from Gauss’s inexplicable silence
towards his old friend is swept away by this letter. Hence we breathe
the clear air of the mountain tops. Gauss would not have failed to
perceive the vast significance of his thoughts, sure to be all the
greater in their effect on future ages from the want of comprehension
of the present. Yet there is not a word or a sign in his writing to
claim the thought for himself. He published no single line on the
subject. By the measure of what he thus silently relinquishes, by
such a measure of a world-transforming thought, we can appreciate his
greatness.
It is a long step from Gauss’s serenity to the disturbed and passionate
life of Johann Bolyai—he and Galois, the two most interesting figures
in the history of mathematics. For Bolyai, the wild soldier, the
duellist, fell at odds with the world. It is related of him that he was
challenged by thirteen officers of his garrison, a thing not unlikely
to happen considering how differently he thought from every one else.
He fought them all in succession—making it his only condition that he
should be allowed to play on his violin for an interval between meeting
each opponent. He disarmed or wounded all his antagonists. It can be
easily imagined that a temperament such as his was one not congenial to
his military superiors. He was retired in 1833.
His epoch-making discovery awoke no attention. He seems to have
conceived the idea that his father had betrayed him in some
inexplicable way by his communications with Gauss, and he challenged
the excellent Wolfgang to a duel. He passed his life in poverty, many a
time, says his biographer, seeking to snatch himself from dissipation
and apply himself again to mathematics. But his efforts had no result.
He died January 27th, 1860, fallen out with the world and with himself.
METAGEOMETRY
The theories which are generally connected with the names of
Lobatchewsky and Bolyai bear a singular and curious relation to the
subject of higher space.
In order to show what this relation is, I must ask the reader to be
at the pains to count carefully the sets of points by which I shall
estimate the volumes of certain figures.
No mathematical processes beyond this simple one of counting will be
necessary.
[Illustration: Fig. 19.]
Let us suppose we have before us in fig. 19 a plane covered with points
at regular intervals, so placed that every four determine a square.
Now it is evident that as four points determine a square, so four
squares meet in a point.
[Illustration: Fig. 20.]
Thus, considering a point inside a square as belonging to it, we may
say that a point on the corner of a square belongs to it and to three
others equally: belongs a quarter of it to each square.
[Illustration: Fig. 21.]
[Illustration: Fig. 22.]
Thus the square ACDE (fig. 21) contains one point, and has four points
at the four corners. Since one-fourth of each of these four belongs to
the square, the four together count as one point, and the point value
of the square is two points—the one inside and the four at the corner
make two points belonging to it exclusively.
Now the area of this square is two unit squares, as can be seen by
drawing two diagonals in fig. 22.
We also notice that the square in question is equal to the sum of the
squares on the sides AB, BC, of the right-angled triangle ABC. Thus we
recognise the proposition that the square on the hypothenuse is equal
to the sum of the squares on the two sides of a right-angled triangle.
Now suppose we set ourselves the question of determining the
whereabouts in the ordered system of points, the end of a line would
come when it turned about a point keeping one extremity fixed at the
point.
We can solve this problem in a particular case. If we can find a square
lying slantwise amongst the dots which is equal to one which goes
regularly, we shall know that the two sides are equal, and that the
slanting side is equal to the straight-way side. Thus the volume and
shape of a figure remaining unchanged will be the test of its having
rotated about the point, so that we can say that its side in its first
position would turn into its side in the second position.
Now, such a square can be found in the one whose side is five units in
length.
[Illustration: Fig. 23.]
In fig. 23, in the square on AB, there are—
9 points interior 9
4 at the corners 1
4 sides with 3 on each side, considered as
1½ on each side, because belonging
equally to two squares 6
The total is 16. There are 9 points in the square on BC.
In the square on AC there are—
24 points inside 24
4 at the corners 1
or 25 altogether.
Hence we see again that the square on the hypothenuse is equal to the
squares on the sides.
Now take the square AFHG, which is larger than the square on AB. It
contains 25 points.
16 inside 16
16 on the sides, counting as 8
4 on the corners 1
making 25 altogether.
If two squares are equal we conclude the sides are equal. Hence, the
line AF turning round A would move so that it would after a certain
turning coincide with AC.
This is preliminary, but it involves all the mathematical difficulties
that will present themselves.
There are two alterations of a body by which its volume is not changed.
One is the one we have just considered, rotation, the other is what is
called shear.
Consider a book, or heap of loose pages. They can be slid so that each
one slips over the preceding one, and the whole assumes the shape _b_
in fig. 24.
[Illustration: Fig. 24.]
This deformation is not shear alone, but shear accompanied by rotation.
Shear can be considered as produced in another way.
Take the square ABCD (fig. 25), and suppose that it is pulled out from
along one of its diagonals both ways, and proportionately compressed
along the other diagonal. It will assume the shape in fig. 26.
This compression and expansion along two lines at right angles is what
is called shear; it is equivalent to the sliding illustrated above,
combined with a turning round.
[Illustration: Fig. 25.] [Illustration: Fig. 26.]
In pure shear a body is compressed and extended in two directions at
right angles to each other, so that its volume remains unchanged.
Now we know that our material bodies resist shear—shear does violence
to the internal arrangement of their particles, but they turn as wholes
without such internal resistance.
But there is an exception. In a liquid shear and rotation take place
equally easily, there is no more resistance against a shear than there
is against a rotation.
Now, suppose all bodies were to be reduced to the liquid state, in
which they yield to shear and to rotation equally easily, and then
were to be reconstructed as solids, but in such a way that shear and
rotation had interchanged places.
That is to say, let us suppose that when they had become solids again
they would shear without offering any internal resistance, but a
rotation would do violence to their internal arrangement.
That is, we should have a world in which shear would have taken the
place of rotation.
A shear does not alter the volume of a body: thus an inhabitant living
in such a world would look on a body sheared as we look on a body
rotated. He would say that it was of the same shape, but had turned a
bit round.
Let us imagine a Pythagoras in this world going to work to investigate,
as is his wont.
[Illustration: Fig. 27.] [Illustration: Fig. 28.]
Fig. 27 represents a square unsheared. Fig. 28 represents a square
sheared. It is not the figure into which the square in fig. 27 would
turn, but the result of shear on some square not drawn. It is a simple
slanting placed figure, taken now as we took a simple slanting placed
square before. Now, since bodies in this world of shear offer no
internal resistance to shearing, and keep their volume when sheared,
an inhabitant accustomed to them would not consider that they altered
their shape under shear. He would call ACDE as much a square as the
square in fig. 27. We will call such figures shear squares. Counting
the dots in ACDE, we find—
2 inside = 2
4 at corners = 1
or a total of 3.
Now, the square on the side AB has 4 points, that on BC has 1 point.
Here the shear square on the hypothenuse has not 5 points but 3; it is
not the sum of the squares on the sides, but the difference.
This relation always holds. Look at fig. 29.
[Illustration: Fig. 29.]
Shear square on hypothenuse—
7 internal 7
4 at corners 1
—
8
[Illustration: Fig. 29 _bis_.]
Square on one side—which the reader can draw for himself—
4 internal 4
8 on sides 4
4 at corners 1
—
9
and the square on the other side is 1. Hence in this case again the
difference is equal to the shear square on the hypothenuse, 9 - 1 = 8.
Thus in a world of shear the square on the hypothenuse would be equal
to the difference of the squares on the sides of a right-angled
triangle.
In fig. 29 _bis_ another shear square is drawn on which the above
relation can be tested.
What now would be the position a line on turning by shear would take up?
We must settle this in the same way as previously with our turning.
Since a body sheared remains the same, we must find two equal bodies,
one in the straight way, one in the slanting way, which have the same
volume. Then the side of one will by turning become the side of the
other, for the two figures are each what the other becomes by a shear
turning.
We can solve the problem in a particular case—
[Illustration: Fig. 30.]
In the figure ACDE (fig. 30) there are—
15 inside 15
4 at corners 1
a total of 16.
Now in the square ABGF, there are 16—
9 inside 9
12 on sides 6
4 at corners 1
—
16
Hence the square on AB would, by the shear turning, become the shear
square ACDE.
And hence the inhabitant of this world would say that the line AB
turned into the line AC. These two lines would be to him two lines of
equal length, one turned a little way round from the other.
That is, putting shear in place of rotation, we get a different kind
of figure, as the result of the shear rotation, from what we got with
our ordinary rotation. And as a consequence we get a position for the
end of a line of invariable length when it turns by the shear rotation,
different from the position which it would assume on turning by our
rotation.
A real material rod in the shear world would, on turning about A, pass
from the position AB to the position AC. We say that its length alters
when it becomes AC, but this transformation of AB would seem to an
inhabitant of the shear world like a turning of AB without altering in
length.
If now we suppose a communication of ideas that takes place between
one of ourselves and an inhabitant of the shear world, there would
evidently be a difference between his views of distance and ours.
We should say that his line AB increased in length in turning to AC. He
would say that our line AF (fig. 23) decreased in length in turning to
AC. He would think that what we called an equal line was in reality a
shorter one.
We should say that a rod turning round would have its extremities in
the positions we call at equal distances. So would he—but the positions
would be different. He could, like us, appeal to the properties of
matter. His rod to him alters as little as ours does to us.
Now, is there any standard to which we could appeal, to say which of
the two is right in this argument? There is no standard.
We should say that, with a change of position, the configuration and
shape of his objects altered. He would say that the configuration and
shape of our objects altered in what we called merely a change of
position. Hence distance independent of position is inconceivable, or
practically distance is solely a property of matter.
There is no principle to which either party in this controversy could
appeal. There is nothing to connect the definition of distance with our
ideas rather than with his, except the behaviour of an actual piece of
matter.
For the study of the processes which go on in our world the definition
of distance given by taking the sum of the squares is of paramount
importance to us. But as a question of pure space without making any
unnecessary assumptions the shear world is just as possible and just as
interesting as our world.
It was the geometry of such conceivable worlds that Lobatchewsky and
Bolyai studied.
This kind of geometry has evidently nothing to do directly with
four-dimensional space.
But a connection arises in this way. It is evident that, instead of
taking a simple shear as I have done, and defining it as that change
of the arrangement of the particles of a solid which they will undergo
without offering any resistance due to their mutual action, I might
take a complex motion, composed of a shear and a rotation together, or
some other kind of deformation.
Let us suppose such an alteration picked out and defined as the one
which means simple rotation, then the type, according to which all
bodies will alter by this rotation, is fixed.
Looking at the movements of this kind, we should say that the objects
were altering their shape as well as rotating. But to the inhabitants
of that world they would seem to be unaltered, and our figures in their
motions would seem to them to alter.
In such a world the features of geometry are different. We have seen
one such difference in the case of our illustration of the world of
shear, where the square on the hypothenuse was equal to the difference,
not the sum, of the squares on the sides.
In our illustration we have the same laws of parallel lines as in our
ordinary rotation world, but in general the laws of parallel lines are
different.
In one of these worlds of a different constitution of matter through
one point there can be two parallels to a given line, in another of
them there can be none, that is, although a line be drawn parallel to
another it will meet it after a time.
Now it was precisely in this respect of parallels that Lobatchewsky and
Bolyai discovered these different worlds. They did not think of them as
worlds of matter, but they discovered that space did not necessarily
mean that our law of parallels is true. They made the distinction
between laws of space and laws of matter, although that is not the
form in which they stated their results.
The way in which they were led to these results was the
following. Euclid had stated the existence of parallel lines as a
postulate—putting frankly this unproved proposition—that one line and
only one parallel to a given straight line can be drawn, as a demand,
as something that must be assumed. The words of his ninth postulate are
these: “If a straight line meeting two other straight lines makes the
interior angles on the same side of it equal to two right angles, the
two straight lines will never meet.”
The mathematicians of later ages did not like this bald assumption, and
not being able to prove the proposition they called it an axiom—the
eleventh axiom.
Many attempts were made to prove the axiom; no one doubted of its
truth, but no means could be found to demonstrate it. At last an
Italian, Sacchieri, unable to find a proof, said: “Let us suppose it
not true.” He deduced the results of there being possibly two parallels
to one given line through a given point, but feeling the waters too
deep for the human reason, he devoted the latter half of his book to
disproving what he had assumed in the first part.
Then Bolyai and Lobatchewsky with firm step entered on the forbidden
path. There can be no greater evidence of the indomitable nature of
the human spirit, or of its manifest destiny to conquer all those
limitations which bind it down within the sphere of sense than this
grand assertion of Bolyai and Lobatchewsky.
───────────────────────────
C D
───────────────────────────────────
A B
Take a line AB and a point C. We say and see and know that through C
can only be drawn one line parallel to AB.
But Bolyai said: “I will draw two.” Let CD be parallel to AB, that
is, not meet AB however far produced, and let lines beyond CD also not
meet AB; let there be a certain region between CD and CE, in which no
line drawn meets AB. CE and CD produced backwards through C will give a
similar region on the other side of C.
[Illustration: Fig. 32.]
Nothing so triumphantly, one may almost say so insolently, ignoring
of sense had ever been written before. Men had struggled against the
limitations of the body, fought them, despised them, conquered them.
But no one had ever thought simply as if the body, the bodily eyes,
the organs of vision, all this vast experience of space, had never
existed. The age-long contest of the soul with the body, the struggle
for mastery, had come to a culmination. Bolyai and Lobatchewsky simply
thought as if the body was not. The struggle for dominion, the strife
and combat of the soul were over; they had mastered, and the Hungarian
drew his line.
Can we point out any connection, as in the case of Parmenides, between
these speculations and higher space? Can we suppose it was any inner
perception by the soul of a motion not known to the senses, which
resulted in this theory so free from the bonds of sense? No such
supposition appears to be possible.
Practically, however, metageometry had a great influence in bringing
the higher space to the front as a working hypothesis. This can
be traced to the tendency the mind has to move in the direction
of least resistance. The results of the new geometry could not be
neglected, the problem of parallels had occupied a place too prominent
in the development of mathematical thought for its final solution
to be neglected. But this utter independence of all mechanical
considerations, this perfect cutting loose from the familiar
intuitions, was so difficult that almost any other hypothesis was
more easy of acceptance, and when Beltrami showed that the geometry
of Lobatchewsky and Bolyai was the geometry of shortest lines drawn
on certain curved surfaces, the ordinary definitions of measurement
being retained, attention was drawn to the theory of a higher space.
An illustration of Beltrami’s theory is furnished by the simple
consideration of hypothetical beings living on a spherical surface.
[Illustration: Fig. 33.]
Let ABCD be the equator of a globe, and AP, BP, meridian lines drawn to
the pole, P. The lines AB, AP, BP would seem to be perfectly straight
to a person moving on the surface of the sphere, and unconscious of its
curvature. Now AP and BP both make right angles with AB. Hence they
satisfy the definition of parallels. Yet they meet in P. Hence a being
living on a spherical surface, and unconscious of its curvature, would
find that parallel lines would meet. He would also find that the angles
in a triangle were greater than two right angles. In the triangle PAB,
for instance, the angles at A and B are right angles, so the three
angles of the triangle PAB are greater than two right angles.
Now in one of the systems of metageometry (for after Lobatchewsky had
shown the way it was found that other systems were possible besides
his) the angles of a triangle are greater than two right angles.
Thus a being on a sphere would form conclusions about his space which
are the same as he would form if he lived on a plane, the matter in
which had such properties as are presupposed by one of these systems
of geometry. Beltrami also discovered a certain surface on which
there could be drawn more than one “straight” line through a point
which would not meet another given line. I use the word straight as
equivalent to the line having the property of giving the shortest path
between any two points on it. Hence, without giving up the ordinary
methods of measurement, it was possible to find conditions in which
a plane being would necessarily have an experience corresponding to
Lobatchewsky’s geometry. And by the consideration of a higher space,
and a solid curved in such a higher space, it was possible to account
for a similar experience in a space of three dimensions.
Now, it is far more easy to conceive of a higher dimensionality to
space than to imagine that a rod in rotating does not move so that
its end describes a circle. Hence, a logical conception having been
found harder than that of a four dimensional space, thought turned
to the latter as a simple explanation of the possibilities to which
Lobatchewsky had awakened it. Thinkers became accustomed to deal with
the geometry of higher space—it was Kant, says Veronese, who first
used the expression of “different spaces”—and with familiarity the
inevitableness of the conception made itself felt.
From this point it is but a small step to adapt the ordinary mechanical
conceptions to a higher spatial existence, and then the recognition of
its objective existence could be delayed no longer. Here, too, as in so
many cases, it turns out that the order and connection of our ideas is
the order and connection of things.
What is the significance of Lobatchewsky’s and Bolyai’s work?
It must be recognised as something totally different from the
conception of a higher space; it is applicable to spaces of any number
of dimensions. By immersing the conception of distance in matter to
which it properly belongs, it promises to be of the greatest aid in
analysis for the effective distance of any two particles is the
product of complex material conditions and cannot be measured by hard
and fast rules. Its ultimate significance is altogether unknown. It
is a cutting loose from the bonds of sense, not coincident with the
recognition of a higher dimensionality, but indirectly contributory
thereto.
Thus, finally, we have come to accept what Plato held in the hollow
of his hand; what Aristotle’s doctrine of the relativity of substance
implies. The vast universe, too, has its higher, and in recognising it
we find that the directing being within us no longer stands inevitably
outside our systematic knowledge.
CHAPTER VI
THE HIGHER WORLD
It is indeed strange, the manner in which we must begin to think about
the higher world.
Those simplest objects analogous to those which are about us on every
side in our daily experience such as a door, a table, a wheel are
remote and uncognisable in the world of four dimensions, while the
abstract ideas of rotation, stress and strain, elasticity into which
analysis resolves the familiar elements of our daily experience are
transferable and applicable with no difficulty whatever. Thus we are
in the unwonted position of being obliged to construct the daily and
habitual experience of a four-dimensional being, from a knowledge of
the abstract theories of the space, the matter, the motion of it;
instead of, as in our case, passing to the abstract theories from the
richness of sensible things.
What would a wheel be in four dimensions? What the shafting for the
transmission of power which a four-dimensional being would use.
The four-dimensional wheel, and the four-dimensional shafting are
what will occupy us for these few pages. And it is no futile or
insignificant enquiry. For in the attempt to penetrate into the nature
of the higher, to grasp within our ken that which transcends all
analogies, because what we know are merely partial views of it, the
purely material and physical path affords a means of approach pursuing
which we are in less likelihood of error than if we use the more
frequently trodden path of framing conceptions which in their elevation
and beauty seem to us ideally perfect.
For where we are concerned with our own thoughts, the development of
our own ideals, we are as it were on a curve, moving at any moment
in a direction of tangency. Whither we go, what we set up and exalt
as perfect, represents not the true trend of the curve, but our own
direction at the present—a tendency conditioned by the past, and by
a vital energy of motion essential but only true when perpetually
modified. That eternal corrector of our aspirations and ideals, the
material universe draws sublimely away from the simplest things we can
touch or handle to the infinite depths of starry space, in one and
all uninfluenced by what we think or feel, presenting unmoved fact
to which, think it good or think it evil, we can but conform, yet
out of all that impassivity with a reference to something beyond our
individual hopes and fears supporting us and giving us our being.
And to this great being we come with the question: “You, too, what is
your higher?”
Or to put it in a form which will leave our conclusions in the shape
of no barren formula, and attacking the problem on its most assailable
side: “What is the wheel and the shafting of the four-dimensional
mechanic?”
In entering on this enquiry we must make a plan of procedure. The
method which I shall adopt is to trace out the steps of reasoning by
which a being confined to movement in a two-dimensional world could
arrive at a conception of our turning and rotation, and then to apply
an analogous process to the consideration of the higher movements. The
plane being must be imagined as no abstract figure, but as a real body
possessing all three dimensions. His limitation to a plane must be the
result of physical conditions.
We will therefore think of him as of a figure cut out of paper placed
on a smooth plane. Sliding over this plane, and coming into contact
with other figures equally thin as he in the third dimension, he will
apprehend them only by their edges. To him they will be completely
bounded by lines. A “solid” body will be to him a two-dimensional
extent, the interior of which can only be reached by penetrating
through the bounding lines.
Now such a plane being can think of our three-dimensional existence in
two ways.
First, he can think of it as a series of sections, each like the solid
he knows of extending in a direction unknown to him, which stretches
transverse to his tangible universe, which lies in a direction at right
angles to every motion which he made.
Secondly, relinquishing the attempt to think of the three-dimensional
solid body in its entirety he can regard it as consisting of a
number of plane sections, each of them in itself exactly like
the two-dimensional bodies he knows, but extending away from his
two-dimensional space.
A square lying in his space he regards as a solid bounded by four
lines, each of which lies in his space.
A square standing at right angles to his plane appears to him as simply
a line in his plane, for all of it except the line stretches in the
third dimension.
He can think of a three-dimensional body as consisting of a number of
such sections, each of which starts from a line in his space.
Now, since in his world he can make any drawing or model which involves
only two dimensions, he can represent each such upright section as it
actually is, and can represent a turning from a known into the unknown
dimension as a turning from one to another of his known dimensions.
To see the whole he must relinquish part of that which he has, and take
the whole portion by portion.
Consider now a plane being in front of a square, fig. 34. The square
can turn about any point in the plane—say the point A. But it cannot
turn about a line, as AB. For, in order to turn about the line AB,
the square must leave the plane and move in the third dimension. This
motion is out of his range of observation, and is therefore, except for
a process of reasoning, inconceivable to him.
[Illustration: Fig. 34.]
Rotation will therefore be to him rotation about a point. Rotation
about a line will be inconceivable to him.
The result of rotation about a line he can apprehend. He can see the
first and last positions occupied in a half-revolution about the line
AC. The result of such a half revolution is to place the square ABCD
on the left hand instead of on the right hand of the line AC. It would
correspond to a pulling of the whole body ABCD through the line AC,
or to the production of a solid body which was the exact reflection
of it in the line AC. It would be as if the square ABCD turned into
its image, the line AB acting as a mirror. Such a reversal of the
positions of the parts of the square would be impossible in his space.
The occurrence of it would be a proof of the existence of a higher
dimensionality.
Let him now, adopting the conception of a three-dimensional body as
a series of sections lying, each removed a little farther than the
preceding one, in direction at right angles to his plane, regard a
cube, fig. 36, as a series of sections, each like the square which
forms its base, all rigidly connected together.
[Illustration: Fig. 35.]
If now he turns the square about the point A in the plane of _xy_,
each parallel section turns with the square he moves. In each of the
sections there is a point at rest, that vertically over A. Hence he
would conclude that in the turning of a three-dimensional body there
is one line which is at rest. That is a three-dimensional turning in a
turning about a line.
* * * * *
In a similar way let us regard ourselves as limited to a
three-dimensional world by a physical condition. Let us imagine that
there is a direction at right angles to every direction in which we can
move, and that we are prevented from passing in this direction by a
vast solid, that against which in every movement we make we slip as the
plane being slips against his plane sheet.
We can then consider a four-dimensional body as consisting of a series
of sections, each parallel to our space, and each a little farther off
than the preceding on the unknown dimension.
Take the simplest four-dimensional body—one which begins as a cube,
fig. 36, in our space, and consists of sections, each a cube like fig.
36, lying away from our space. If we turn the cube which is its base in
our space about a line, if, _e.g._, in fig. 36 we turn the cube about
the line AB, not only it but each of the parallel cubes moves about a
line. The cube we see moves about the line AB, the cube beyond it about
a line parallel to AB and so on. Hence the whole four-dimensional body
moves about a plane, for the assemblage of these lines is our way of
thinking about the plane which, starting from the line AB in our space,
runs off in the unknown direction.
[Illustration: Fig. 36.]
In this case all that we see of the plane about which the turning takes
place is the line AB.
But it is obvious that the axis plane may lie in our space. A point
near the plane determines with it a three-dimensional space. When it
begins to rotate round the plane it does not move anywhere in this
three-dimensional space, but moves out of it. A point can no more
rotate round a plane in three-dimensional space than a point can move
round a line in two-dimensional space.
We will now apply the second of the modes of representation to this
case of turning about a plane, building up our analogy step by step
from the turning in a plane about a point and that in space about a
line, and so on.
In order to reduce our considerations to those of the greatest
simplicity possible, let us realise how the plane being would think of
the motion by which a square is turned round a line.
Let, fig. 34, ABCD be a square on his plane, and represent the two
dimensions of his space by the axes A_x_ A_y_.
Now the motion by which the square is turned over about the line AC
involves the third dimension.
He cannot represent the motion of the whole square in its turning,
but he can represent the motions of parts of it. Let the third axis
perpendicular to the plane of the paper be called the axis of _z_. Of
the three axes _x_, _y_, _z_, the plane being can represent any two in
his space. Let him then draw, in fig. 35, two axes, _x_ and _z_. Here
he has in his plane a representation of what exists in the plane which
goes off perpendicularly to his space.
In this representation the square would not be shown, for in the plane
of _xz_ simply the line AB of the square is contained.
The plane being then would have before him, in fig. 35, the
representation of one line AB of his square and two axes, _x_ and _z_,
at right angles. Now it would be obvious to him that, by a turning
such as he knows, by a rotation about a point, the line AB can turn
round A, and occupying all the intermediate positions, such as AB_{1},
come after half a revolution to lie as A_x_ produced through A.
Again, just as he can represent the vertical plane through AB, so he
can represent the vertical plane through A´B´, fig. 34, and in a like
manner can see that the line A´B´ can turn about the point A´ till it
lies in the opposite direction from that which it ran in at first.
Now these two turnings are not inconsistent. In his plane, if AB
turned about A, and A´B´ about A´, the consistency of the square would
be destroyed, it would be an impossible motion for a rigid body to
perform. But in the turning which he studies portion by portion there
is nothing inconsistent. Each line in the square can turn in this way,
hence he would realise the turning of the whole square as the sum of
a number of turnings of isolated parts. Such turnings, if they took
place in his plane, would be inconsistent, but by virtue of a third
dimension they are consistent, and the result of them all is that the
square turns about the line AC and lies in a position in which it is
the mirror image of what it was in its first position. Thus he can
realise a turning about a line by relinquishing one of his axes, and
representing his body part by part.
Let us apply this method to the turning of a cube so as to become the
mirror image of itself. In our space we can construct three independent
axes, _x_, _y_, _z_, shown in fig. 36. Suppose that there is a fourth
axis, _w_, at right angles to each and every one of them. We cannot,
keeping all three axes, _x_, _y_, _z_, represent _w_ in our space; but
if we relinquish one of our three axes we can let the fourth axis take
its place, and we can represent what lies in the space, determined by
the two axes we retain and the fourth axis.
[Illustration: Fig. 37.]
Let us suppose that we let the _y_ axis drop, and that we represent
the _w_ axis as occupying its direction. We have in fig. 37 a drawing
of what we should then see of the cube. The square ABCD, remains
unchanged, for that is in the plane of _xz_, and we still have that
plane. But from this plane the cube stretches out in the direction of
the _y_ axis. Now the _y_ axis is gone, and so we have no more of the
cube than the face ABCD. Considering now this face ABCD, we see that
it is free to turn about the line AB. It can rotate in the _x_ to _w_
direction about this line. In fig. 38 it is shown on its way, and it
can evidently continue this rotation till it lies on the other side of
the _z_ axis in the plane of _xz_.
We can also take a section parallel to the face ABCD, and then letting
drop all of our space except the plane of that section, introduce
the _w_ axis, running in the old _y_ direction. This section can be
represented by the same drawing, fig. 38, and we see that it can rotate
about the line on its left until it swings half way round and runs in
the opposite direction to that which it ran in before. These turnings
of the different sections are not inconsistent, and taken all together
they will bring the cube from the position shown in fig. 36 to that
shown in fig. 41.
[Illustration: Fig. 38.]
Since we have three axes at our disposal in our space, we are not
obliged to represent the _w_ axis by any particular one. We may let any
axis we like disappear, and let the fourth axis take its place.
[Illustration: Fig. 39.]
[Illustration: Fig. 40.]
[Illustration: Fig. 41.]
In fig. 36 suppose the _z_ axis to go. We have then simply the plane of
_xy_ and the square base of the cube ACEG, fig. 39, is all that could
be seen of it. Let now the _w_ axis take the place of the _z_ axis and
we have, in fig. 39 again, a representation of the space of _xyw_, in
which all that exists of the cube is its square base. Now, by a turning
of _x_ to _w_, this base can rotate around the line AE, it is shown
on its way in fig. 40, and finally it will, after half a revolution,
lie on the other side of the _y_ axis. In a similar way we may rotate
sections parallel to the base of the _xw_ rotation, and each of them
comes to run in the opposite direction from that which they occupied at
first.
Thus again the cube comes from the position of fig. 36. to that of
fig. 41. In this _x_ to _w_ turning, we see that it takes place by
the rotations of sections parallel to the front face about lines
parallel to AB, or else we may consider it as consisting of the
rotation of sections parallel to the base about lines parallel to AE.
It is a rotation of the whole cube about the plane ABEF. Two separate
sections could not rotate about two separate lines in our space without
conflicting, but their motion is consistent when we consider another
dimension. Just, then, as a plane being can think of rotation about
a line as a rotation about a number of points, these rotations not
interfering as they would if they took place in his two-dimensional
space, so we can think of a rotation about a plane as the rotation
of a number of sections of a body about a number of lines in a plane,
these rotations not being inconsistent in a four-dimensional space as
they are in three-dimensional space.
We are not limited to any particular direction for the lines in the
plane about which we suppose the rotation of the particular sections to
take place. Let us draw the section of the cube, fig. 36, through A,
F, C, H, forming a sloping plane. Now since the fourth dimension is at
right angles to every line in our space it is at right angles to this
section also. We can represent our space by drawing an axis at right
angles to the plane ACEG, our space is then determined by the plane
ACEG, and the perpendicular axis. If we let this axis drop and suppose
the fourth axis, _w_, to take its place, we have a representation of
the space which runs off in the fourth dimension from the plane ACEG.
In this space we shall see simply the section ACEG of the cube, and
nothing else, for one cube does not extend to any distance in the
fourth dimension.
If, keeping this plane, we bring in the fourth dimension, we shall have
a space in which simply this section of the cube exists and nothing
else. The section can turn about the line AF, and parallel sections can
turn about parallel lines. Thus in considering the rotation about a
plane we can draw any lines we like and consider the rotation as taking
place in sections about them.
[Illustration: Fig. 42.]
To bring out this point more clearly let us take two parallel lines,
A and B, in the space of _xyz_, and let CD and EF be two rods running
above and below the plane of _xy_, from these lines. If we turn these
rods in our space about the lines A and B, as the upper end of one,
F, is going down, the lower end of the other, C, will be coming up.
They will meet and conflict. But it is quite possible for these two
rods each of them to turn about the two lines without altering their
relative distances.
To see this suppose the _y_ axis to go, and let the _w_ axis take its
place. We shall see the lines A and B no longer, for they run in the
_y_ direction from the points G and H.
[Illustration: Fig. 43.]
Fig. 43 is a picture of the two rods seen in the space of _xzw_. If
they rotate in the direction shown by the arrows—in the _z_ to _w_
direction—they move parallel to one another, keeping their relative
distances. Each will rotate about its own line, but their rotation will
not be inconsistent with their forming part of a rigid body.
Now we have but to suppose a central plane with rods crossing it
at every point, like CD and EF cross the plane of _xy_, to have an
image of a mass of matter extending equal distances on each side of a
diametral plane. As two of these rods can rotate round, so can all, and
the whole mass of matter can rotate round its diametral plane.
This rotation round a plane corresponds, in four dimensions, to the
rotation round an axis in three dimensions. Rotation of a body round a
plane is the analogue of rotation of a rod round an axis.
In a plane we have rotation round a point, in three-space rotation
round an axis line, in four-space rotation round an axis plane.
The four-dimensional being’s shaft by which he transmits power is a
disk rotating round its central plane—the whole contour corresponds
to the ends of an axis of rotation in our space. He can impart the
rotation at any point and take it off at any other point on the
contour, just as rotation round a line can in three-space be imparted
at one end of a rod and taken off at the other end.
A four-dimensional wheel can easily be described from the analogy of
the representation which a plane being would form for himself of one of
our wheels.
Suppose a wheel to move transverse to a plane, so that the whole disk,
which I will consider to be solid and without spokes, came at the same
time into contact with the plane. It would appear as a circular portion
of plane matter completely enclosing another and smaller portion—the
axle.
This appearance would last, supposing the motion of the wheel to
continue until it had traversed the plane by the extent of its
thickness, when there would remain in the plane only the small disk
which is the section of the axle. There would be no means obvious in
the plane at first by which the axle could be reached, except by going
through the substance of the wheel. But the possibility of reaching it
without destroying the substance of the wheel would be shown by the
continued existence of the axle section after that of the wheel had
disappeared.
In a similar way a four-dimensional wheel moving transverse to our
space would appear first as a solid sphere, completely surrounding
a smaller solid sphere. The outer sphere would represent the wheel,
and would last until the wheel has traversed our space by a distance
equal to its thickness. Then the small sphere alone would remain,
representing the section of the axle. The large sphere could move
round the small one quite freely. Any line in space could be taken as
an axis, and round this line the outer sphere could rotate, while the
inner sphere remained still. But in all these directions of revolution
there would be in reality one line which remained unaltered, that is
the line which stretches away in the fourth direction, forming the
axis of the axle. The four-dimensional wheel can rotate in any number
of planes, but all these planes are such that there is a line at right
angles to them all unaffected by rotation in them.
An objection is sometimes experienced as to this mode of reasoning from
a plane world to a higher dimensionality. How artificial, it is argued,
this conception of a plane world is. If any real existence confined to
a superficies could be shown to exist, there would be an argument for
one relative to which our three-dimensional existence is superficial.
But, both on the one side and the other of the space we are familiar
with, spaces either with less or more than three dimensions are merely
arbitrary conceptions.
In reply to this I would remark that a plane being having one less
dimension than our three would have one-third of our possibilities of
motion, while we have only one-fourth less than those of the higher
space. It may very well be that there may be a certain amount of
freedom of motion which is demanded as a condition of an organised
existence, and that no material existence is possible with a more
limited dimensionality than ours. This is well seen if we try to
construct the mechanics of a two-dimensional world. No tube could
exist, for unless joined together completely at one end two parallel
lines would be completely separate. The possibility of an organic
structure, subject to conditions such as this, is highly problematical;
yet, possibly in the convolutions of the brain there may be a mode of
existence to be described as two-dimensional.
We have but to suppose the increase in surface and the diminution in
mass carried on to a certain extent to find a region which, though
without mobility of the constituents, would have to be described as
two-dimensional.
But, however artificial the conception of a plane being may be, it is
none the less to be used in passing to the conception of a greater
dimensionality than ours, and hence the validity of the first part of
this objection altogether disappears directly we find evidence for such
a state of being.
The second part of the objection has more weight. How is it possible
to conceive that in a four-dimensional space any creatures should be
confined to a three-dimensional existence?
In reply I would say that we know as a matter of fact that life is
essentially a phenomenon of surface. The amplitude of the movements
which we can make is much greater along the surface of the earth than
it is up or down.
Now we have but to conceive the extent of a solid surface increased,
while the motions possible tranverse to it are diminished in the
same proportion, to obtain the image of a three-dimensional world in
four-dimensional space.
And as our habitat is the meeting of air and earth on the world, so
we must think of the meeting place of two as affording the condition
for our universe. The meeting of what two? What can that vastness be
in the higher space which stretches in such a perfect level that our
astronomical observations fail to detect the slightest curvature?
The perfection of the level suggests a liquid—a lake amidst what vast
scenery!—whereon the matter of the universe floats speck-like.
But this aspect of the problem is like what are called in mathematics
boundary conditions.
We can trace out all the consequences of four-dimensional movements
down to their last detail. Then, knowing the mode of action which
would be characteristic of the minutest particles, if they were
free, we can draw conclusions from what they actually do of what the
constraint on them is. Of the two things, the material conditions and
the motion, one is known, and the other can be inferred. If the place
of this universe is a meeting of two, there would be a one-sideness
to space. If it lies so that what stretches away in one direction in
the unknown is unlike what stretches away in the other, then, as far
as the movements which participate in that dimension are concerned,
there would be a difference as to which way the motion took place. This
would be shown in the dissimilarity of phenomena, which, so far as
all three-space movements are concerned, were perfectly symmetrical.
To take an instance, merely, for the sake of precising our ideas,
not for any inherent probability in it; if it could be shown that
the electric current in the positive direction were exactly like the
electric current in the negative direction, except for a reversal of
the components of the motion in three-dimensional space, then the
dissimilarity of the discharge from the positive and negative poles
would be an indication of a one-sideness to our space. The only cause
of difference in the two discharges would be due to a component in
the fourth dimension, which directed in one direction transverse to
our space, met with a different resistance to that which it met when
directed in the opposite direction.
CHAPTER VII
THE EVIDENCES FOR A FOURTH DIMENSION
The method necessarily to be employed in the search for the evidences
of a fourth dimension, consists primarily in the formation of the
conceptions of four-dimensional shapes and motions. When we are in
possession of these it is possible to call in the aid of observation,
without them we may have been all our lives in the familiar presence of
a four-dimensional phenomenon without ever recognising its nature.
To take one of the conceptions we have already formed, the turning of a
real thing into its mirror image would be an occurrence which it would
be hard to explain, except on the assumption of a fourth dimension.
We know of no such turning. But there exist a multitude of forms which
show a certain relation to a plane, a relation of symmetry, which
indicates more than an accidental juxtaposition of parts. In organic
life the universal type is of right- and left-handed symmetry, there
is a plane on each side of which the parts correspond. Now we have
seen that in four dimensions a plane takes the place of a line in
three dimensions. In our space, rotation about an axis is the type of
rotation, and the origin of bodies symmetrical about a line as the
earth is symmetrical about an axis can easily be explained. But where
there is symmetry about a plane no simple physical motion, such as we
are accustomed to, suffices to explain it. In our space a symmetrical
object must be built up by equal additions on each side of a central
plane. Such additions about such a plane are as little likely as any
other increments. The probability against the existence of symmetrical
form in inorganic nature is overwhelming in our space, and in organic
forms they would be as difficult of production as any other variety
of configuration. To illustrate this point we may take the child’s
amusement of making from dots of ink on a piece of paper a lifelike
representation of an insect by simply folding the paper over. The
dots spread out on a symmetrical line, and give the impression of a
segmented form with antennæ and legs.
Now seeing a number of such figures we should naturally infer a folding
over. Can, then, a folding over in four-dimensional space account for
the symmetry of organic forms? The folding cannot of course be of the
bodies we see, but it may be of those minute constituents, the ultimate
elements of living matter which, turned in one way or the other, become
right- or left-handed, and so produce a corresponding structure.
There is something in life not included in our conceptions of
mechanical movement. Is this something a four-dimensional movement?
If we look at it from the broadest point of view, there is something
striking in the fact that where life comes in there arises an entirely
different set of phenomena to those of the inorganic world.
The interest and values of life as we know it in ourselves, as we
know it existing around us in subordinate forms, is entirely and
completely different to anything which inorganic nature shows. And in
living beings we have a kind of form, a disposition of matter which
is entirely different from that shown in inorganic matter. Right-
and left-handed symmetry does not occur in the configurations of dead
matter. We have instances of symmetry about an axis, but not about
a plane. It can be argued that the occurrence of symmetry in two
dimensions involves the existence of a three-dimensional process, as
when a stone falls into water and makes rings of ripples, or as when
a mass of soft material rotates about an axis. It can be argued that
symmetry in any number of dimensions is the evidence of an action in
a higher dimensionality. Thus considering living beings, there is an
evidence both in their structure, and their different mode of activity,
of a something coming in from without into the inorganic world.
And the objections which will readily occur, such as those derived from
the forms of twin crystals and the theoretical structure of chemical
molecules, do not invalidate the argument; for in these forms too the
presumable seat of the activity producing them lies in that very minute
region in which we necessarily place the seat of a four-dimensional
mobility.
In another respect also the existence of symmetrical forms is
noteworthy. It is puzzling to conceive how two shapes exactly equal can
exist which are not superposible. Such a pair of symmetrical figures
as the two hands, right and left, show either a limitation in our
power of movement, by which we cannot superpose the one on the other,
or a definite influence and compulsion of space on matter, inflicting
limitations which are additional to those of the proportions of the
parts.
We will, however, put aside the arguments to be drawn from the
consideration of symmetry as inconclusive, retaining one valuable
indication which they afford. If it is in virtue of a four-dimensional
motion that symmetry exists, it is only in the very minute particles
of bodies that that motion is to be found, for there is no such thing
as a bending over in four dimensions of any object of a size which we
can observe. The region of the extremely minute is the one, then, which
we shall have to investigate. We must look for some phenomenon which,
occasioning movements of the kind we know, still is itself inexplicable
as any form of motion which we know.
Now in the theories of the actions of the minute particles of bodies
on one another, and in the motions of the ether, mathematicians
have tacitly assumed that the mechanical principles are the same as
those which prevail in the case of bodies which can be observed, it
has been assumed without proof that the conception of motion being
three-dimensional, holds beyond the region from observations in which
it was formed.
Hence it is not from any phenomenon explained by mathematics that we
can derive a proof of four dimensions. Every phenomenon that has been
explained is explained as three-dimensional. And, moreover, since in
the region of the very minute we do not find rigid bodies acting on
each other at a distance, but elastic substances and continuous fluids
such as ether, we shall have a double task.
We must form the conceptions of the possible movements of elastic and
liquid four-dimensional matter, before we can begin to observe. Let
us, therefore, take the four-dimensional rotation about a plane, and
enquire what it becomes in the case of extensible fluid substances. If
four-dimensional movements exist, this kind of rotation must exist, and
the finer portions of matter must exhibit it.
Consider for a moment a rod of flexible and extensible material. It can
turn about an axis, even if not straight; a ring of india rubber can
turn inside out.
What would this be in the case of four dimensions?
Let us consider a sphere of our three-dimensional matter having a
definite thickness. To represent this thickness let us suppose that
from every point of the sphere in fig. 44 rods project both ways, in
and out, like D and F. We can only see the external portion, because
the internal parts are hidden by the sphere.
[Illustration: Fig. 44.
_Axis of x running towards the observer._]
In this sphere the axis of _x_ is supposed to come towards the
observer, the axis of _z_ to run up, the axis of _y_ to go to the right.
[Illustration: Fig. 45.]
Now take the section determined by the _zy_ plane. This will be a
circle as shown in fig. 45. If we let drop the _x_ axis, this circle
is all we have of the sphere. Letting the _w_ axis now run in the
place of the old _x_ axis we have the space _yzw_, and in this space
all that we have of the sphere is the circle. Fig. 45 then represents
all that there is of the sphere in the space of _yzw_. In this space
it is evident that the rods CD and EF can turn round the circumference
as an axis. If the matter of the spherical shell is sufficiently
extensible to allow the particles C and E to become as widely separated
as they would be in the positions D and F, then the strip of matter
represented by CD and EF and a multitude of rods like them can turn
round the circular circumference.
Thus this particular section of the sphere can turn inside out, and
what holds for any one section holds for all. Hence in four dimensions
the whole sphere can, if extensible turn inside out. Moreover, any part
of it—a bowl-shaped portion, for instance—can turn inside out, and so
on round and round.
This is really no more than we had before in the rotation about a
plane, except that we see that the plane can, in the case of extensible
matter, be curved, and still play the part of an axis.
If we suppose the spherical shell to be of four-dimensional matter, our
representation will be a little different. Let us suppose there to be a
small thickness to the matter in the fourth dimension. This would make
no difference in fig. 44, for that merely shows the view in the _xyz_
space. But when the _x_ axis is let drop, and the _w_ axis comes in,
then the rods CD and EF which represent the matter of the shell, will
have a certain thickness perpendicular to the plane of the paper on
which they are drawn. If they have a thickness in the fourth dimension
they will show this thickness when looked at from the direction of the
_w_ axis.
Supposing these rods, then, to be small slabs strung on the
circumference of the circle in fig. 45, we see that there will not
be in this case either any obstacle to their turning round the
circumference. We can have a shell of extensible material or of fluid
material turning inside out in four dimensions.
And we must remember that in four dimensions there is no such thing as
rotation round an axis. If we want to investigate the motion of fluids
in four dimensions we must take a movement about an axis in our space,
and find the corresponding movement about a plane in four space.
Now, of all the movements which take place in fluids, the most
important from a physical point of view is vortex motion.
A vortex is a whirl or eddy—it is shown in the gyrating wreaths of
dust seen on a summer day; it is exhibited on a larger scale in the
destructive march of a cyclone.
A wheel whirling round will throw off the water on it. But when
this circling motion takes place in a liquid itself it is strangely
persistent. There is, of course, a certain cohesion between the
particles of water by which they mutually impede their motions. But
in a liquid devoid of friction, such that every particle is free from
lateral cohesion on its path of motion, it can be shown that a vortex
or eddy separates from the mass of the fluid a certain portion, which
always remain in that vortex.
The shape of the vortex may alter, but it always consists of the same
particles of the fluid.
Now, a very remarkable fact about such a vortex is that the ends of the
vortex cannot remain suspended and isolated in the fluid. They must
always run to the boundary of the fluid. An eddy in water that remains
half way down without coming to the top is impossible.
The ends of a vortex must reach the boundary of a fluid—the boundary
may be external or internal—a vortex may exist between two objects
in the fluid, terminating one end on each object, the objects being
internal boundaries of the fluid. Again, a vortex may have its ends
linked together, so that it forms a ring. Circular vortex rings of
this description are often seen in puffs of smoke, and that the smoke
travels on in the ring is a proof that the vortex always consists of
the same particles of air.
Let us now enquire what a vortex would be in a four-dimensional fluid.
We must replace the line axis by a plane axis. We should have therefore
a portion of fluid rotating round a plane.
We have seen that the contour of this plane corresponds with the ends
of the axis line. Hence such a four-dimensional vortex must have its
rim on a boundary of the fluid. There would be a region of vorticity
with a contour. If such a rotation were started at one part of a
circular boundary, its edges would run round the boundary in both
directions till the whole interior region was filled with the vortex
sheet.
A vortex in a three-dimensional liquid may consist of a number of
vortex filaments lying together producing a tube, or rod of vorticity.
In the same way we can have in four dimensions a number of vortex
sheets alongside each other, each of which can be thought of as a
bowl-shaped portion of a spherical shell turning inside out. The
rotation takes place at any point not in the space occupied by the
shell, but from that space to the fourth dimension and round back again.
Is there anything analogous to this within the range of our observation?
An electric current answers this description in every respect.
Electricity does not flow through a wire. Its effect travels both ways
from the starting point along the wire. The spark which shows its
passing midway in its circuit is later than that which occurs at points
near its starting point on either side of it.
Moreover, it is known that the action of the current is not in the
wire. It is in the region enclosed by the wire, this is the field of
force, the locus of the exhibition of the effects of the current.
And the necessity of a conducting circuit for a current is exactly
that which we should expect if it were a four-dimensional vortex.
According to Maxwell every current forms a closed circuit, and this,
from the four-dimensional point of view, is the same as saying a vortex
must have its ends on a boundary of the fluid.
Thus, on the hypothesis of a fourth dimension, the rotation of the
fluid ether would give the phenomenon of an electric current. We must
suppose the ether to be full of movement, for the more we examine into
the conditions which prevail in the obscurity of the minute, the more
we find that an unceasing and perpetual motion reigns. Thus we may say
that the conception of the fourth dimension means that there must be a
phenomenon which presents the characteristics of electricity.
We know now that light is an electro-magnetic action, and that so far
from being a special and isolated phenomenon this electric action is
universal in the realm of the minute. Hence, may we not conclude that,
so far from the fourth dimension being remote and far away, being a
thing of symbolic import, a term for the explanation of dubious facts
by a more obscure theory, it is really the most important fact within
our knowledge. Our three-dimensional world is superficial. These
processes, which really lie at the basis of all phenomena of matter,
escape our observation by their minuteness, but reveal to our intellect
an amplitude of motion surpassing any that we can see. In such shapes
and motions there is a realm of the utmost intellectual beauty, and one
to which our symbolic methods apply with a better grace than they do to
those of three dimensions.
CHAPTER VIII
THE USE OF FOUR DIMENSIONS IN THOUGHT
Having held before ourselves this outline of a conjecture of the world
as four-dimensional, having roughly thrown together those facts of
movement which we can see apply to our actual experience, let us pass
to another branch of our subject.
The engineer uses drawings, graphical constructions, in a variety of
manners. He has, for instance, diagrams which represent the expansion
of steam, the efficiency of his valves. These exist alongside the
actual plans of his machines. They are not the pictures of anything
really existing, but enable him to think about the relations which
exist in his mechanisms.
And so, besides showing us the actual existence of that world which
lies beneath the one of visible movements, four-dimensional space
enables us to make ideal constructions which serve to represent the
relations of things, and throw what would otherwise be obscure into a
definite and suggestive form.
From amidst the great variety of instances which lies before me I will
select two, one dealing with a subject of slight intrinsic interest,
which however gives within a limited field a striking example of the
method of drawing conclusions and the use of higher space figures.[1]
[1] It is suggestive also in another respect, because it shows very
clearly that in our processes of thought there are in play faculties
other than logical; in it the origin of the idea which proves to be
justified is drawn from the consideration of symmetry, a branch of the
beautiful.
The other instance is chosen on account of the bearing it has on our
fundamental conceptions. In it I try to discover the real meaning of
Kant’s theory of experience.
The investigation of the properties of numbers is much facilitated
by the fact that relations between numbers are themselves able to be
represented as numbers—_e.g._, 12, and 3 are both numbers, and the
relation between them is 4, another number. The way is thus opened for
a process of constructive theory, without there being any necessity for
a recourse to another class of concepts besides that which is given in
the phenomena to be studied.
The discipline of number thus created is of great and varied
applicability, but it is not solely as quantitative that we learn to
understand the phenomena of nature. It is not possible to explain the
properties of matter by number simply, but all the activities of matter
are energies in space. They are numerically definite and also, we may
say, directedly definite, _i.e._ definite in direction.
Is there, then, a body of doctrine about space which, like that of
number, is available in science? It is needless to answer: Yes;
geometry. But there is a method lying alongside the ordinary methods of
geometry, which tacitly used and presenting an analogy to the method of
numerical thought deserves to be brought into greater prominence than
it usually occupies.
The relation of numbers is a number.
Can we say in the same way that the relation of shapes is a shape?
We can.
To take an instance chosen on account of its ready availability. Let
us take two right-angled triangles of a given hypothenuse, but having
sides of different lengths (fig. 46). These triangles are shapes which
have a certain relation to each other. Let us exhibit their relation as
a figure.
[Illustration: Fig. 46.]
Draw two straight lines at right angles to each other, the one HL a
horizontal level, the other VL a vertical level (fig. 47). By means
of these two co-ordinating lines we can represent a double set of
magnitudes; one set as distances to the right of the vertical level,
the other as distances above the horizontal level, a suitable unit
being chosen.
[Illustration: Fig. 47.]
Thus the line marked 7 will pick out the assemblage of points whose
distance from the vertical level is 7, and the line marked 1 will pick
out the points whose distance above the horizontal level is 1. The
meeting point of these two lines, 7 and 1, will define a point which
with regard to the one set of magnitudes is 7, with regard to the
other is 1. Let us take the sides of our triangles as the two sets of
magnitudes in question.
Then the point 7, 1, will represent the triangle whose sides are 7 and
1. Similarly the point 5, 5—5, that is, to the right of the vertical
level and 5 above the horizontal level—will represent the triangle
whose sides are 5 and 5 (fig. 48).
[Illustration: Fig. 48.]
Thus we have obtained a figure consisting of the two points 7, 1, and
5, 5, representative of our two triangles. But we can go further,
and, drawing an arc of a circle about O, the meeting point of the
horizontal and vertical levels, which passes through 7, 1, and 5,
5, assert that all the triangles which are right-angled and have a
hypothenuse whose square is 50 are represented by the points on this
arc.
Thus, each individual of a class being represented by a point, the
whole class is represented by an assemblage of points forming a
figure. Accepting this representation we can attach a definite and
calculable significance to the expression, resemblance, or similarity
between two individuals of the class represented, the difference being
measured by the length of the line between two representative points.
It is needless to multiply examples, or to show how, corresponding to
different classes of triangles, we obtain different curves.
A representation of this kind in which an object, a thing in space,
is represented as a point, and all its properties are left out, their
effect remaining only in the relative position which the representative
point bears to the representative points of the other objects, may be
called, after the analogy of Sir William R. Hamilton’s hodograph, a
“Poiograph.”
Representations thus made have the character of natural objects;
they have a determinate and definite character of their own. Any
lack of completeness in them is probably due to a failure in point
of completeness of those observations which form the ground of their
construction.
Every system of classification is a poiograph. In Mendeléeff’s scheme
of the elements, for instance, each element is represented by a point,
and the relations between the elements are represented by the relations
between the points.
So far I have simply brought into prominence processes and
considerations with which we are all familiar. But it is worth while
to bring into the full light of our attention our habitual assumptions
and processes. It often happens that we find there are two of them
which have a bearing on each other, which, without this dragging into
the light, we should have allowed to remain without mutual influence.
There is a fact which it concerns us to take into account in discussing
the theory of the poiograph.
With respect to our knowledge of the world we are far from that
condition which Laplace imagined when he asserted that an all-knowing
mind could determine the future condition of every object, if he knew
the co-ordinates of its particles in space, and their velocity at any
particular moment.
On the contrary, in the presence of any natural object, we have a great
complexity of conditions before us, which we cannot reduce to position
in space and date in time.
There is mass, attraction apparently spontaneous, electrical and
magnetic properties which must be superadded to spatial configuration.
To cut the list short we must say that practically the phenomena of the
world present us problems involving many variables, which we must take
as independent.
From this it follows that in making poiographs we must be prepared
to use space of more than three dimensions. If the symmetry and
completeness of our representation is to be of use to us we must be
prepared to appreciate and criticise figures of a complexity greater
than of those in three dimensions. It is impossible to give an example
of such a poiograph which will not be merely trivial, without going
into details of some kind irrelevant to our subject. I prefer to
introduce the irrelevant details rather than treat this part of the
subject perfunctorily.
To take an instance of a poiograph which does not lead us into the
complexities incident on its application in classificatory science,
let us follow Mrs. Alicia Boole Stott in her representation of the
syllogism by its means. She will be interested to find that the curious
gap she detected has a significance.
A syllogism consists of two statements, the major and the minor
premiss, with the conclusion that can be drawn from them. Thus, to take
an instance, fig. 49. It is evident, from looking at the successive
figures that, if we know that the region M lies altogether within the
region P, and also know that the region S lies altogether within the
region M, we can conclude that the region S lies altogether within
the region P. M is P, major premiss; S is M, minor premiss; S is P,
conclusion. Given the first two data we must conclude that S lies
in P. The conclusion S is P involves two terms, S and P, which are
respectively called the subject and the predicate, the letters S and
P being chosen with reference to the parts the notions they designate
play in the conclusion. S is the subject of the conclusion, P is the
predicate of the conclusion. The major premiss we take to be, that
which does not involve S, and here we always write it first.
[Illustration: Fig. 49.]
There are several varieties of statement possessing different degrees
of universality and manners of assertiveness. These different forms of
statement are called the moods.
We will take the major premiss as one variable, as a thing capable of
different modifications of the same kind, the minor premiss as another,
and the different moods we will consider as defining the variations
which these variables undergo.
There are four moods:—
1. The universal affirmative; all M is P, called mood A.
2. The universal negative; no M is P, mood E.
3. The particular affirmative; some M is P, mood I.
4. The particular negative; some M is not P, mood O.
[Illustration: 1. 2. 3. 4. Mood A. Mood E. Mood I. Mood O.
Fig. 50.]
The dotted lines in 3 and 4, fig. 50, denote that it is not known
whether or no any objects exist, corresponding to the space of which
the dotted line forms one delimiting boundary; thus, in mood I we do
not know if there are any M’S which are not P, we only know some M’S
are P.
[Illustration: Fig. 51.]
Representing the first premiss in its various moods by regions marked
by vertical lines to the right of PQ, we have in fig. 51, running up
from the four letters AEIO, four columns, each of which indicates that
the major premiss is in the mood denoted by the respective letter. In
the first column to the right of PQ is the mood A. Now above the line
RS let there be marked off four regions corresponding to the four moods
of the minor premiss. Thus, in the first row above RS all the region
between RS and the first horizontal line above it denotes that the
minor premiss is in the mood A. The letters E, I, O, in the same way
show the mood characterising the minor premiss in the rows opposite
these letters.
We have still to exhibit the conclusion. To do this we must consider
the conclusion as a third variable, characterised in its different
varieties by four moods—this being the syllogistic classification. The
introduction of a third variable involves a change in our system of
representation.
Before we started with the regions to the right of a certain line as
representing successively the major premiss in its moods; now we must
start with the regions to the right of a certain plane. Let LMNR be
the plane face of a cube, fig. 52, and let the cube be divided into
four parts by vertical sections parallel to LMNR. The variable, the
major premiss, is represented by the successive regions which occur to
the right of the plane LMNR—that region to which A stands opposite,
that slice of the cube, is significative of the mood A. This whole
quarter-part of the cube represents that for every part of it the major
premiss is in the mood A.
[Illustration: Fig. 52.]
In a similar manner the next section, the second with the letter E
opposite it, represents that for every one of the sixteen small cubic
spaces in it, the major premiss is in the mood E. The third and fourth
compartments made by the vertical sections denote the major premiss in
the moods I and O. But the cube can be divided in other ways by other
planes. Let the divisions, of which four stretch from the front face,
correspond to the minor premiss. The first wall of sixteen cubes,
facing the observer, has as its characteristic that in each of the
small cubes, whatever else may be the case, the minor premiss is in the
mood A. The variable—the minor premiss—varies through the phases A, E,
I, O, away from the front face of the cube, or the front plane of which
the front face is a part.
And now we can represent the third variable in a precisely similar way.
We can take the conclusion as the third variable, going through its
four phases from the ground plane upwards. Each of the small cubes at
the base of the whole cube has this true about it, whatever else may
be the case, that the conclusion is, in it, in the mood A. Thus, to
recapitulate, the first wall of sixteen small cubes, the first of the
four walls which, proceeding from left to right, build up the whole
cube, is characterised in each part of it by this, that the major
premiss is in the mood A.
The next wall denotes that the major premiss is in the mood E, and
so on. Proceeding from the front to the back the first wall presents
a region in every part of which the minor premiss is in the mood A.
The second wall is a region throughout which the minor premiss is in
the mood E, and so on. In the layers, from the bottom upwards, the
conclusion goes through its various moods beginning with A in the
lowest, E in the second, I in the third, O in the fourth.
In the general case, in which the variables represented in the
poiograph pass through a wide range of values, the planes from which we
measure their degrees of variation in our representation are taken to
be indefinitely extended. In this case, however, all we are concerned
with is the finite region.
We have now to represent, by some limitation of the complex we have
obtained, the fact that not every combination of premisses justifies
any kind of conclusion. This can be simply effected by marking the
regions in which the premisses, being such as are defined by the
positions, a conclusion which is valid is found.
Taking the conjunction of the major premiss, all M is P, and the minor,
all S is M, we conclude that all S is P. Hence, that region must be
marked in which we have the conjunction of major premiss in mood A;
minor premiss, mood A; conclusion, mood A. This is the cube occupying
the lowest left-hand corner of the large cube.
[Illustration: Fig. 53.]
Proceeding in this way, we find that the regions which must be marked
are those shown in fig. 53. To discuss the case shown in the marked
cube which appears at the top of fig. 53. Here the major premiss is
in the second wall to the right—it is in the mood E and is of the
type no M is P. The minor premiss is in the mood characterised by the
third wall from the front. It is of the type some S is M. From these
premisses we draw the conclusion that some S is not P, a conclusion in
the mood O. Now the mood O of the conclusion is represented in the top
layer. Hence we see that the marking is correct in this respect.
[Illustration: Fig. 54.]
It would, of course, be possible to represent the cube on a plane by
means of four squares, as in fig. 54, if we consider each square to
represent merely the beginning of the region it stands for. Thus the
whole cube can be represented by four vertical squares, each standing
for a kind of vertical tray, and the markings would be as shown. In No.
1 the major premiss is in mood A for the whole of the region indicated
by the vertical square of sixteen divisions; in No. 2 it is in the mood
E, and so on.
A creature confined to a plane would have to adopt some such
disjunctive way of representing the whole cube. He would be obliged to
represent that which we see as a whole in separate parts, and each part
would merely represent, would not be, that solid content which we see.
The view of these four squares which the plane creature would have
would not be such as ours. He would not see the interior of the four
squares represented above, but each would be entirely contained within
its outline, the internal boundaries of the separate small squares he
could not see except by removing the outer squares.
We are now ready to introduce the fourth variable involved in the
syllogism.
In assigning letters to denote the terms of the syllogism we have taken
S and P to represent the subject and predicate in the conclusion, and
thus in the conclusion their order is invariable. But in the premisses
we have taken arbitrarily the order all M is P, and all S is M. There
is no reason why M instead of P should not be the predicate of the
major premiss, and so on.
Accordingly we take the order of the terms in the premisses as the
fourth variable. Of this order there are four varieties, and these
varieties are called figures.
Using the order in which the letters are written to denote that the
letter first written is subject, the one written second is predicate,
we have the following possibilities:—
1st Figure. 2nd Figure. 3rd Figure. 4th Figure.
Major M P P M M P P M
Minor S M S M M S M S
There are therefore four possibilities with regard to this fourth
variable as with regard to the premisses.
We have used up our dimensions of space in representing the phases of
the premisses and the conclusion in respect of mood, and to represent
in an analogous manner the variations in figure we require a fourth
dimension.
Now in bringing in this fourth dimension we must make a change in our
origins of measurement analogous to that which we made in passing from
the plane to the solid.
This fourth dimension is supposed to run at right angles to any of the
three space dimensions, as the third space dimension runs at right
angles to the two dimensions of a plane, and thus it gives us the
opportunity of generating a new kind of volume. If the whole cube moves
in this dimension, the solid itself traces out a path, each section of
which, made at right angles to the direction in which it moves, is a
solid, an exact repetition of the cube itself.
The cube as we see it is the beginning of a solid of such a kind. It
represents a kind of tray, as the square face of the cube is a kind of
tray against which the cube rests.
Suppose the cube to move in this fourth dimension in four stages,
and let the hyper-solid region traced out in the first stage of its
progress be characterised by this, that the terms of the syllogism
are in the first figure, then we can represent in each of the three
subsequent stages the remaining three figures. Thus the whole cube
forms the basis from which we measure the variation in figure. The
first figure holds good for the cube as we see it, and for that
hyper-solid which lies within the first stage; the second figure holds
good in the second stage, and so on.
Thus we measure from the whole cube as far as figures are concerned.
But we saw that when we measured in the cube itself having three
variables, namely, the two premisses and the conclusion, we measured
from three planes. The base from which we measured was in every case
the same.
Hence, in measuring in this higher space we should have bases of the
same kind to measure from, we should have solid bases.
The first solid base is easily seen, it is the cube itself. The other
can be found from this consideration.
That solid from which we measure figure is that in which the remaining
variables run through their full range of varieties.
Now, if we want to measure in respect of the moods of the major
premiss, we must let the minor premiss, the conclusion, run through
their range, and also the order of the terms. That is we must take as
basis of measurement in respect to the moods of the major that which
represents the variation of the moods of the minor, the conclusion and
the variation of the figures.
Now the variation of the moods of the minor and of the conclusion are
represented in the square face on the left of the cube. Here are all
varieties of the minor premiss and the conclusion. The varieties of
the figures are represented by stages in a motion proceeding at right
angles to all space directions, at right angles consequently to the
face in question, the left-hand face of the cube.
Consequently letting the left-hand face move in this direction we get
a cube, and in this cube all the varieties of the minor premiss, the
conclusion, and the figure are represented.
Thus another cubic base of measurement is given to the cube, generated
by movement of the left-hand square in the fourth dimension.
We find the other bases in a similar manner, one is the cube generated
by the front square moved in the fourth dimension so as to generate a
cube. From this cube variations in the mood of the minor are measured.
The fourth base is that found by moving the bottom square of the cube
in the fourth dimension. In this cube the variations of the major,
the minor, and the figure are given. Considering this as a basis in
the four stages proceeding from it, the variation in the moods of the
conclusion are given.
Any one of these cubic bases can be represented in space, and then the
higher solid generated from them lies out of our space. It can only
be represented by a device analogous to that by which the plane being
represents a cube.
He represents the cube shown above, by taking four square sections and
placing them arbitrarily at convenient distances the one from the other.
So we must represent this higher solid by four cubes: each cube
represents only the beginning of the corresponding higher volume.
It is sufficient for us, then, if we draw four cubes, the first
representing that region in which the figure is of the first kind,
the second that region in which the figure is of the second kind,
and so on. These cubes are the beginnings merely of the respective
regions—they are the trays, as it were, against which the real solids
must be conceived as resting, from which they start. The first one, as
it is the beginning of the region of the first figure, is characterised
by the order of the terms in the premisses being that of the first
figure. The second similarly has the terms of the premisses in the
order of the second figure, and so on.
These cubes are shown below.
For the sake of showing the properties of the method of representation,
not for the logical problem, I will make a digression. I will represent
in space the moods of the minor and of the conclusion and the different
figures, keeping the major always in mood A. Here we have three
variables in different stages, the minor, the conclusion, and the
figure. Let the square of the left-hand side of the original cube be
imagined to be standing by itself, without the solid part of the cube,
represented by (2) fig. 55. The A, E, I, O, which run away represent
the moods of the minor, the A, E, I, O, which run up represent the
moods of the conclusion. The whole square, since it is the beginning
of the region in the major premiss, mood A, is to be considered as in
major premiss, mood A.
From this square, let it be supposed that that direction in which the
figures are represented runs to the left hand. Thus we have a cube (1)
running from the square above, in which the square itself is hidden,
but the letters A, E, I, O, of the conclusion are seen. In this cube
we have the minor premiss and the conclusion in all their moods, and
all the figures represented. With regard to the major premiss, since
the face (2) belongs to the first wall from the left in the original
arrangement, and in this arrangement was characterised by the major
premiss in the mood A, we may say that the whole of the cube we now
have put up represents the mood A of the major premiss.
[Illustration: Fig. 55.]
Hence the small cube at the bottom to the right in 1, nearest to the
spectator, is major premiss, mood A; minor premiss, mood A; conclusion,
mood A; and figure the first. The cube next to it, running to the left,
is major premiss, mood A; minor premiss, mood A; conclusion, mood A;
figure 2.
So in this cube we have the representations of all the combinations
which can occur when the major premiss, remaining in the mood A, the
minor premiss, the conclusion, and the figures pass through their
varieties.
In this case there is no room in space for a natural representation of
the moods of the major premiss. To represent them we must suppose as
before that there is a fourth dimension, and starting from this cube as
base in the fourth direction in four equal stages, all the first volume
corresponds to major premiss A, the second to major premiss, mood E,
the next to the mood I, and the last to mood O.
The cube we see is as it were merely a tray against which the
four-dimensional figure rests. Its section at any stage is a cube. But
a transition in this direction being transverse to the whole of our
space is represented by no space motion. We can exhibit successive
stages of the result of transference of the cube in that direction, but
cannot exhibit the product of a transference, however small, in that
direction.
[Illustration: Fig. 56.]
To return to the original method of representing our variables,
consider fig. 56. These four cubes represent four sections of the
figure derived from the first of them by moving it in the fourth
dimension. The first portion of the motion, which begins with 1, traces
out a more than solid body, which is all in the first figure. The
beginning of this body is shown in 1. The next portion of the motion
traces out a more than solid body, all of which is in the second
figure; the beginning of this body is shown in 2; 3 and 4 follow on in
like manner. Here, then, in one four-dimensional figure we have all
the combinations of the four variables, major premiss, minor premiss,
figure, conclusion, represented, each variable going through its four
varieties. The disconnected cubes drawn are our representation in space
by means of disconnected sections of this higher body.
Now it is only a limited number of conclusions which are true—their
truth depends on the particular combinations of the premisses and
figures which they accompany. The total figure thus represented may be
called the universe of thought in respect to these four constituents,
and out of the universe of possibly existing combinations it is the
province of logic to select those which correspond to the results of
our reasoning faculties.
We can go over each of the premisses in each of the moods, and find out
what conclusion logically follows. But this is done in the works on
logic; most simply and clearly I believe in “Jevon’s Logic.” As we are
only concerned with a formal presentation of the results we will make
use of the mnemonic lines printed below, in which the words enclosed in
brackets refer to the figures, and are not significative:—
Barbara celarent Darii ferio_que_ [prioris].
Caesare Camestris Festino Baroko [secundae].
[Tertia] darapti disamis datisi felapton.
Bokardo ferisson _habet_ [Quarta insuper addit].
Bramantip camenes dimaris ferapton fresison.
In these lines each significative word has three vowels, the first
vowel refers to the major premiss, and gives the mood of that premiss,
“a” signifying, for instance, that the major mood is in mood _a_. The
second vowel refers to the minor premiss, and gives its mood. The third
vowel refers to the conclusion, and gives its mood. Thus (prioris)—of
the first figure—the first mnemonic word is “barbara,” and this gives
major premiss, mood A; minor premiss, mood A; conclusion, mood A.
Accordingly in the first of our four cubes we mark the lowest left-hand
front cube. To take another instance in the third figure “Tertia,”
the word “ferisson” gives us major premiss mood E—_e.g._, no M is P,
minor premiss mood I; some M is S, conclusion, mood O; some S is not P.
The region to be marked then in the third representative cube is the
one in the second wall to the right for the major premiss, the third
wall from the front for the minor premiss, and the top layer for the
conclusion.
It is easily seen that in the diagram this cube is marked, and so with
all the valid conclusions. The regions marked in the total region show
which combinations of the four variables, major premiss, minor premiss,
figure, and conclusion exist.
That is to say, we objectify all possible conclusions, and build up an
ideal manifold, containing all possible combinations of them with the
premisses, and then out of this we eliminate all that do not satisfy
the laws of logic. The residue is the syllogism, considered as a canon
of reasoning.
Looking at the shape which represents the totality of the valid
conclusions, it does not present any obvious symmetry, or easily
characterisable nature. A striking configuration, however, is
obtained, if we project the four-dimensional figure obtained into a
three-dimensional one; that is, if we take in the base cube all those
cubes which have a marked space anywhere in the series of four regions
which start from that cube.
This corresponds to making abstraction of the figures, giving all the
conclusions which are valid whatever the figure may be.
[Illustration: Fig. 57.]
Proceeding in this way we obtain the arrangement of marked cubes shown
in fig. 57. We see that the valid conclusions are arranged almost
symmetrically round one cube—the one on the top of the column starting
from AAA. There is one breach of continuity however in this scheme.
One cube is unmarked, which if marked would give symmetry. It is the
one which would be denoted by the letters I, E, O, in the third
wall to the right, the second wall away, the topmost layer. Now this
combination of premisses in the mood IE, with a conclusion in the mood
O, is not noticed in any book on logic with which I am familiar. Let
us look at it for ourselves, as it seems that there must be something
curious in connection with this break of continuity in the poiograph.
[Illustration: Fig. 58.]
The propositions I, E, in the various figures are the following, as
shown in the accompanying scheme, fig. 58:—First figure: some M is P;
no S is M. Second figure: some P is M; no S is M. Third figure: some M
is P; no M is S. Fourth figure: some P is M; no M is S.
Examining these figures, we see, taking the first, that if some M is P
and no S is M, we have no conclusion of the form S is P in the various
moods. It is quite indeterminate how the circle representing S lies
with regard to the circle representing P. It may lie inside, outside,
or partly inside P. The same is true in the other figures 2 and 3.
But when we come to the fourth figure, since M and S lie completely
outside each other, there cannot lie inside S that part of P which lies
inside M. Now we know by the major premiss that some of P does lie in
M. Hence S cannot contain the whole of P. In words, some P is M, no
M is S, therefore S does not contain the whole of P. If we take P as
the subject, this gives us a conclusion in the mood O about P. Some
P is not S. But it does not give us conclusion about S in any one of
the four forms recognised in the syllogism and called its moods. Hence
the breach of the continuity in the poiograph has enabled us to detect
a lack of completeness in the relations which are considered in the
syllogism.
To take an instance:—Some Americans (P) are of African stock (M); No
Aryans (S) are of African stock (M); Aryans (S) do not include all of
Americans (P).
In order to draw a conclusion about S we have to admit the statement,
“S does not contain the whole of P,” as a valid logical form—it is a
statement about S which can be made. The logic which gives us the form,
“some P is not S,” and which does not allow us to give the exactly
equivalent and equally primary form, “S does not contain the whole of
P,” is artificial.
And I wish to point out that this artificiality leads to an error.
If one trusted to the mnemonic lines given above, one would conclude
that no logical conclusion about S can be drawn from the statement,
“some P are M, no M are S.”
But a conclusion can be drawn: S does not contain the whole of P.
It is not that the result is given expressed in another form. The
mnemonic lines deny that any conclusion can be drawn from premisses in
the moods I, E, respectively.
Thus a simple four-dimensional poiograph has enabled us to detect a
mistake in the mnemonic lines which have been handed down unchallenged
from mediæval times. To discuss the subject of these lines more fully a
logician defending them would probably say that a particular statement
cannot be a major premiss; and so deny the existence of the fourth
figure in the combination of moods.
To take our instance: some Americans are of African stock; no Aryans
are of African stock. He would say that the conclusion is some
Americans are not Aryans; and that the second statement is the major.
He would refuse to say anything about Aryans, condemning us to an
eternal silence about them, as far as these premisses are concerned!
But, if there is a statement involving the relation of two classes, it
must be expressible as a statement about either of them.
To bar the conclusion, “Aryans do not include the whole of Americans,”
is purely a makeshift in favour of a false classification.
And the argument drawn from the universality of the major premiss
cannot be consistently maintained. It would preclude such combinations
as major O, minor A, conclusion O—_i.e._, such as some mountains (M)
are not permanent (P); all mountains (M) are scenery (S); some scenery
(S) is not permanent (P).
This is allowed in “Jevon’s Logic,” and his omission to discuss I, E,
O, in the fourth figure, is inexplicable. A satisfactory poiograph
of the logical scheme can be made by admitting the use of the words
some, none, or all, about the predicate as well as about the subject.
Then we can express the statement, “Aryans do not include the whole of
Americans,” clumsily, but, when its obscurity is fathomed, correctly,
as “Some Aryans are not all Americans.” And this method is what is
called the “quantification of the predicate.”
The laws of formal logic are coincident with the conclusions which
can be drawn about regions of space, which overlap one another in the
various possible ways. It is not difficult so to state the relations
or to obtain a symmetrical poiograph. But to enter into this branch
of geometry is beside our present purpose, which is to show the
application of the poiograph in a finite and limited region, without
any of those complexities which attend its use in regard to natural
objects.
If we take the latter—plants, for instance—and, without assuming
fixed directions in space as representative of definite variations,
arrange the representative points in such a manner as to correspond to
the similarities of the objects, we obtain configuration of singular
interest; and perhaps in this way, in the making of shapes of shapes,
bodies with bodies omitted, some insight into the structure of the
species and genera might be obtained.
CHAPTER IX
APPLICATION TO KANT’S THEORY OF EXPERIENCE
When we observe the heavenly bodies we become aware that they all
participate in one universal motion—a diurnal revolution round the
polar axis.
In the case of fixed stars this is most unqualifiedly true, but in the
case of the sun, and the planets also, the single motion of revolution
can be discerned, modified, and slightly altered by other and secondary
motions.
Hence the universal characteristic of the celestial bodies is that they
move in a diurnal circle.
But we know that this one great fact which is true of them all has in
reality nothing to do with them. The diurnal revolution which they
visibly perform is the result of the condition of the observer. It is
because the observer is on a rotating earth that a universal statement
can be made about all the celestial bodies.
The universal statement which is valid about every one of the celestial
bodies is that which does not concern them at all, and is but a
statement of the condition of the observer.
Now there are universal statements of other kinds which we can make. We
can say that all objects of experience are in space and subject to the
laws of geometry.
Does this mean that space and all that it means is due to a condition
of the observer?
If a universal law in one case means nothing affecting the objects
themselves, but only a condition of observation, is this true in every
case? There is shown us in astronomy a _vera causa_ for the assertion
of a universal. Is the same cause to be traced everywhere?
Such is a first approximation to the doctrine of Kant’s critique.
It is the apprehension of a relation into which, on the one side and
the other, perfectly definite constituents enter—the human observer and
the stars—and a transference of this relation to a region in which the
constituents on either side are perfectly unknown.
If spatiality is due to a condition of the observer, the observer
cannot be this bodily self of ours—the body, like the objects around
it, are equally in space.
This conception Kant applied, not only to the intuitions of sense, but
to the concepts of reason—wherever a universal statement is made there
is afforded him an opportunity for the application of his principle.
He constructed a system in which one hardly knows which the most to
admire, the architectonic skill, or the reticence with regard to things
in themselves, and the observer in himself.
His system can be compared to a garden, somewhat formal perhaps, but
with the charm of a quality more than intellectual, a _besonnenheit_,
an exquisite moderation over all. And from the ground he so carefully
prepared with that buried in obscurity, which it is fitting should be
obscure, science blossoms and the tree of real knowledge grows.
The critique is a storehouse of ideas of profound interest. The one
of which I have given a partial statement leads, as we shall see
on studying it in detail, to a theory of mathematics suggestive of
enquiries in many directions.
The justification for my treatment will be found amongst other passages
in that part of the transcendental analytic, in which Kant speaks of
objects of experience subject to the forms of sensibility, not subject
to the concepts of reason.
Kant asserts that whenever we think we think of objects in space and
time, but he denies that the space and time exist as independent
entities. He goes about to explain them, and their universality, not by
assuming them, as most other philosophers do, but by postulating their
absence. How then does it come to pass that the world is in space and
time to us?
Kant takes the same position with regard to what we call nature—a great
system subject to law and order. “How do you explain the law and order
in nature?” we ask the philosophers. All except Kant reply by assuming
law and order somewhere, and then showing how we can recognise it.
In explaining our notions, philosophers from other than the Kantian
standpoint, assume the notions as existing outside us, and then it is
no difficult task to show how they come to us, either by inspiration or
by observation.
We ask “Why do we have an idea of law in nature?” “Because natural
processes go according to law,” we are answered, “and experience
inherited or acquired, gives us this notion.”
But when we speak about the law in nature we are speaking about a
notion of our own. So all that these expositors do is to explain our
notion by an assumption of it.
Kant is very different. He supposes nothing. An experience such as ours
is very different from experience in the abstract. Imagine just simply
experience, succession of states, of consciousness! Why, there would be
no connecting any two together, there would be no personal identity,
no memory. It is out of a general experience such as this, which, in
respect to anything we call real, is less than a dream, that Kant shows
the genesis of an experience such as ours.
Kant takes up the problem of the explanation of space, time, order, and
so quite logically does not presuppose them.
But how, when every act of thought is of things in space, and time,
and ordered, shall we represent to ourselves that perfectly indefinite
somewhat which is Kant’s necessary hypothesis—that which is not in
space or time and is not ordered. That is our problem, to represent
that which Kant assumes not subject to any of our forms of thought, and
then show some function which working on that makes it into a “nature”
subject to law and order, in space and time. Such a function Kant
calls the “Unity of Apperception”; _i.e._, that which makes our state
of consciousness capable of being woven into a system with a self, an
outer world, memory, law, cause, and order.
The difficulty that meets us in discussing Kant’s hypothesis is that
everything we think of is in space and time—how then shall we represent
in space an existence not in space, and in time an existence not in
time? This difficulty is still more evident when we come to construct
a poiograph, for a poiograph is essentially a space structure. But
because more evident the difficulty is nearer a solution. If we always
think in space, _i.e._ using space concepts, the first condition
requisite for adapting them to the representation of non-spatial
existence, is to be aware of the limitation of our thought, and so be
able to take the proper steps to overcome it. The problem before us,
then, is to represent in space an existence not in space.
The solution is an easy one. It is provided by the conception of
alternativity.
To get our ideas clear let us go right back behind the distinctions of
an inner and an outer world. Both of these, Kant says, are products.
Let us take merely states of consciousness, and not ask the question
whether they are produced or superinduced—to ask such a question is to
have got too far on, to have assumed something of which we have not
traced the origin. Of these states let us simply say that they occur.
Let us now use the word a “posit” for a phase of consciousness reduced
to its last possible stage of evanescence; let a posit be that phase of
consciousness of which all that can be said is that it occurs.
Let _a_, _b_, _c_, be three such posits. We cannot represent them in
space without placing them in a certain order, as _a_, _b_, _c_. But
Kant distinguishes between the forms of sensibility and the concepts
of reason. A dream in which everything happens at haphazard would be
an experience subject to the form of sensibility and only partially
subject to the concepts of reason. It is partially subject to the
concepts of reason because, although there is no order of sequence,
still at any given time there is order. Perception of a thing as in
space is a form of sensibility, the perception of an order is a concept
of reason.
We must, therefore, in order to get at that process which Kant supposes
to be constitutive of an ordered experience imagine the posits as in
space without order.
As we know them they must be in some order, _abc_, _bca_, _cab_, _acb_,
_cba_, _bac_, one or another.
To represent them as having no order conceive all these different
orders as equally existing. Introduce the conception of
alternativity—let us suppose that the order _abc_, and _bac_, for
example, exist equally, so that we cannot say about _a_ that it comes
before or after _b_. This would correspond to a sudden and arbitrary
change of _a_ into _b_ and _b_ into _a_, so that, to use Kant’s words,
it would be possible to call one thing by one name at one time and at
another time by another name.
In an experience of this kind we have a kind of chaos, in which no
order exists; it is a manifold not subject to the concepts of reason.
Now is there any process by which order can be introduced into such a
manifold—is there any function of consciousness in virtue of which an
ordered experience could arise?
In the precise condition in which the posits are, as described above,
it does not seem to be possible. But if we imagine a duality to exist
in the manifold, a function of consciousness can be easily discovered
which will produce order out of no order.
Let us imagine each posit, then, as having, a dual aspect. Let _a_ be
1_a_ in which the dual aspect is represented by the combination of
symbols. And similarly let _b_ be 2_b_, _c_ be 3_c_, in which 2 and _b_
represent the dual aspects of _b_, 3 and _c_ those of _c_.
Since _a_ can arbitrarily change into _b_, or into _c_, and so on, the
particular combinations written above cannot be kept. We have to assume
the equally possible occurrence of form such as 2_a_, 2_b_, and so on;
and in order to get a representation of all those combinations out of
which any set is alternatively possible, we must take every aspect with
every aspect. We must, that is, have every letter with every number.
Let us now apply the method of space representation.
_Note._—At the beginning of the next chapter the same structures as
those which follow are exhibited in more detail and a reference to
them will remove any obscurity which may be found in the immediately
following passages. They are there carried on to a greater
multiplicity of dimensions, and the significance of the process here
briefly explained becomes more apparent.
[Illustration: Fig. 59.]
Take three mutually rectangular axes in space 1, 2, 3 (fig. 59), and
on each mark three points, the common meeting point being the first on
each axis. Then by means of these three points on each axis we define
27 positions, 27 points in a cubical cluster, shown in fig. 60, the
same method of co-ordination being used as has been described before.
Each of these positions can be named by means of the axes and the
points combined.
[Illustration: Fig. 60.]
Thus, for instance, the one marked by an asterisk can be called 1_c_,
2_b_, 3_c_, because it is opposite to _c_ on 1, to _b_ on 2, to _c_ on
3.
Let us now treat of the states of consciousness corresponding to
these positions. Each point represents a composite of posits, and
the manifold of consciousness corresponding to them is of a certain
complexity.
Suppose now the constituents, the points on the axes, to interchange
arbitrarily, any one to become any other, and also the axes 1, 2, and
3, to interchange amongst themselves, any one to become any other, and
to be subject to no system or law, that is to say, that order does not
exist, and that the points which run _abc_ on each axis may run _bac_,
and so on.
Then any one of the states of consciousness represented by the points
in the cluster can become any other. We have a representation of a
random consciousness of a certain degree of complexity.
Now let us examine carefully one particular case of arbitrary
interchange of the points, _a_, _b_, _c_; as one such case, carefully
considered, makes the whole clear.
[Illustration: Fig. 61.]
Consider the points named in the figure 1_c_, 2_a_, 3_c_; 1_c_, 2_c_,
3_a_; 1_a_, 2_c_, 3_c_, and examine the effect on them when a change of
order takes place. Let us suppose, for instance, that _a_ changes into
_b_, and let us call the two sets of points we get, the one before and
the one after, their change conjugates.
Before the change 1_c_ 2_a_ 3_c_ 1_c_ 2_c_ 3_a_ 1_a_ 2_c_ 3_c_}Conjug-
After the change 1_c_ 2_b_ 3_c_ 1_c_ 2_c_ 3_b_ 1_b_ 2_c_ 3_c_} ates.
The points surrounded by rings represent the conjugate points.
It is evident that as consciousness, represented first by the first
set of points and afterwards by the second set of points, would have
nothing in common in its two phases. It would not be capable of giving
an account of itself. There would be no identity.
If, however, we can find any set of points in the cubical cluster,
which, when any arbitrary change takes place in the points on the
axes, or in the axes themselves, repeats itself, is reproduced, then a
consciousness represented by those points would have a permanence. It
would have a principle of identity. Despite the no law, the no order,
of the ultimate constituents, it would have an order, it would form a
system, the condition of a personal identity would be fulfilled.
The question comes to this, then. Can we find a system of points
which is self-conjugate which is such that when any posit on the axes
becomes any other, or when any axis becomes any other, such a set
is transformed into itself, its identity is not submerged, but rises
superior to the chaos of its constituents?
[Illustration: Fig. 62.]
Such a set can be found. Consider the set represented in fig. 62, and
written down in the first of the two lines—
Self- {1_a_ 2_b_ 3_c_ 1_b_ 2_a_ 3_c_ 1_c_ 2_a_ 3_b_
conjugate. {1_c_ 2_b_ 3_a_ 1_b_ 2_c_ 3_a_ 1_a_ 2_c_ 3_b_
Self- {1_c_ 2_b_ 3_a_ 1_b_ 2_c_ 3_a_ 1_a_ 2_c_ 3_b_
conjugate. {1_a_ 2_b_ 3_c_ 1_b_ 2_a_ 3_c_ 1_c_ 2_a_ 3_b_
If now _a_ change into _c_ and _c_ into _a_, we get the set in the
second line, which has the same members as are in the upper line.
Looking at the diagram we see that it would correspond simply to the
turning of the figures as a whole.[2] Any arbitrary change of the
points on the axes, or of the axes themselves, reproduces the same set.
[2] These figures are described more fully, and extended, in the next
chapter.
Thus, a function, by which a random, an unordered, consciousness
could give an ordered and systematic one, can be represented. It
is noteworthy that it is a system of selection. If out of all the
alternative forms that only is attended to which is self-conjugate,
an ordered consciousness is formed. A selection gives a feature of
permanence.
Can we say that the permanent consciousness is this selection?
An analogy between Kant and Darwin comes into light. That which is
swings clear of the fleeting, in virtue of its presenting a feature of
permanence. There is no need to suppose any function of “attending to.”
A consciousness capable of giving an account of itself is one which is
characterised by this combination. All combinations exist—of this kind
is the consciousness which can give an account of itself. And the very
duality which we have presupposed may be regarded as originated by a
process of selection.
Darwin set himself to explain the origin of the fauna and flora of
the world. He denied specific tendencies. He assumed an indefinite
variability—that is, chance—but a chance confined within narrow limits
as regards the magnitude of any consecutive variations. He showed that
organisms possessing features of permanence, if they occurred would be
preserved. So his account of any structure or organised being was that
it possessed features of permanence.
Kant, undertaking not the explanation of any particular phenomena but
of that which we call nature as a whole, had an origin of species
of his own, an account of the flora and fauna of consciousness. He
denied any specific tendency of the elements of consciousness, but
taking our own consciousness, pointed out that in which it resembled
any consciousness which could survive, which could give an account of
itself.
He assumes a chance or random world, and as great and small were not
to him any given notions of which he could make use, he did not limit
the chance, the randomness, in any way. But any consciousness which
is permanent must possess certain features—those attributes namely
which give it permanence. Any consciousness like our own is simply a
consciousness which possesses those attributes. The main thing is that
which he calls the unity of apperception, which we have seen above is
simply the statement that a particular set of phases of consciousness
on the basis of complete randomness will be self-conjugate, and so
permanent.
As with Darwin so with Kant, the reason for existence of any feature
comes to this—show that it tends to the permanence of that which
possesses it.
We can thus regard Kant as the creator of the first of the modern
evolution theories. And, as is so often the case, the first effort was
the most stupendous in its scope. Kant does not investigate the origin
of any special part of the world, such as its organisms, its chemical
elements, its social communities of men. He simply investigates the
origin of the whole—of all that is included in consciousness, the
origin of that “thought thing” whose progressive realisation is the
knowable universe.
This point of view is very different from the ordinary one, in which a
man is supposed to be placed in a world like that which he has come to
think of it, and then to learn what he has found out from this model
which he himself has placed on the scene.
We all know that there are a number of questions in attempting an
answer to which such an assumption is not allowable.
Mill, for instance, explains our notion of “law” by an invariable
sequence in nature. But what we call nature is something given in
thought. So he explains a thought of law and order by a thought of an
invariable sequence. He leaves the problem where he found it.
Kant’s theory is not unique and alone. It is one of a number of
evolution theories. A notion of its import and significance can be
obtained by a comparison of it with other theories.
Thus in Darwin’s theoretical world of natural selection a certain
assumption is made, the assumption of indefinite variability—slight
variability it is true, over any appreciable lapse of time, but
indefinite in the postulated epochs of transformation—and a whole chain
of results is shown to follow.
This element of chance variation is not, however, an ultimate resting
place. It is a preliminary stage. This supposing the all is a
preliminary step towards finding out what is. If every kind of organism
can come into being, those that do survive will present such and such
characteristics. This is the necessary beginning for ascertaining what
kinds of organisms do come into existence. And so Kant’s hypothesis
of a random consciousness is the necessary beginning for the rational
investigation of consciousness as it is. His assumption supplies, as
it were, the space in which we can observe the phenomena. It gives the
general laws constitutive of any experience. If, on the assumption
of absolute randomness in the constituents, such and such would be
characteristic of the experience, then, whatever the constituents,
these characteristics must be universally valid.
We will now proceed to examine more carefully the poiograph,
constructed for the purpose of exhibiting an illustration of Kant’s
unity of apperception.
In order to show the derivation order out of non-order it has been
necessary to assume a principle of duality—we have had the axes and the
posits on the axes—there are two sets of elements, each non-ordered,
and it is in the reciprocal relation of them that the order, the
definite system, originates.
Is there anything in our experience of the nature of a duality?
There certainly are objects in our experience which have order and
those which are incapable of order. The two roots of a quadratic
equation have no order. No one can tell which comes first. If a body
rises vertically and then goes at right angles to its former course,
no one can assign any priority to the direction of the north or to
the east. There is no priority in directions of turning. We associate
turnings with no order progressions in a line with order. But in the
axes and points we have assumed above there is no such distinction.
It is the same, whether we assume an order among the turnings, and no
order among the points on the axes, or, _vice versa_, an order in the
points and no order in the turnings. A being with an infinite number of
axes mutually at right angles, with a definite sequence between them
and no sequence between the points on the axes, would be in a condition
formally indistinguishable from that of a creature who, according to an
assumption more natural to us, had on each axis an infinite number of
ordered points and no order of priority amongst the axes. A being in
such a constituted world would not be able to tell which was turning
and which was length along an axis, in order to distinguish between
them. Thus to take a pertinent illustration, we may be in a world
of an infinite number of dimensions, with three arbitrary points on
each—three points whose order is indifferent, or in a world of three
axes of arbitrary sequence with an infinite number of ordered points on
each. We can’t tell which is which, to distinguish it from the other.
Thus it appears the mode of illustration which we have used is not an
artificial one. There really exists in nature a duality of the kind
which is necessary to explain the origin of order out of no order—the
duality, namely, of dimension and position. Let us use the term group
for that system of points which remains unchanged, whatever arbitrary
change of its constituents takes place. We notice that a group involves
a duality, is inconceivable without a duality.
Thus, according to Kant, the primary element of experience is the
group, and the theory of groups would be the most fundamental branch
of science. Owing to an expression in the critique the authority of
Kant is sometimes adduced against the assumption of more than three
dimensions to space. It seems to me, however, that the whole tendency
of his theory lies in the opposite direction, and points to a perfect
duality between dimension and position in a dimension.
If the order and the law we see is due to the conditions of conscious
experience, we must conceive nature as spontaneous, free, subject to no
predication that we can devise, but, however apprehended, subject to
our logic.
And our logic is simply spatiality in the general sense—that resultant
of a selection of the permanent from the unpermanent, the ordered from
the unordered, by the means of the group and its underlying duality.
We can predicate nothing about nature, only about the way in which
we can apprehend nature. All that we can say is that all that which
experience gives us will be conditioned as spatial, subject to our
logic. Thus, in exploring the facts of geometry from the simplest
logical relations to the properties of space of any number of
dimensions, we are merely observing ourselves, becoming aware of the
conditions under which we must perceive. Do any phenomena present
themselves incapable of explanation under the assumption of the space
we are dealing with, then we must habituate ourselves to the conception
of a higher space, in order that our logic may be equal to the task
before us.
We gain a repetition of the thought that came before, experimentally
suggested. If the laws of the intellectual comprehension of nature are
those derived from considering her as absolute chance, subject to no
law save that derived from a process of selection, then, perhaps, the
order of nature requires different faculties from the intellectual to
apprehend it. The source and origin of ideas may have to be sought
elsewhere than in reasoning.
The total outcome of the critique is to leave the ordinary man just
where he is, justified in his practical attitude towards nature,
liberated from the fetters of his own mental representations.
The truth of a picture lies in its total effect. It is vain to seek
information about the landscape from an examination of the pigments.
And in any method of thought it is the complexity of the whole that
brings us to a knowledge of nature. Dimensions are artificial enough,
but in the multiplicity of them we catch some breath of nature.
We must therefore, and this seems to me the practical conclusion of the
whole matter, proceed to form means of intellectual apprehension of a
greater and greater degree of complexity, both dimensionally and in
extent in any dimension. Such means of representation must always be
artificial, but in the multiplicity of the elements with which we deal,
however incipiently arbitrary, lies our chance of apprehending nature.
And as a concluding chapter to this part of the book, I will extend
the figures, which have been used to represent Kant’s theory, two
steps, so that the reader may have the opportunity of looking at a
four-dimensional figure which can be delineated without any of the
special apparatus, to the consideration of which I shall subsequently
pass on.
CHAPTER X
A FOUR-DIMENSIONAL FIGURE
The method used in the preceding chapter to illustrate the problem
of Kant’s critique, gives a singularly easy and direct mode of
constructing a series of important figures in any number of dimensions.
We have seen that to represent our space a plane being must give up one
of his axes, and similarly to represent the higher shapes we must give
up one amongst our three axes.
But there is another kind of giving up which reduces the construction
of higher shapes to a matter of the utmost simplicity.
Ordinarily we have on a straight line any number of positions. The
wealth of space in position is illimitable, while there are only three
dimensions.
I propose to give up this wealth of positions, and to consider the
figures obtained by taking just as many positions as dimensions.
In this way I consider dimensions and positions as two “kinds,” and
applying the simple rule of selecting every one of one kind with every
other of every other kind, get a series of figures which are noteworthy
because they exactly fill space of any number of dimensions (as the
hexagon fills a plane) by equal repetitions of themselves.
The rule will be made more evident by a simple application.
Let us consider one dimension and one position. I will call the axis
_i_, and the position _o_.
———————————————-_i_
_o_
Here the figure is the position _o_ on the line _i_. Take now two
dimensions and two positions on each.
[Illustration: Fig. 63.]
We have the two positions _o_; 1 on _i_, and the two positions _o_, 1
on _j_, fig. 63. These give rise to a certain complexity. I will let
the two lines _i_ and _j_ meet in the position I call _o_ on each, and
I will consider _i_ as a direction starting equally from every position
on _j_, and _j_ as starting equally from every position on _i_. We thus
obtain the following figure:—A is both _oi_ and _oj_, B is 1_i_ and
_oj_, and so on as shown in fig. 63_b_. The positions on AC are all
_oi_ positions. They are, if we like to consider it in that way, points
at no distance in the _i_ direction from the line AC. We can call the
line AC the _oi_ line. Similarly the points on AB are those no distance
from AB in the _j_ direction, and we can call them _oj_ points and the
line AB the _oj_ line. Again, the line CD can be called the 1_j_ line
because the points on it are at a distance, 1 in the _j_ direction.
[Illustration: Fig. 63_b_.]
We have then four positions or points named as shown, and, considering
directions and positions as “kinds,” we have the combination of two
kinds with two kinds. Now, selecting every one of one kind with every
other of every other kind will mean that we take 1 of the kind _i_ and
with it _o_ of the kind _j_; and then, that we take _o_ of the kind _i_
and with it 1 of the kind _j_.
Thus we get a pair of positions lying in the straight line BC, fig.
64. We can call this pair 10 and 01 if we adopt the plan of mentally,
adding an _i_ to the first and a _j_ to the second of the symbols
written thus—01 is a short expression for O_i_, 1_j_.
[Illustration: Fig. 64.]
Coming now to our space, we have three dimensions, so we take three
positions on each. These positions I will suppose to be at equal
distances along each axis. The three axes and the three positions on
each are shown in the accompanying diagrams, fig. 65, of which the
first represents a cube with the front faces visible, the second the
rear faces of the same cube; the positions I will call 0, 1, 2; the
axes, _i_, _j_, _k_. I take the base ABC as the starting place, from
which to determine distances in the _k_ direction, and hence every
point in the base ABC will be an _ok_ position, and the base ABC can be
called an _ok_ plane.
[Illustration: Fig. 65.]
In the same way, measuring the distances from the face ADC, we see
that every position in the face ADC is an _oi_ position, and the whole
plane of the face may be called an _oi_ plane. Thus we see that with
the introduction of a new dimension the signification of a compound
symbol, such as “_oi_,” alters. In the plane it meant the line AC. In
space it means the whole plane ACD.
Now, it is evident that we have twenty-seven positions, each of them
named. If the reader will follow this nomenclature in respect of the
positions marked in the figures he will have no difficulty in assigning
names to each one of the twenty-seven positions. A is _oi_, _oj_, _ok_.
It is at the distance 0 along _i_, 0 along _j_, 0 along _k_, and _io_
can be written in short 000, where the _ijk_ symbols are omitted.
The point immediately above is 001, for it is no distance in the _i_
direction, and a distance of 1 in the _k_ direction. Again, looking at
B, it is at a distance of 2 from A, or from the plane ADC, in the _i_
direction, 0 in the _j_ direction from the plane ABD, and 0 in the _k_
direction, measured from the plane ABC. Hence it is 200 written for
2_i_, 0_j_, 0_k_.
Now, out of these twenty-seven “things” or compounds of position and
dimension, select those which are given by the rule, every one of one
kind with every other of every other kind.
Take 2 of the _i_ kind. With this we must have a 1 of the _j_ kind, and
then by the rule we can only have a 0 of the _k_ kind, for if we had
any other of the _k_ kind we should repeat one of the kinds we already
had. In 2_i_, 1_j_, 1_k_, for instance, 1 is repeated. The point we
obtain is that marked 210, fig. 66.
[Illustration: Fig. 66.]
Proceeding in this way, we pick out the following cluster of points,
fig. 67. They are joined by lines, dotted where they are hidden by the
body of the cube, and we see that they form a figure—a hexagon which
could be taken out of the cube and placed on a plane. It is a figure
which will fill a plane by equal repetitions of itself. The plane being
representing this construction in his plane would take three squares to
represent the cube. Let us suppose that he takes the _ij_ axes in his
space and _k_ represents the axis running out of his space, fig. 68.
In each of the three squares shown here as drawn separately he could
select the points given by the rule, and he would then have to try to
discover the figure determined by the three lines drawn. The line from
210 to 120 is given in the figure, but the line from 201 to 102 or GK
is not given. He can determine GK by making another set of drawings and
discovering in them what the relation between these two extremities is.
[Illustration: Fig. 67.]
[Illustration: Fig. 68.]
[Illustration: Fig. 69.]
Let him draw the _i_ and _k_ axes in his plane, fig. 69. The _j_ axis
then runs out and he has the accompanying figure. In the first of these
three squares, fig. 69, he can pick out by the rule the two points
201, 102—G, and K. Here they occur in one plane and he can measure the
distance between them. In his first representation they occur at G and
K in separate figures.
Thus the plane being would find that the ends of each of the lines was
distant by the diagonal of a unit square from the corresponding end
of the last and he could then place the three lines in their right
relative position. Joining them he would have the figure of a hexagon.
[Illustration: Fig. 70.]
We may also notice that the plane being could make a representation of
the whole cube simultaneously. The three squares, shown in perspective
in fig. 70, all lie in one plane, and on these the plane being could
pick out any selection of points just as well as on three separate
squares. He would obtain a hexagon by joining the points marked. This
hexagon, as drawn, is of the right shape, but it would not be so if
actual squares were used instead of perspective, because the relation
between the separate squares as they lie in the plane figure is not
their real relation. The figure, however, as thus constructed, would
give him an idea of the correct figure, and he could determine it
accurately by remembering that distances in each square were correct,
but in passing from one square to another their distance in the third
dimension had to be taken into account.
Coming now to the figure made by selecting according to our rule from
the whole mass of points given by four axes and four positions in each,
we must first draw a catalogue figure in which the whole assemblage is
shown.
We can represent this assemblage of points by four solid figures. The
first giving all those positions which are at a distance O from our
space in the fourth dimension, the second showing all those that are at
a distance 1, and so on.
These figures will each be cubes. The first two are drawn showing the
front faces, the second two the rear faces. We will mark the points 0,
1, 2, 3, putting points at those distances along each of these axes,
and suppose all the points thus determined to be contained in solid
models of which our drawings in fig. 71 are representatives. Here we
notice that as on the plane 0_i_ meant the whole line from which the
distances in the _i_ direction was measured, and as in space 0_i_
means the whole plane from which distances in the _i_ direction are
measured, so now 0_h_ means the whole space in which the first cube
stands—measuring away from that space by a distance of one we come to
the second cube represented.
[Illustration: Fig. 71.]
Now selecting according to the rule every one of one kind with every
other of every other kind, we must take, for instance, 3_i_, 2_j_,
1_k_, 0_h_. This point is marked 3210 at the lower star in the figure.
It is 3 in the _i_ direction, 2 in the _j_ direction, 1 in the _k_
direction, 0 in the _h_ direction.
With 3_i_ we must also take 1_j_, 2_k_, 0_h_. This point is shown by
the second star in the cube 0_h_.
[Illustration: Fig. 72.]
In the first cube, since all the points are 0_h_ points, we can only
have varieties in which _i_, _j_, _k_, are accompanied by 3, 2, 1.
The points determined are marked off in the diagram fig. 72, and lines
are drawn joining the adjacent pairs in each figure, the lines being
dotted when they pass within the substance of the cube in the first two
diagrams.
Opposite each point, on one side or the other of each cube, is written
its name. It will be noticed that the figures are symmetrical right and
left; and right and left the first two numbers are simply interchanged.
Now this being our selection of points, what figure do they make when
all are put together in their proper relative positions?
To determine this we must find the distance between corresponding
corners of the separate hexagons.
[Illustration: Fig. 73.]
To do this let us keep the axes _i_, _j_, in our space, and draw _h_
instead of _k_, letting _k_ run out in the fourth dimension, fig. 73.
Here we have four cubes again, in the first of which all the points are
0_k_ points; that is, points at a distance zero in the _k_ direction
from the space of the three dimensions _ijh_. We have all the points
selected before, and some of the distances, which in the last diagram
led from figure to figure are shown here in the same figure, and so
capable of measurement. Take for instance the points 3120 to 3021,
which in the first diagram (fig. 72) lie in the first and second
figures. Their actual relation is shown in fig. 73 in the cube marked
2K, where the points in question are marked with a *. We see that the
distance in question is the diagonal of a unit square. In like manner
we find that the distance between corresponding points of any two
hexagonal figures is the diagonal of a unit square. The total figure
is now easily constructed. An idea of it may be gained by drawing all
the four cubes in the catalogue figure in one (fig. 74). These cubes
are exact repetitions of one another, so one drawing will serve as a
representation of the whole series, if we take care to remember where
we are, whether in a 0_h_, a 1_h_, a 2_h_, or a 3_h_ figure, when we
pick out the points required. Fig. 74 is a representation of all the
catalogue cubes put in one. For the sake of clearness the front faces
and the back faces of this cube are represented separately.
[Illustration: Fig. 74.]
The figure determined by the selected points is shown below.
In putting the sections together some of the outlines in them
disappear. The line TW for instance is not wanted.
We notice that PQTW and TWRS are each the half of a hexagon. Now QV and
VR lie in one straight line. Hence these two hexagons fit together,
forming one hexagon, and the line TW is only wanted when we consider a
section of the whole figure, we thus obtain the solid represented in
the lower part of fig. 74. Equal repetitions of this figure, called a
tetrakaidecagon, will fill up three-dimensional space.
To make the corresponding four-dimensional figure we have to take five
axes mutually at right angles with five points on each. A catalogue of
the positions determined in five-dimensional space can be found thus.
Take a cube with five points on each of its axes, the fifth point is
at a distance of four units of length from the first on any one of
the axes. And since the fourth dimension also stretches to a distance
of four we shall need to represent the successive sets of points at
distances 0, 1, 2, 3, 4, in the fourth dimensions, five cubes. Now
all of these extend to no distance at all in the fifth dimension. To
represent what lies in the fifth dimension we shall have to draw,
starting from each of our cubes, five similar cubes to represent the
four steps on in the fifth dimension. By this assemblage we get a
catalogue of all the points shown in fig. 75, in which _L_ represents
the fifth dimension.
[Illustration: Fig. 75.]
Now, as we saw before, there is nothing to prevent us from putting all
the cubes representing the different stages in the fourth dimension in
one figure, if we take note when we look at it, whether we consider
it as a 0_h_, a 1_h_, a 2_h_, etc., cube. Putting then the 0_h_, 1_h_,
2_h_, 3_h_, 4_h_ cubes of each row in one, we have five cubes with the
sides of each containing five positions, the first of these five cubes
represents the 0_l_ points, and has in it the _i_ points from 0 to 4,
the _j_ points from 0 to 4, the _k_ points from 0 to 4, while we have
to specify with regard to any selection we make from it, whether we
regard it as a 0_h_, a 1_h_, a 2_h_, a 3_h_, or a 4_h_ figure. In fig.
76 each cube is represented by two drawings, one of the front part, the
other of the rear part.
Let then our five cubes be arranged before us and our selection be made
according to the rule. Take the first figure in which all points are
0_l_ points. We cannot have 0 with any other letter. Then, keeping in
the first figure, which is that of the 0_l_ positions, take first of
all that selection which always contains 1_h_. We suppose, therefore,
that the cube is a 1_h_ cube, and in it we take _i_, _j_, _k_ in
combination with 4, 3, 2 according to the rule.
The figure we obtain is a hexagon, as shown, the one in front. The
points on the right hand have the same figures as those on the left,
with the first two numerals interchanged. Next keeping still to the
0_l_ figure let us suppose that the cube before us represents a section
at a distance of 2 in the _h_ direction. Let all the points in it be
considered as 2_h_ points. We then have a 0_l_, 2_h_ region, and have
the sets _ijk_ and 431 left over. We must then pick out in accordance
with our rule all such points as 4_i_, 3_j_, 1_k_.
These are shown in the figure and we find that we can draw them without
confusion, forming the second hexagon from the front. Going on in this
way it will be seen that in each of the five figures a set of hexagons
is picked out, which put together form a three-space figure something
like the tetrakaidecagon.
[Illustration: Fig. 76.]
These separate figures are the successive stages in which the whole
four-dimensional figure in which they cohere can be apprehended.
The first figure and the last are tetrakaidecagons. These are two
of the solid boundaries of the figure. The other solid boundaries
can be traced easily. Some of them are complete from one face in the
figure to the corresponding face in the next, as for instance the
solid which extends from the hexagonal base of the first figure to the
equal hexagonal base of the second figure. This kind of boundary is a
hexagonal prism. The hexagonal prism also occurs in another sectional
series, as for instance, in the square at the bottom of the first
figure, the oblong at the base of the second and the square at the base
of the third figure.
Other solid boundaries can be traced through four of the five sectional
figures. Thus taking the hexagon at the top of the first figure we
find in the next a hexagon also, of which some alternate sides are
elongated. The top of the third figure is also a hexagon with the other
set of alternate rules elongated, and finally we come in the fourth
figure to a regular hexagon.
These four sections are the sections of a tetrakaidecagon as can
be recognised from the sections of this figure which we have had
previously. Hence the boundaries are of two kinds, hexagonal prisms and
tetrakaidecagons.
These four-dimensional figures exactly fill four-dimensional space by
equal repetitions of themselves.
CHAPTER XI
NOMENCLATURE AND ANALOGIES PRELIMINARY TO THE STUDY OF FOUR-DIMENSIONAL
FIGURES
In the following pages a method of designating different regions of
space by a systematic colour scheme has been adopted. The explanations
have been given in such a manner as to involve no reference to models,
the diagrams will be found sufficient. But to facilitate the study a
description of a set of models is given in an appendix which the reader
can either make for himself or obtain. If models are used the diagrams
in Chapters XI. and XII. will form a guide sufficient to indicate their
use. Cubes of the colours designated by the diagrams should be picked
out and used to reinforce the diagrams. The reader, in the following
description, should suppose that a board or wall stretches away from
him, against which the figures are placed.
[Illustration: Fig. 77.]
Take a square, one of those shown in Fig. 77 and give it a neutral
colour, let this colour be called “null,” and be such that it makes no
appreciable difference to any colour with which it mixed. If there is
no such real colour let us imagine such a colour, and assign to it the
properties of the number zero, which makes no difference in any number
to which it is added.
Above this square place a red square. Thus we symbolise the going up by
adding red to null.
Away from this null square place a yellow square, and represent going
away by adding yellow to null.
To complete the figure we need a fourth square. Colour this orange,
which is a mixture of red and yellow, and so appropriately represents a
going in a direction compounded of up and away. We have thus a colour
scheme which will serve to name the set of squares drawn. We have two
axes of colours—red and yellow—and they may occupy as in the figure
the direction up and away, or they may be turned about; in any case
they enable us to name the four squares drawn in their relation to one
another.
Now take, in Fig. 78, nine squares, and suppose that at the end of the
going in any direction the colour started with repeats itself.
[Illustration: Fig. 78.]
We obtain a square named as shown.
Let us now, in fig. 79, suppose the number of squares to be increased,
keeping still to the principle of colouring already used.
Here the nulls remain four in number. There are three reds between the
first null and the null above it, three yellows between the first null
and the null beyond it, while the oranges increase in a double way.
[Illustration: Fig. 79.]
Suppose this process of enlarging the number of the squares to be
indefinitely pursued and the total figure obtained to be reduced in
size, we should obtain a square of which the interior was all orange,
while the lines round it were red and yellow, and merely the points
null colour, as in fig. 80. Thus all the points, lines, and the area
would have a colour.
[Illustration: Fig. 80.]
We can consider this scheme to originate thus:—Let a null point move
in a yellow direction and trace out a yellow line and end in a null
point. Then let the whole line thus traced move in a red direction. The
null points at the ends of the line will produce red lines, and end in
null points. The yellow line will trace out a yellow and red, or orange
square.
Now, turning back to fig. 78, we see that these two ways of naming, the
one we started with and the one we arrived at, can be combined.
By its position in the group of four squares, in fig. 77, the null
square has a relation to the yellow and to the red directions. We can
speak therefore of the red line of the null square without confusion,
meaning thereby the line AB, fig. 81, which runs up from the initial
null point A in the figure as drawn. The yellow line of the null square
is its lower horizontal line AC as it is situated in the figure.
[Illustration: Fig. 81.]
If we wish to denote the upper yellow line BD, fig. 81, we can speak
of it as the yellow γ line, meaning the yellow line which is separated
from the primary yellow line by the red movement.
In a similar way each of the other squares has null points, red and
yellow lines. Although the yellow square is all yellow, its line CD,
for instance, can be referred to as its red line.
This nomenclature can be extended.
If the eight cubes drawn, in fig. 82, are put close together, as on
the right hand of the diagram, they form a cube, and in them, as thus
arranged, a going up is represented by adding red to the zero, or
null colour, a going away by adding yellow, a going to the right by
adding white. White is used as a colour, as a pigment, which produces
a colour change in the pigments with which it is mixed. From whatever
cube of the lower set we start, a motion up brings us to a cube showing
a change to red, thus light yellow becomes light yellow red, or light
orange, which is called ochre. And going to the right from the null on
the left we have a change involving the introduction of white, while
the yellow change runs from front to back. There are three colour
axes—the red, the white, the yellow—and these run in the position the
cubes occupy in the drawing—up, to the right, away—but they could be
turned about to occupy any positions in space.
[Illustration: Fig. 82.]
[Illustration: Fig. 83. The three layers.]
We can conveniently represent a block of cubes by three sets of
squares, representing each the base of a cube.
Thus the block, fig. 83, can be represented by the layers on the
right. Here, as in the case of the plane, the initial colours repeat
themselves at the end of the series.
Proceeding now to increase the number of the cubes we obtain fig. 84,
in which the initial letters of the colours are given instead of their
full names.
Here we see that there are four null cubes as before, but the series
which spring from the initial corner will tend to become lines of
cubes, as also the sets of cubes parallel to them, starting from other
corners. Thus, from the initial null springs a line of red cubes, a
line of white cubes, and a line of yellow cubes.
If the number of the cubes is largely increased, and the size of the
whole cube is diminished, we get a cube with null points, and the edges
coloured with these three colours.
[Illustration: Fig. 84.]
The light yellow cubes increase in two ways, forming ultimately a sheet
of cubes, and the same is true of the orange and pink sets. Hence,
ultimately the cube thus formed would have red, white, and yellow
lines surrounding pink, orange, and light yellow faces. The ochre cubes
increase in three ways, and hence ultimately the whole interior of the
cube would be coloured ochre.
We have thus a nomenclature for the points, lines, faces, and solid
content of a cube, and it can be named as exhibited in fig. 85.
[Illustration: Fig. 85.]
We can consider the cube to be produced in the following way. A null
point moves in a direction to which we attach the colour indication
yellow; it generates a yellow line and ends in a null point. The yellow
line thus generated moves in a direction to which we give the colour
indication red. This lies up in the figure. The yellow line traces out
a yellow, red, or orange square, and each of its null points trace out
a red line, and ends in a null point.
This orange square moves in a direction to which we attribute the
colour indication white, in this case the direction is the right. The
square traces out a cube coloured orange, red, or ochre, the red lines
trace out red to white or pink squares, and the yellow lines trace out
light yellow squares, each line ending in a line of its own colour.
While the points each trace out a null + white, or white line to end in
a null point.
Now returning to the first block of eight cubes we can name each point,
line, and square in them by reference to the colour scheme, which they
determine by their relation to each other.
Thus, in fig. 86, the null cube touches the red cube by a light yellow
square; it touches the yellow cube by a pink square, and touches the
white cube by an orange square.
There are three axes to which the colours red, yellow, and white are
assigned, the faces of each cube are designated by taking these colours
in pairs. Taking all the colours together we get a colour name for the
solidity of a cube.
[Illustration: Fig. 86.]
Let us now ask ourselves how the cube could be presented to the plane
being. Without going into the question of how he could have a real
experience of it, let us see how, if we could turn it about and show it
to him, he, under his limitations, could get information about it. If
the cube were placed with its red and yellow axes against a plane, that
is resting against it by its orange face, the plane being would observe
a square surrounded by red and yellow lines, and having null points.
See the dotted square, fig. 87.
[Illustration: Fig. 87.]
We could turn the cube about the red line so that a different face
comes into juxtaposition with the plane.
Suppose the cube turned about the red line. As it is turning from its
first position all of it except the red line leaves the plane—goes
absolutely out of the range of the plane being’s apprehension. But when
the yellow line points straight out from the plane then the pink face
comes into contact with it. Thus the same red line remaining as he saw
it at first, now towards him comes a face surrounded by white and red
lines.
If we call the direction to the right the unknown direction, then
the line he saw before, the yellow line, goes out into this unknown
direction, and the line which before went into the unknown direction,
comes in. It comes in in the opposite direction to that in which the
yellow line ran before; the interior of the face now against the plane
is pink. It is a property of two lines at right angles that, if one
turns out of a given direction and stands at right angles to it, then
the other of the two lines comes in, but runs the opposite way in that
given direction, as in fig. 88.
[Illustration: Fig. 88.]
Now these two presentations of the cube would seem, to the plane
creature like perfectly different material bodies, with only that line
in common in which they both meet.
Again our cube can be turned about the yellow line. In this case the
yellow square would disappear as before, but a new square would come
into the plane after the cube had rotated by an angle of 90° about this
line. The bottom square of the cube would come in thus in figure 89.
The cube supposed in contact with the plane is rotated about the lower
yellow line and then the bottom face is in contact with the plane.
Here, as before, the red line going out into the unknown dimension,
the white line which before ran in the unknown dimension would come
in downwards in the opposite sense to that in which the red line ran
before.
[Illustration: Fig. 89.]
Now if we use _i_, _j_, _k_, for the three space directions, _i_ left
to right, _j_ from near away, _k_ from below up; then, using the colour
names for the axes, we have that first of all white runs _i_, yellow
runs _j_, red runs _k_; then after the first turning round the _k_
axis, white runs negative _j_, yellow runs _i_, red runs _k_; thus we
have the table:—
_i_ _j_ _k_
1st position white yellow red
2nd position yellow white— red
3rd position red yellow white—
Here white with a negative sign after it in the column under _j_ means
that white runs in the negative sense of the _j_ direction.
We may express the fact in the following way:— In the plane there is
room for two axes while the body has three. Therefore in the plane we
can represent any two. If we want to keep the axis that goes in the
unknown dimension always running in the positive sense, then the axis
which originally ran in the unknown dimension (the white axis) must
come in in the negative sense of that axis which goes out of the plane
into the unknown dimension.
It is obvious that the unknown direction, the direction in which the
white line runs at first, is quite distinct from any direction which
the plane creature knows. The white line may come in towards him, or
running down. If he is looking at a square, which is the face of a cube
(looking at it by a line), then any one of the bounding lines remaining
unmoved, another face of the cube may come in, any one of the faces,
namely, which have the white line in them. And the white line comes
sometimes in one of the space directions he knows, sometimes in another.
Now this turning which leaves a line unchanged is something quite
unlike any turning he knows in the plane. In the plane a figure turns
round a point. The square can turn round the null point in his plane,
and the red and yellow lines change places, only of course, as with
every rotation of lines at right angles, if red goes where yellow went,
yellow comes in negative of red’s old direction.
This turning, as the plane creature conceives it, we should call
turning about an axis perpendicular to the plane. What he calls turning
about the null point we call turning about the white line as it stands
out from his plane. There is no such thing as turning about a point,
there is always an axis, and really much more turns than the plane
being is aware of.
Taking now a different point of view, let us suppose the cubes to be
presented to the plane being by being passed transverse to his plane.
Let us suppose the sheet of matter over which the plane being and all
objects in his world slide, to be of such a nature that objects can
pass through it without breaking it. Let us suppose it to be of the
same nature as the film of a soap bubble, so that it closes around
objects pushed through it, and, however the object alters its shape as
it passes through it, let us suppose this film to run up to the contour
of the object in every part, maintaining its plane surface unbroken.
Then we can push a cube or any object through the film and the plane
being who slips about in the film will know the contour of the cube
just and exactly where the film meets it.
[Illustration: Fig. 90.]
Fig. 90 represents a cube passing through a plane film. The plane being
now comes into contact with a very thin slice of the cube somewhere
between the left and right hand faces. This very thin slice he thinks
of as having no thickness, and consequently his idea of it is what we
call a section. It is bounded by him by pink lines front and back,
coming from the part of the pink face he is in contact with, and above
and below, by light yellow lines. Its corners are not null-coloured
points, but white points, and its interior is ochre, the colour of the
interior of the cube.
If now we suppose the cube to be an inch in each dimension, and to pass
across, from right to left, through the plane, then we should explain
the appearances presented to the plane being by saying: First of all
you have the face of a cube, this lasts only a moment; then you have a
figure of the same shape but differently coloured. This, which appears
not to move to you in any direction which you know of, is really moving
transverse to your plane world. Its appearance is unaltered, but each
moment it is something different—a section further on, in the white,
the unknown dimension. Finally, at the end of the minute, a face comes
in exactly like the face you first saw. This finishes up the cube—it is
the further face in the unknown dimension.
The white line, which extends in length just like the red or the
yellow, you do not see as extensive; you apprehend it simply as an
enduring white point. The null point, under the condition of movement
of the cube, vanishes in a moment, the lasting white point is really
your apprehension of a white line, running in the unknown dimension.
In the same way the red line of the face by which the cube is first in
contact with the plane lasts only a moment, it is succeeded by the pink
line, and this pink line lasts for the inside of a minute. This lasting
pink line in your apprehension of a surface, which extends in two
dimensions just like the orange surface extends, as you know it, when
the cube is at rest.
But the plane creature might answer, “This orange object is substance,
solid substance, bounded completely and on every side.”
Here, of course, the difficulty comes in. His solid is our surface—his
notion of a solid is our notion of an abstract surface with no
thickness at all.
We should have to explain to him that, from every point of what he
called a solid, a new dimension runs away. From every point a line
can be drawn in a direction unknown to him, and there is a solidity
of a kind greater than that which he knows. This solidity can only
be realised by him by his supposing an unknown direction, by motion
in which what he conceives to be solid matter instantly disappears.
The higher solid, however, which extends in this dimension as well
as in those which he knows, lasts when a motion of that kind takes
place, different sections of it come consecutively in the plane
of his apprehension, and take the place of the solid which he at
first conceives to be all. Thus, the higher solid—our solid in
contradistinction to his area solid, his two-dimensional solid, must
be conceived by him as something which has duration in it, under
circumstances in which his matter disappears out of his world.
We may put the matter thus, using the conception of motion.
A null point moving in a direction away generates a yellow line, and
the yellow line ends in a null point. We suppose, that is, a point
to move and mark out the products of this motion in such a manner.
Now suppose this whole line as thus produced to move in an upward
direction; it traces out the two-dimensional solid, and the plane being
gets an orange square. The null point moves in a red line and ends in
a null point, the yellow line moves and generates an orange square and
ends in a yellow line, the farther null point generates a red line and
ends in a null point. Thus, by movement in two successive directions
known to him, he can imagine his two-dimensional solid produced with
all its boundaries.
Now we tell him: “This whole two-dimensional solid can move in a third
or unknown dimension to you. The null point moving in this dimension
out of your world generates a white line and ends in a null point. The
yellow line moving generates a light yellow two-dimensional solid and
ends in a yellow line, and this two-dimensional solid, lying end on to
your plane world, is bounded on the far side by the other yellow line.
In the same way each of the lines surrounding your square traces out an
area, just like the orange area you know. But there is something new
produced, something which you had no idea of before; it is that which
is produced by the movement of the orange square. That, than which you
can imagine nothing more solid, itself moves in a direction open to it
and produces a three-dimensional solid. Using the addition of white
to symbolise the products of this motion this new kind of solid will
be light orange or ochre, and it will be bounded on the far side by
the final position of the orange square which traced it out, and this
final position we suppose to be coloured like the square in its first
position, orange with yellow and red boundaries and null corners.”
This product of movement, which it is so easy for us to describe, would
be difficult for him to conceive. But this difficulty is connected
rather with its totality than with any particular part of it.
Any line, or plane of this, to him higher, solid we could show to him,
and put in his sensible world.
We have already seen how the pink square could be put in his world by
a turning of the cube about the red line. And any section which we can
conceive made of the cube could be exhibited to him. You have simply to
turn the cube and push it through, so that the plane of his existence
is the plane which cuts out the given section of the cube, then the
section would appear to him as a solid. In his world he would see the
contour, get to any part of it by digging down into it.
THE PROCESS BY WHICH A PLANE BEING WOULD GAIN A NOTION OF A SOLID.
If we suppose the plane being to have a general idea of the existence
of a higher solid—our solid—we must next trace out in detail the
method, the discipline, by which he would acquire a working familiarity
with our space existence. The process begins with an adequate
realisation of a simple solid figure. For this purpose we will suppose
eight cubes forming a larger cube, and first we will suppose each cube
to be coloured throughout uniformly. Let the cubes in fig. 91 be the
eight making a larger cube.
[Illustration: Fig. 91.]
Now, although each cube is supposed to be coloured entirely through
with the colour, the name of which is written on it, still we can
speak of the faces, edges, and corners of each cube as if the colour
scheme we have investigated held for it. Thus, on the null cube we can
speak of a null point, a red line, a white line, a pink face, and so
on. These colour designations are shown on No. 1 of the views of the
tesseract in the plate. Here these colour names are used simply in
their geometrical significance. They denote what the particular line,
etc., referred to would have as its colour, if in reference to the
particular cube the colour scheme described previously were carried out.
If such a block of cubes were put against the plane and then passed
through it from right to left, at the rate of an inch a minute, each
cube being an inch each way, the plane being would have the following
appearances:—
First of all, four squares null, yellow, red, orange, lasting each a
minute; and secondly, taking the exact places of these four squares,
four others, coloured white, light yellow, pink, ochre. Thus, to make
a catalogue of the solid body, he would have to put side by side in
his world two sets of four squares each, as in fig. 92. The first are
supposed to last a minute, and then the others to come in in place of
them, and also last a minute.
[Illustration: Fig. 92.]
In speaking of them he would have to denote what part of the respective
cube each square represents. Thus, at the beginning he would have null
cube orange face, and after the motion had begun he would have null
cube ochre section. As he could get the same coloured section whichever
way the cube passed through, it would be best for him to call this
section white section, meaning that it is transverse to the white axis.
These colour-names, of course, are merely used as names, and do not
imply in this case that the object is really coloured. Finally, after
a minute, as the first cube was passing beyond his plane he would have
null cube orange face again.
The same names will hold for each of the other cubes, describing what
face or section of them the plane being has before him; and the second
wall of cubes will come on, continue, and go out in the same manner. In
the area he thus has he can represent any movement which we carry out
in the cubes, as long as it does not involve a motion in the direction
of the white axis. The relation of parts that succeed one another in
the direction of the white axis is realised by him as a consecution of
states.
Now, his means of developing his space apprehension lies in this, that
that which is represented as a time sequence in one position of the
cubes, can become a real co-existence, _if something that has a real
co-existence becomes a time sequence_.
We must suppose the cubes turned round each of the axes, the red line,
and the yellow line, then something, which was given as time before,
will now be given as the plane creature’s space; something, which was
given as space before, will now be given as a time series as the cube
is passed through the plane.
The three positions in which the cubes must be studied are the one
given above and the two following ones. In each case the original null
point which was nearest to us at first is marked by an asterisk. In
figs. 93 and 94 the point marked with a star is the same in the cubes
and in the plane view.
[Illustration: Fig. 93. The cube swung round the red line, so that the
white line points towards us.]
In fig. 93 the cube is swung round the red line so as to point towards
us, and consequently the pink face comes next to the plane. As it
passes through there are two varieties of appearance designated by
the figures 1 and 2 in the plane. These appearances are named in the
figure, and are determined by the order in which the cubes come in the
motion of the whole block through the plane.
With regard to these squares severally, however, different names must
be used, determined by their relations in the block.
Thus, in fig. 93, when the cube first rests against the plane the null
cube is in contact by its pink face; as the block passes through we get
an ochre section of the null cube, but this is better called a yellow
section, as it is made by a plane perpendicular to the yellow line.
When the null cube has passed through the plane, as it is leaving it,
we get again a pink face.
[Illustration: Fig. 94. The cube swung round yellow line, with red line
running from left to right, and white line running down.]
The same series of changes take place with the cube appearances which
follow on those of the null cube. In this motion the yellow cube
follows on the null cube, and the square marked yellow in 2 in the
plane will be first “yellow pink face,” then “yellow yellow section,”
then “yellow pink face.”
In fig. 94, in which the cube is turned about the yellow line, we have
a certain difficulty, for the plane being will find that the position
his squares are to be placed in will lie below that which they first
occupied. They will come where the support was on which he stood his
first set of squares. He will get over this difficulty by moving his
support.
Then, since the cubes come upon his plane by the light yellow face, he
will have, taking the null cube as before for an example, null, light
yellow face; null, red section, because the section is perpendicular
to the red line; and finally, as the null cube leaves the plane, null,
light yellow face. Then, in this case red following on null, he will
have the same series of views of the red as he had of the null cube.
[Illustration: Fig. 95.]
There is another set of considerations which we will briefly allude to.
Suppose there is a hollow cube, and a string is stretched across it
from null to null, _r_, _y_, _wh_, as we may call the far diagonal
point, how will this string appear to the plane being as the cube moves
transverse to his plane?
Let us represent the cube as a number of sections, say 5, corresponding
to 4 equal divisions made along the white line perpendicular to it.
We number these sections 0, 1, 2, 3, 4, corresponding to the distances
along the white line at which they are taken, and imagine each section
to come in successively, taking the place of the preceding one.
These sections appear to the plane being, counting from the first, to
exactly coincide each with the preceding one. But the section of the
string occupies a different place in each to that which it does in the
preceding section. The section of the string appears in the position
marked by the dots. Hence the slant of the string appears as a motion
in the frame work marked out by the cube sides. If we suppose the
motion of the cube not to be recognised, then the string appears to the
plane being as a moving point. Hence extension on the unknown dimension
appears as duration. Extension sloping in the unknown direction appears
as continuous movement.
CHAPTER XII
THE SIMPLEST FOUR-DIMENSIONAL SOLID
A plane being, in learning to apprehend solid existence, must first
of all realise that there is a sense of direction altogether wanting
to him. That which we call right and left does not exist in his
perception. He must assume a movement in a direction, and a distinction
of positive and negative in that direction, which has no reality
corresponding to it in the movements he can make. This direction, this
new dimension, he can only make sensible to himself by bringing in
time, and supposing that changes, which take place in time, are due
to objects of a definite configuration in three dimensions passing
transverse to his plane, and the different sections of it being
apprehended as changes of one and the same plane figure.
He must also acquire a distinct notion about his plane world, he must
no longer believe that it is the all of space, but that space extends
on both sides of it. In order, then, to prevent his moving off in this
unknown direction, he must assume a sheet, an extended solid sheet, in
two dimensions, against which, in contact with which, all his movements
take place.
When we come to think of a four-dimensional solid, what are the
corresponding assumptions which we must make?
We must suppose a sense which we have not, a sense of direction
wanting in us, something which a being in a four-dimensional world
has, and which we have not. It is a sense corresponding to a new space
direction, a direction which extends positively and negatively from
every point of our space, and which goes right away from any space
direction we know of. The perpendicular to a plane is perpendicular,
not only to two lines in it, but to every line, and so we must conceive
this fourth dimension as running perpendicularly to each and every line
we can draw in our space.
And as the plane being had to suppose something which prevented his
moving off in the third, the unknown dimension to him, so we have to
suppose something which prevents us moving off in the direction unknown
to us. This something, since we must be in contact with it in every one
of our movements, must not be a plane surface, but a solid; it must be
a solid, which in every one of our movements we are against, not in.
It must be supposed as stretching out in every space dimension that we
know; but we are not in it, we are against it, we are next to it, in
the fourth dimension.
That is, as the plane being conceives himself as having a very small
thickness in the third dimension, of which he is not aware in his
sense experience, so we must suppose ourselves as having a very small
thickness in the fourth dimension, and, being thus four-dimensional
beings, to be prevented from realising that we are such beings by a
constraint which keeps us always in contact with a vast solid sheet,
which stretches on in every direction. We are against that sheet, so
that, if we had the power of four-dimensional movement, we should
either go away from it or through it; all our space movements as we
know them being such that, performing them, we keep in contact with
this solid sheet.
Now consider the exposition a plane being would make for himself as to
the question of the enclosure of a square, and of a cube.
He would say the square A, in Fig. 96, is completely enclosed by the
four squares, A far, A near, A above, A below, or as they are written
A_n_, A_f_, A_a_, A_b_.
[Illustration: Fig. 96.]
If now he conceives the square A to move in the, to him, unknown
dimension it will trace out a cube, and the bounding squares will
form cubes. Will these completely surround the cube generated by A?
No; there will be two faces of the cube made by A left uncovered;
the first, that face which coincides with the square A in its first
position; the next, that which coincides with the square A in its
final position. Against these two faces cubes must be placed in order
to completely enclose the cube A. These may be called the cubes left
and right or A_l_ and A_r_. Thus each of the enclosing squares of the
square A becomes a cube and two more cubes are wanted to enclose the
cube formed by the movement of A in the third dimension.
[Illustration: Fig. 97.]
The plane being could not see the square A with the squares A_n_, A_f_,
etc., placed about it, because they completely hide it from view; and
so we, in the analogous case in our three-dimensional world, cannot
see a cube A surrounded by six other cubes. These cubes we will call A
near A_n_, A far A_f_, A above A_a_, A below A_b_, A left A_l_, A right
A_r_, shown in fig. 97. If now the cube A moves in the fourth dimension
right out of space, it traces out a higher cube—a tesseract, as it may
be called. Each of the six surrounding cubes carried on in the same
motion will make a tesseract also, and these will be grouped around the
tesseract formed by A. But will they enclose it completely?
All the cubes A_n_, A_f_, etc., lie in our space. But there is nothing
between the cube A and that solid sheet in contact with which every
particle of matter is. When the cube A moves in the fourth direction
it starts from its position, say A_k_, and ends in a final position
A_n_ (using the words “ana” and “kata” for up and down in the fourth
dimension). Now the movement in this fourth dimension is not bounded by
any of the cubes A_n_, A_f_, nor by what they form when thus moved. The
tesseract which A becomes is bounded in the positive and negative ways
in this new direction by the first position of A and the last position
of A. Or, if we ask how many tesseracts lie around the tesseract which
A forms, there are eight, of which one meets it by the cube A, and
another meets it by a cube like A at the end of its motion.
We come here to a very curious thing. The whole solid cube A is to be
looked on merely as a boundary of the tesseract.
Yet this is exactly analogous to what the plane being would come to in
his study of the solid world. The square A (fig. 96), which the plane
being looks on as a solid existence in his plane world, is merely the
boundary of the cube which he supposes generated by its motion.
The fact is that we have to recognise that, if there is another
dimension of space, our present idea of a solid body, as one which
has three dimensions only, does not correspond to anything real,
but is the abstract idea of a three-dimensional boundary limiting a
four-dimensional solid, which a four-dimensional being would form. The
plane being’s thought of a square is not the thought of what we should
call a possibly existing real square, but the thought of an abstract
boundary, the face of a cube.
Let us now take our eight coloured cubes, which form a cube in
space, and ask what additions we must make to them to represent
the simplest collection of four-dimensional bodies—namely, a group
of them of the same extent in every direction. In plane space we
have four squares. In solid space we have eight cubes. So we should
expect in four-dimensional space to have sixteen four-dimensional
bodies-bodies which in four-dimensional space correspond to cubes in
three-dimensional space, and these bodies we call tesseracts.
Given then the null, white, red, yellow cubes, and those which make up
the block, we notice that we represent perfectly well the extension
in three directions (fig. 98). From the null point of the null cube,
travelling one inch, we come to the white cube; travelling one inch
away we come to the yellow cube; travelling one inch up we come to the
red cube. Now, if there is a fourth dimension, then travelling from the
same null point for one inch in that direction, we must come to the
body lying beyond the null region.
[Illustration: Fig. 98.]
I say null region, not cube; for with the introduction of the fourth
dimension each of our cubes must become something different from cubes.
If they are to have existence in the fourth dimension, they must be
“filled up from” in this fourth dimension.
Now we will assume that as we get a transference from null to white
going in one way, from null to yellow going in another, so going
from null in the fourth direction we have a transference from null
to blue, using thus the colours white, yellow, red, blue, to denote
transferences in each of the four directions—right, away, up, unknown
or fourth dimension.
[Illustration: Fig. 99.
A plane being’s representation of a block of eight cubes by two sets of
four squares.]
Hence, as the plane being must represent the solid regions, he would
come to by going right, as four squares lying in some position in his
plane, arbitrarily chosen, side by side with his original four squares,
so we must represent those eight four-dimensional regions, which we
should come to by going in the fourth dimension from each of our eight
cubes, by eight cubes placed in some arbitrary position relative to our
first eight cubes.
[Illustration: Fig. 100.]
Our representation of a block of sixteen tesseracts by two blocks of
eight cubes.[3]
[3] The eight cubes used here in 2 can be found in the second of the
model blocks. They can be taken out and used.
Hence, of the two sets of eight cubes, each one will serve us as a
representation of one of the sixteen tesseracts which form one single
block in four-dimensional space. Each cube, as we have it, is a tray,
as it were, against which the real four-dimensional figure rests—just
as each of the squares which the plane being has is a tray, so to
speak, against which the cube it represents could rest.
If we suppose the cubes to be one inch each way, then the original
eight cubes will give eight tesseracts of the same colours, or the
cubes, extending each one inch in the fourth dimension.
But after these there come, going on in the fourth dimension, eight
other bodies, eight other tesseracts. These must be there, if we
suppose the four-dimensional body we make up to have two divisions, one
inch each in each of four directions.
The colour we choose to designate the transference to this second
region in the fourth dimension is blue. Thus, starting from the null
cube and going in the fourth dimension, we first go through one inch of
the null tesseract, then we come to a blue cube, which is the beginning
of a blue tesseract. This blue tesseract stretches one inch farther on
in the fourth dimension.
Thus, beyond each of the eight tesseracts, which are of the same colour
as the cubes which are their bases, lie eight tesseracts whose colours
are derived from the colours of the first eight by adding blue. Thus—
Null gives blue
Yellow ” green
Red ” purple
Orange ” brown
White ” light blue
Pink ” light purple
Light yellow ” light green
Ochre ” light brown
The addition of blue to yellow gives green—this is a natural
supposition to make. It is also natural to suppose that blue added to
red makes purple. Orange and blue can be made to give a brown, by using
certain shades and proportions. And ochre and blue can be made to give
a light brown.
But the scheme of colours is merely used for getting a definite and
realisable set of names and distinctions visible to the eye. Their
naturalness is apparent to any one in the habit of using colours, and
may be assumed to be justifiable, as the sole purpose is to devise a
set of names which are easy to remember, and which will give us a set
of colours by which diagrams may be made easy of comprehension. No
scientific classification of colours has been attempted.
Starting, then, with these sixteen colour names, we have a catalogue of
the sixteen tesseracts, which form a four-dimensional block analogous
to the cubic block. But the cube which we can put in space and look at
is not one of the constituent tesseracts; it is merely the beginning,
the solid face, the side, the aspect, of a tesseract.
We will now proceed to derive a name for each region, point, edge,
plane face, solid and a face of the tesseract.
The system will be clear, if we look at a representation in the plane
of a tesseract with three, and one with four divisions in its side.
The tesseract made up of three tesseracts each way corresponds to the
cube made up of three cubes each way, and will give us a complete
nomenclature.
In this diagram, fig. 101, 1 represents a cube of 27 cubes, each of
which is the beginning of a tesseract. These cubes are represented
simply by their lowest squares, the solid content must be understood. 2
represents the 27 cubes which are the beginnings of the 27 tesseracts
one inch on in the fourth dimension. These tesseracts are represented
as a block of cubes put side by side with the first block, but in
their proper positions they could not be in space with the first set. 3
represents 27 cubes (forming a larger cube) which are the beginnings of
the tesseracts, which begin two inches in the fourth direction from our
space and continue another inch.
[Illustration: Fig. 101.]
[Illustration: Fig. 102[4]]
[4] The coloured plate, figs. 1, 2, 3, shows these relations more
conspicuously.
In fig. 102, we have the representation of a block of 4 × 4 × 4 × 4
or 256 tesseracts. They are given in four consecutive sections, each
supposed to be taken one inch apart in the fourth dimension, and so
giving four blocks of cubes, 64 in each block. Here we see, comparing
it with the figure of 81 tesseracts, that the number of the different
regions show a different tendency of increase. By taking five blocks of
five divisions each way this would become even more clear.
We see, fig. 102, that starting from the point at any corner, the white
coloured regions only extend out in a line. The same is true for the
yellow, red, and blue. With regard to the latter it should be noticed
that the line of blues does not consist in regions next to each other
in the drawing, but in portions which come in in different cubes.
The portions which lie next to one another in the fourth dimension
must always be represented so, when we have a three-dimensional
representation. Again, those regions such as the pink one, go on
increasing in two dimensions. About the pink region this is seen
without going out of the cube itself, the pink regions increase in
length and height, but in no other dimension. In examining these
regions it is sufficient to take one as a sample.
The purple increases in the same manner, for it comes in in a
succession from below to above in block 2, and in a succession from
block to block in 2 and 3. Now, a succession from below to above
represents a continuous extension upwards, and a succession from block
to block represents a continuous extension in the fourth dimension.
Thus the purple regions increase in two dimensions, the upward and
the fourth, so when we take a very great many divisions, and let each
become very small, the purple region forms a two-dimensional extension.
In the same way, looking at the regions marked l. b. or light blue,
which starts nearest a corner, we see that the tesseracts occupying
it increase in length from left to right, forming a line, and that
there are as many lines of light blue tesseracts as there are sections
between the first and last section. Hence the light blue tesseracts
increase in number in two ways—in the right and left, and in the fourth
dimension. They ultimately form what we may call a plane surface.
Now all those regions which contain a mixture of two simple colours,
white, yellow, red, blue, increase in two ways. On the other hand,
those which contain a mixture of three colours increase in three ways.
Take, for instance, the ochre region; this has three colours, white,
yellow, red; and in the cube itself it increases in three ways.
Now regard the orange region; if we add blue to this we get a brown.
The region of the brown tesseracts extends in two ways on the left of
the second block, No. 2 in the figure. It extends also from left to
right in succession from one section to another, from section 2 to
section 3 in our figure.
Hence the brown tesseracts increase in number in three dimensions
upwards, to and fro, fourth dimension. Hence they form a cubic, a
three-dimensional region; this region extends up and down, near
and far, and in the fourth direction, but is thin in the direction
from left to right. It is a cube which, when the complete tesseract
is represented in our space, appears as a series of faces on the
successive cubic sections of the tesseract. Compare fig. 103 in which
the middle block, 2, stands as representing a great number of sections
intermediate between 1 and 3.
In a similar way from the pink region by addition of blue we have
the light purple region, which can be seen to increase in three ways
as the number of divisions becomes greater. The three ways in which
this region of tesseracts extends is up and down, right and left,
fourth dimension. Finally, therefore, it forms a cubic mass of very
small tesseracts, and when the tesseract is given in space sections
it appears on the faces containing the upward and the right and left
dimensions.
We get then altogether, as three-dimensional regions, ochre, brown,
light purple, light green.
Finally, there is the region which corresponds to a mixture of all the
colours; there is only one region such as this. It is the one that
springs from ochre by the addition of blue—this colour we call light
brown.
Looking at the light brown region we see that it increases in four
ways. Hence, the tesseracts of which it is composed increase in
number in each of four dimensions, and the shape they form does not
remain thin in any of the four dimensions. Consequently this region
becomes the solid content of the block of tesseracts, itself; it
is the real four-dimensional solid. All the other regions are then
boundaries of this light brown region. If we suppose the process
of increasing the number of tesseracts and diminishing their size
carried on indefinitely, then the light brown coloured tesseracts
become the whole interior mass, the three-coloured tesseracts become
three-dimensional boundaries, thin in one dimension, and form the
ochre, the brown, the light purple, the light green. The two-coloured
tesseracts become two-dimensional boundaries, thin in two dimensions,
_e.g._, the pink, the green, the purple, the orange, the light blue,
the light yellow. The one-coloured tesseracts become bounding lines,
thin in three dimensions, and the null points become bounding corners,
thin in four dimensions. From these thin real boundaries we can pass in
thought to the abstractions—points, lines, faces, solids—bounding the
four-dimensional solid, which in this case is light brown coloured, and
under this supposition the light brown coloured region is the only real
one, is the only one which is not an abstraction.
It should be observed that, in taking a square as the representation
of a cube on a plane, we only represent one face, or the section
between two faces. The squares, as drawn by a plane being, are not the
cubes themselves, but represent the faces or the sections of a cube.
Thus in the plane being’s diagram a cube of twenty-seven cubes “null”
represents a cube, but is really, in the normal position, the orange
square of a null cube, and may be called null, orange square.
A plane being would save himself confusion if he named his
representative squares, not by using the names of the cubes simply, but
by adding to the names of the cubes a word to show what part of a cube
his representative square was.
Thus a cube null standing against his plane touches it by null orange
face, passing through his plane it has in the plane a square as trace,
which is null white section, if we use the phrase white section to
mean a section drawn perpendicular to the white line. In the same way
the cubes which we take as representative of the tesseract are not
the tesseract itself, but definite faces or sections of it. In the
preceding figures we should say then, not null, but “null tesseract
ochre cube,” because the cube we actually have is the one determined by
the three axes, white, red, yellow.
There is another way in which we can regard the colour nomenclature of
the boundaries of a tesseract.
Consider a null point to move tracing out a white line one inch in
length, and terminating in a null point, see fig. 103 or in the
coloured plate.
Then consider this white line with its terminal points itself to move
in a second dimension, each of the points traces out a line, the line
itself traces out an area, and gives two lines as well, its initial and
its final position.
Thus, if we call “a region” any element of the figure, such as a point,
or a line, etc., every “region” in moving traces out a new kind of
region, “a higher region,” and gives two regions of its own kind, an
initial and a final position. The “higher region” means a region with
another dimension in it.
Now the square can move and generate a cube. The square light yellow
moves and traces out the mass of the cube. Letting the addition of
red denote the region made by the motion in the upward direction we
get an ochre solid. The light yellow face in its initial and terminal
positions give the two square boundaries of the cube above and below.
Then each of the four lines of the light yellow square—white, yellow,
and the white, yellow opposite them—trace out a bounding square. So
there are in all six bounding squares, four of these squares being
designated in colour by adding red to the colour of the generating
lines. Finally, each point moving in the up direction gives rise to
a line coloured null + red, or red, and then there are the initial
and terminal positions of the points giving eight points. The number
of the lines is evidently twelve, for the four lines of this light
yellow square give four lines in their initial, four lines in their
final position, while the four points trace out four lines, that is
altogether twelve lines.
Now the squares are each of them separate boundaries of the cube, while
the lines belong, each of them, to two squares, thus the red line is
that which is common to the orange and pink squares.
Now suppose that there is a direction, the fourth dimension, which is
perpendicular alike to every one of the space dimensions already used—a
dimension perpendicular, for instance, to up and to right hand, so that
the pink square moving in this direction traces out a cube.
A dimension, moreover, perpendicular to the up and away directions,
so that the orange square moving in this direction also traces out
a cube, and the light yellow square, too, moving in this direction
traces out a cube. Under this supposition, the whole cube moving in
the unknown dimension, traces out something new—a new kind of volume,
a higher volume. This higher volume is a four-dimensional volume, and
we designate it in colour by adding blue to the colour of that which by
moving generates it.
It is generated by the motion of the ochre solid, and hence it is
of the colour we call light brown (white, yellow, red, blue, mixed
together). It is represented by a number of sections like 2 in fig. 103.
Now this light brown higher solid has for boundaries: first, the ochre
cube in its initial position, second, the same cube in its final
position, 1 and 3, fig. 103. Each of the squares which bound the cube,
moreover, by movement in this new direction traces out a cube, so we
have from the front pink faces of the cube, third, a pink blue or
light purple cube, shown as a light purple face on cube 2 in fig. 103,
this cube standing for any number of intermediate sections; fourth,
a similar cube from the opposite pink face; fifth, a cube traced out
by the orange face—this is coloured brown and is represented by the
brown face of the section cube in fig. 103; sixth, a corresponding
brown cube on the right hand; seventh, a cube starting from the light
yellow square below; the unknown dimension is at right angles to this
also. This cube is coloured light yellow and blue or light green; and,
finally, eighth, a corresponding cube from the upper light yellow face,
shown as the light green square at the top of the section cube.
The tesseract has thus eight cubic boundaries. These completely enclose
it, so that it would be invisible to a four-dimensional being. Now, as
to the other boundaries, just as the cube has squares, lines, points,
as boundaries, so the tesseract has cubes, squares, lines, points, as
boundaries.
The number of squares is found thus—round the cube are six squares,
these will give six squares in their initial and six in their final
positions. Then each of the twelve lines of the cube trace out a square
in the motion in the fourth dimension. Hence there will be altogether
12 + 12 = 24 squares.
If we look at any one of these squares we see that it is the meeting
surface of two of the cubic sides. Thus, the red line by its movement
in the fourth dimension, traces out a purple square—this is common
to two cubes, one of which is traced out by the pink square moving
in the fourth dimension, and the other is traced out by the orange
square moving in the same way. To take another square, the light yellow
one, this is common to the ochre cube and the light green cube. The
ochre cube comes from the light yellow square by moving it in the up
direction, the light green cube is made from the light yellow square by
moving it in the fourth dimension. The number of lines is thirty-two,
for the twelve lines of the cube give twelve lines of the tesseract
in their initial position, and twelve in their final position, making
twenty-four, while each of the eight points traces out a line, thus
forming thirty-two lines altogether.
The lines are each of them common to three cubes, or to three square
faces; take, for instance, the red line. This is common to the orange
face, the pink face, and that face which is formed by moving the red
line in the sixth dimension, namely, the purple face. It is also common
to the ochre cube, the pale purple cube, and the brown cube.
The points are common to six square faces and to four cubes; thus,
the null point from which we start is common to the three square
faces—pink, light yellow, orange, and to the three square faces made by
moving the three lines white, yellow, red, in the fourth dimension,
namely, the light blue, the light green, the purple faces—that is, to
six faces in all. The four cubes which meet in it are the ochre cube,
the light purple cube, the brown cube, and the light green cube.
[Illustration: Fig. 103.
The tesseract, red, white, yellow axes in space. In the lower line the
three rear faces are shown, the interior being removed.]
[Illustration: Fig. 104.
The tesseract, red, yellow, blue axes in space, the blue axis running
to the left, opposite faces are coloured identically.]
A complete view of the tesseract in its various space presentations
is given in the following figures or catalogue cubes, figs. 103-106.
The first cube in each figure represents the view of a tesseract
coloured as described as it begins to pass transverse to our space.
The intermediate figure represents a sectional view when it is partly
through, and the final figure represents the far end as it is just
passing out. These figures will be explained in detail in the next
chapter.
[Illustration: Fig. 105.
The tesseract, with red, white, blue axes in space. Opposite faces are
coloured identically.]
[Illustration: Fig. 106.
The tesseract, with blue, white, yellow axes in space. The blue axis
runs downward from the base of the ochre cube as it stands originally.
Opposite faces are coloured identically.]
We have thus obtained a nomenclature for each of the regions of a
tesseract; we can speak of any one of the eight bounding cubes, the
twenty square faces, the thirty-two lines, the sixteen points.
CHAPTER XIII
REMARKS ON THE FIGURES
An inspection of above figures will give an answer to many questions
about the tesseract. If we have a tesseract one inch each way, then it
can be represented as a cube—a cube having white, yellow, red axes,
and from this cube as a beginning, a volume extending into the fourth
dimension. Now suppose the tesseract to pass transverse to our space,
the cube of the red, yellow, white axes disappears at once, it is
indefinitely thin in the fourth dimension. Its place is occupied by
those parts of the tesseract which lie further away from our space in
the fourth dimension. Each one of these sections will last only for
one moment, but the whole of them will take up some appreciable time
in passing. If we take the rate of one inch a minute the sections will
take the whole of the minute in their passage across our space, they
will take the whole of the minute except the moment which the beginning
cube and the end cube occupy in their crossing our space. In each one
of the cubes, the section cubes, we can draw lines in all directions
except in the direction occupied by the blue line, the fourth
dimension; lines in that direction are represented by the transition
from one section cube to another. Thus to give ourselves an adequate
representation of the tesseract we ought to have a limitless number of
section cubes intermediate between the first bounding cube, the ochre
cube, and the last bounding cube, the other ochre cube. Practically
three intermediate sectional cubes will be found sufficient for most
purposes. We will take then a series of five figures—two terminal
cubes, and three intermediate sections—and show how the different
regions appear in our space when we take each set of three out of the
four axes of the tesseract as lying in our space.
In fig. 107 initial letters are used for the colours. A reference to
fig. 103 will show the complete nomenclature, which is merely indicated
here.
[Illustration: Fig. 107.]
In this figure the tesseract is shown in five stages distant from our
space: first, zero; second, 1/4 in.; third, 2/4 in.; fourth, 3/4 in.;
fifth, 1 in.; which are called _b_0, _b_1, _b_2, _b_3, _b_4, because
they are sections taken at distances 0, 1, 2, 3, 4 quarter inches along
the blue line. All the regions can be named from the first cube, the
_b_0 cube, as before, simply by remembering that transference along
the b axis gives the addition of blue to the colour of the region in
the ochre, the _b_0 cube. In the final cube _b_4, the colouring of the
original _b_0 cube is repeated. Thus the red line moved along the blue
axis gives a red and blue or purple square. This purple square appears
as the three purple lines in the sections _b_1, _b_2, _b_3, taken at
1/4, 2/4, 3/4 of an inch in the fourth dimension. If the tesseract
moves transverse to our space we have then in this particular region,
first of all a red line which lasts for a moment, secondly a purple
line which takes its place. This purple line lasts for a minute—that
is, all of a minute, except the moment taken by the crossing our space
of the initial and final red line. The purple line having lasted for
this period is succeeded by a red line, which lasts for a moment; then
this goes and the tesseract has passed across our space. The final red
line we call red bl., because it is separated from the initial red
line by a distance along the axis for which we use the colour blue.
Thus a line that lasts represents an area duration; is in this mode
of presentation equivalent to a dimension of space. In the same way
the white line, during the crossing our space by the tesseract, is
succeeded by a light blue line which lasts for the inside of a minute,
and as the tesseract leaves our space, having crossed it, the white bl.
line appears as the final termination.
Take now the pink face. Moved in the blue direction it traces out a
light purple cube. This light purple cube is shown in sections in
_b__{1}, _b__{2}, _b__{3}, and the farther face of this cube in the
blue direction is shown in _b__{4}—a pink face, called pink _b_ because
it is distant from the pink face we began with in the blue direction.
Thus the cube which we colour light purple appears as a lasting square.
The square face itself, the pink face, vanishes instantly the tesseract
begins to move, but the light purple cube appears as a lasting square.
Here also duration is the equivalent of a dimension of space—a lasting
square is a cube. It is useful to connect these diagrams with the views
given in the coloured plate.
Take again the orange face, that determined by the red and yellow axes;
from it goes a brown cube in the blue direction, for red and yellow
and blue are supposed to make brown. This brown cube is shown in three
sections in the faces _b__{1}, _b__{2}, _b__{3}. In _b__{4} is the
opposite orange face of the brown cube, the face called orange _b_,
for it is distant in the blue direction from the orange face. As the
tesseract passes transverse to our space, we have then in this region
an instantly vanishing orange square, followed by a lasting brown
square, and finally an orange face which vanishes instantly.
Now, as any three axes will be in our space, let us send the white
axis out into the unknown, the fourth dimension, and take the blue
axis into our known space dimension. Since the white and blue axes are
perpendicular to each other, if the white axis goes out into the fourth
dimension in the positive sense, the blue axis will come into the
direction the white axis occupied, in the negative sense.
[Illustration: Fig. 108.]
Hence, not to complicate matters by having to think of two senses in
the unknown direction, let us send the white line into the positive
sense of the fourth dimension, and take the blue one as running in the
negative sense of that direction which the white line has left; let the
blue line, that is, run to the left. We have now the row of figures
in fig. 108. The dotted cube shows where we had a cube when the white
line ran in our space—now it has turned out of our space, and another
solid boundary, another cubic face of the tesseract comes into our
space. This cube has red and yellow axes as before; but now, instead
of a white axis running to the right, there is a blue axis running to
the left. Here we can distinguish the regions by colours in a perfectly
systematic way. The red line traces out a purple square in the
transference along the blue axis by which this cube is generated from
the orange face. This purple square made by the motion of the red line
is the same purple face that we saw before as a series of lines in the
sections _b__{1}, _b__{2}, _b__{3}. Here, since both red and blue axes
are in our space, we have no need of duration to represent the area
they determine. In the motion of the tesseract across space this purple
face would instantly disappear.
From the orange face, which is common to the initial cubes in fig. 107
and fig. 108, there goes in the blue direction a cube coloured brown.
This brown cube is now all in our space, because each of its three axes
run in space directions, up, away, to the left. It is the same brown
cube which appeared as the successive faces on the sections _b__{1},
_b__{2}, _b__{3}. Having all its three axes in our space, it is given
in extension; no part of it needs to be represented as a succession.
The tesseract is now in a new position with regard to our space, and
when it moves across our space the brown cube instantly disappears.
In order to exhibit the other regions of the tesseract we must remember
that now the white line runs in the unknown dimension. Where shall we
put the sections at distances along the line? Any arbitrary position in
our space will do: there is no way by which we can represent their real
position.
However, as the brown cube comes off from the orange face to the left,
let us put these successive sections to the left. We can call them
_wh__{0}, _wh__{1}, _wh__{2}, _wh__{3}, _wh__{4}, because they are
sections along the white axis, which now runs in the unknown dimension.
Running from the purple square in the white direction we find the light
purple cube. This is represented in the sections _wh__{1}, _wh__{2},
_wh__{3}, _wh__{4}, fig. 108. It is the same cube that is represented
in the sections _b__{1}, _b__{2}, _b__{3}: in fig. 107 the red and
white axes are in our space, the blue out of it; in the other case, the
red and blue are in our space, the white out of it. It is evident that
the face pink _y_, opposite the pink face in fig. 107, makes a cube
shown in squares in _b__{1}, _b__{2}, _b__{3}, _b__{4}, on the opposite
side to the _l_ purple squares. Also the light yellow face at the base
of the cube _b__{0}, makes a light green cube, shown as a series of
base squares.
The same light green cube can be found in fig. 107. The base square in
_wh__{0} is a green square, for it is enclosed by blue and yellow axes.
From it goes a cube in the white direction, this is then a light green
cube and the same as the one just mentioned as existing in the sections
_b__{0}, _b__{1}, _b__{2}, _b__{3}, _b__{4}.
The case is, however, a little different with the brown cube. This cube
we have altogether in space in the section _wh__{0}, fig. 108, while
it exists as a series of squares, the left-hand ones, in the sections
_b__{0}, _b__{1}, _b__{2}, _b__{3}, _b__{4}. The brown cube exists as a
solid in our space, as shown in fig. 108. In the mode of representation
of the tesseract exhibited in fig. 107, the same brown cube appears as
a succession of squares. That is, as the tesseract moves across space,
the brown cube would actually be to us a square—it would be merely
the lasting boundary of another solid. It would have no thickness at
all, only extension in two dimensions, and its duration would show its
solidity in three dimensions.
It is obvious that, if there is a four-dimensional space, matter in
three dimensions only is a mere abstraction; all material objects
must then have a slight four-dimensional thickness. In this case the
above statement will undergo modification. The material cube which is
used as the model of the boundary of a tesseract will have a slight
thickness in the fourth dimension, and when the cube is presented to
us in another aspect, it would not be a mere surface. But it is most
convenient to regard the cubes we use as having no extension at all in
the fourth dimension. This consideration serves to bring out a point
alluded to before, that, if there is a fourth dimension, our conception
of a solid is the conception of a mere abstraction, and our talking
about real three-dimensional objects would seem to a four-dimensional
being as incorrect as a two-dimensional being’s telling about real
squares, real triangles, etc., would seem to us.
The consideration of the two views of the brown cube shows that any
section of a cube can be looked at by a presentation of the cube in
a different position in four-dimensional space. The brown faces in
_b__{1}, _b__{2}, _b__{3}, are the very same brown sections that would
be obtained by cutting the brown cube, _wh__{0}, across at the right
distances along the blue line, as shown in fig. 108. But as these
sections are placed in the brown cube, _wh__{0}, they come behind one
another in the blue direction. Now, in the sections _wh__{1}, _wh__{2},
_wh__{3}, we are looking at these sections from the white direction—the
blue direction does not exist in these figures. So we see them in
a direction at right angles to that in which they occur behind one
another in _wh__{0}. There are intermediate views, which would come in
the rotation of a tesseract. These brown squares can be looked at from
directions intermediate between the white and blue axes. It must be
remembered that the fourth dimension is perpendicular equally to all
three space axes. Hence we must take the combinations of the blue axis,
with each two of our three axes, white, red, yellow, in turn.
In fig. 109 we take red, white, and blue axes in space, sending yellow
into the fourth dimension. If it goes into the positive sense of the
fourth dimension the blue line will come in the opposite direction to
that in which the yellow line ran before. Hence, the cube determined
by the white, red, blue axes, will start from the pink plane and run
towards us. The dotted cube shows where the ochre cube was. When it is
turned out of space, the cube coming towards from its front face is
the one which comes into our space in this turning. Since the yellow
line now runs in the unknown dimension we call the sections _y__{0},
_y__{1}, _y__{2}, _y__{3}, _y__{4}, as they are made at distances 0, 1,
2, 3, 4, quarter inches along the yellow line. We suppose these cubes
arranged in a line coming towards us—not that that is any more natural
than any other arbitrary series of positions, but it agrees with the
plan previously adopted.
[Illustration: Fig. 109.]
The interior of the first cube, _y__{0}, is that derived from pink by
adding blue, or, as we call it, light purple. The faces of the cube are
light blue, purple, pink. As drawn, we can only see the face nearest to
us, which is not the one from which the cube starts—but the face on the
opposite side has the same colour name as the face towards us.
The successive sections of the series, _y__{0}, _y__{1}, _y__{2}, etc.,
can be considered as derived from sections of the _b__{0} cube made at
distances along the yellow axis. What is distant a quarter inch from
the pink face in the yellow direction? This question is answered by
taking a section from a point a quarter inch along the yellow axis in
the cube _b__{0}, fig. 107. It is an ochre section with lines orange
and light yellow. This section will therefore take the place of the
pink face in _y__{1} when we go on in the yellow direction. Thus, the
first section, _y__{1}, will begin from an ochre face with light yellow
and orange lines. The colour of the axis which lies in space towards
us is blue, hence the regions of this section-cube are determined in
nomenclature, they will be found in full in fig. 105.
There remains only one figure to be drawn, and that is the one in which
the red axis is replaced by the blue. Here, as before, if the red axis
goes out into the positive sense of the fourth dimension, the blue line
must come into our space in the negative sense of the direction which
the red line has left. Accordingly, the first cube will come in beneath
the position of our ochre cube, the one we have been in the habit of
starting with.
[Illustration: Fig. 110.]
To show these figures we must suppose the ochre cube to be on a movable
stand. When the red line swings out into the unknown dimension, and the
blue line comes in downwards, a cube appears below the place occupied
by the ochre cube. The dotted cube shows where the ochre cube was.
That cube has gone and a different cube runs downwards from its base.
This cube has white, yellow, and blue axes. Its top is a light yellow
square, and hence its interior is light yellow + blue or light green.
Its front face is formed by the white line moving along the blue axis,
and is therefore light blue, the left-hand side is formed by the yellow
line moving along the blue axis, and therefore green.
As the red line now runs in the fourth dimension, the successive
sections can he called _r__{0}, _r__{1}, _r__{2}, _r__{3}, _r__{4},
these letters indicating that at distances 0, 1/4, 2/4, 3/4, 1 inch
along the red axis we take all of the tesseract that can be found in a
three-dimensional space, this three-dimensional space extending not at
all in the fourth dimension, but up and down, right and left, far and
near.
We can see what should replace the light yellow face of _r__{0}, when
the section _r__{1} comes in, by looking at the cube _b__{0}, fig. 107.
What is distant in it one-quarter of an inch from the light yellow face
in the red direction? It is an ochre section with orange and pink lines
and red points; see also fig. 103.
This square then forms the top square of _r__{1}. Now we can determine
the nomenclature of all the regions of _r__{1} by considering what
would be formed by the motion of this square along a blue axis.
But we can adopt another plan. Let us take a horizontal section of
_r__{0}, and finding that section in the figures, of fig. 107 or fig.
103, from them determine what will replace it, going on in the red
direction.
A section of the _r__{0} cube has green, light blue, green, light blue
sides and blue points.
Now this square occurs on the base of each of the section figures,
_b__{1}, _b__{2}, etc. In them we see that 1/4 inch in the red
direction from it lies a section with brown and light purple lines and
purple corners, the interior being of light brown. Hence this is the
nomenclature of the section which in _r__{1} replaces the section of
_r__{0} made from a point along the blue axis.
Hence the colouring as given can be derived.
We have thus obtained a perfectly named group of tesseracts. We can
take a group of eighty-one of them 3 × 3 × 3 × 3, in four dimensions,
and each tesseract will have its name null, red, white, yellow, blue,
etc., and whatever cubic view we take of them we can say exactly
what sides of the tesseracts we are handling, and how they touch each
other.[5]
[5] At this point the reader will find it advantageous, if he has the
models, to go through the manipulations described in the appendix.
Thus, for instance, if we have the sixteen tesseracts shown below, we
can ask how does null touch blue.
[Illustration: Fig. 111.]
In the arrangement given in fig. 111 we have the axes white, red,
yellow, in space, blue running in the fourth dimension. Hence we have
the ochre cubes as bases. Imagine now the tesseractic group to pass
transverse to our space—we have first of all null ochre cube, white
ochre cube, etc.; these instantly vanish, and we get the section shown
in the middle cube in fig. 103, and finally, just when the tesseract
block has moved one inch transverse to our space, we have null ochre
cube, and then immediately afterwards the ochre cube of blue comes in.
Hence the tesseract null touches the tesseract blue by its ochre cube,
which is in contact, each and every point of it, with the ochre cube of
blue.
How does null touch white, we may ask? Looking at the beginning A, fig.
111, where we have the ochre cubes, we see that null ochre touches
white ochre by an orange face. Now let us generate the null and white
tesseracts by a motion in the blue direction of each of these cubes.
Each of them generates the corresponding tesseract, and the plane of
contact of the cubes generates the cube by which the tesseracts are
in contact. Now an orange plane carried along a blue axis generates a
brown cube. Hence null touches white by a brown cube.
[Illustration: Fig. 112.]
If we ask again how red touches light blue tesseract, let us rearrange
our group, fig. 112, or rather turn it about so that we have a
different space view of it; let the red axis and the white axis run
up and right, and let the blue axis come in space towards us, then
the yellow axis runs in the fourth dimension. We have then two blocks
in which the bounding cubes of the tesseracts are given, differently
arranged with regard to us—the arrangement is really the same, but it
appears different to us. Starting from the plane of the red and white
axes we have the four squares of the null, white, red, pink tesseracts
as shown in A, on the red, white plane, unaltered, only from them now
comes out towards us the blue axis. Hence we have null, white, red,
pink tesseracts in contact with our space by their cubes which have
the red, white, blue axis in them, that is by the light purple cubes.
Following on these four tesseracts we have that which comes next to
them in the blue direction, that is the four blue, light blue, purple,
light purple. These are likewise in contact with our space by their
light purple cubes, so we see a block as named in the figure, of which
each cube is the one determined by the red, white, blue, axes.
The yellow line now runs out of space; accordingly one inch on in the
fourth dimension we come to the tesseracts which follow on the eight
named in C, fig. 112, in the yellow direction.
These are shown in C.y_{1}, fig. 112. Between figure C and C.y_{1} is
that four-dimensional mass which is formed by moving each of the cubes
in C one inch in the fourth dimension—that is, along a yellow axis; for
the yellow axis now runs in the fourth dimension.
In the block C we observe that red (light purple cube) touches light
blue (light purple cube) by a point. Now these two cubes moving
together remain in contact during the period in which they trace out
the tesseracts red and light blue. This motion is along the yellow
axis, consequently red and light blue touch by a yellow line.
We have seen that the pink face moved in a yellow direction traces out
a cube; moved in the blue direction it also traces out a cube. Let us
ask what the pink face will trace out if it is moved in a direction
within the tesseract lying equally between the yellow and blue
directions. What section of the tesseract will it make?
We will first consider the red line alone. Let us take a cube with the
red line in it and the yellow and blue axes.
The cube with the yellow, red, blue axes is shown in fig. 113. If the
red line is moved equally in the yellow and in the blue direction by
four equal motions of ¼ inch each, it takes the positions 11, 22, 33,
and ends as a red line.
[Illustration: Fig. 113.]
Now, the whole of this red, yellow, blue, or brown cube appears as a
series of faces on the successive sections of the tesseract starting
from the ochre cube and letting the blue axis run in the fourth
dimension. Hence the plane traced out by the red line appears as a
series of lines in the successive sections, in our ordinary way of
representing the tesseract; these lines are in different places in each
successive section.
[Illustration: Fig. 114.]
Thus drawing our initial cube and the successive sections, calling them
_b__{0}, _b__{1}, _b__{2}, _b__{3}, _b__{4}, fig. 115, we have the red
line subject to this movement appearing in the positions indicated.
We will now investigate what positions in the tesseract another line in
the pink face assumes when it is moved in a similar manner.
Take a section of the original cube containing a vertical line, 4,
in the pink plane, fig. 115. We have, in the section, the yellow
direction, but not the blue.
From this section a cube goes off in the fourth dimension, which is
formed by moving each point of the section in the blue direction.
[Illustration: Fig. 115.]
[Illustration: Fig. 116.]
Drawing this cube we have fig. 116.
Now this cube occurs as a series of sections in our original
representation of the tesseract. Taking four steps as before this cube
appears as the sections drawn in _b__{0}, _b__{1}, _b__{2}, _b__{3},
_b__{4}, fig. 117, and if the line 4 is subjected to a movement equal
in the blue and yellow directions, it will occupy the positions
designated by 4, 4_{1}, 4_{2}, 4_{3}, 4_{4}.
[Illustration: Fig. 117.]
Hence, reasoning in a similar manner about every line, it is evident
that, moved equally in the blue and yellow directions, the pink plane
will trace out a space which is shown by the series of section planes
represented in the diagram.
Thus the space traced out by the pink face, if it is moved equally in
the yellow and blue directions, is represented by the set of planes
delineated in Fig. 118, pink face or 0, then 1, 2, 3, and finally pink
face or 4. This solid is a diagonal solid of the tesseract, running
from a pink face to a pink face. Its length is the length of the
diagonal of a square, its side is a square.
Let us now consider the unlimited space which springs from the pink
face extended.
This space, if it goes off in the yellow direction, gives us in it the
ochre cube of the tesseract. Thus, if we have the pink face given and a
point in the ochre cube, we have determined this particular space.
Similarly going off from the pink face in the blue direction is another
space, which gives us the light purple cube of the tesseract in it. And
any point being taken in the light purple cube, this space going off
from the pink face is fixed.
[Illustration: Fig. 118.]
The space we are speaking of can be conceived as swinging round the
pink face, and in each of its positions it cuts out a solid figure from
the tesseract, one of which we have seen represented in fig. 118.
Each of these solid figures is given by one position of the swinging
space, and by one only. Hence in each of them, if one point is taken,
the particular one of the slanting spaces is fixed. Thus we see that
given a plane and a point out of it a space is determined.
Now, two points determine a line.
Again, think of a line and a point outside it. Imagine a plane rotating
round the line. At some time in its rotation it passes through the
point. Thus a line and a point, or three points, determine a plane.
And finally four points determine a space. We have seen that a plane
and a point determine a space, and that three points determine a plane;
so four points will determine a space.
These four points may be any points, and we can take, for instance, the
four points at the extremities of the red, white, yellow, blue axes, in
the tesseract. These will determine a space slanting with regard to the
section spaces we have been previously considering. This space will cut
the tesseract in a certain figure.
One of the simplest sections of a cube by a plane is that in which the
plane passes through the extremities of the three edges which meet in a
point. We see at once that this plane would cut the cube in a triangle,
but we will go through the process by which a plane being would most
conveniently treat the problem of the determination of this shape, in
order that we may apply the method to the determination of the figure
in which a space cuts a tesseract when it passes through the 4 points
at unit distance from a corner.
We know that two points determine a line, three points determine a
plane, and given any two points in a plane the line between them lies
wholly in the plane.
[Illustration: Fig. 119.]
Let now the plane being study the section made by a plane passing
through the null _r_, null _wh_, and null _y_ points, fig. 119. Looking
at the orange square, which, as usual, we suppose to be initially in
his plane, he sees that the line from null _r_ to null _y_, which is
a line in the section plane, the plane, namely, through the three
extremities of the edges meeting in null, cuts the orange face in an
orange line with null points. This then is one of the boundaries of the
section figure.
Let now the cube be so turned that the pink face comes in his plane.
The points null _r_ and null _wh_ are now visible. The line between
them is pink with null points, and since this line is common to the
surface of the cube and the cutting plane, it is a boundary of the
figure in which the plane cuts the cube.
Again, suppose the cube turned so that the light yellow face is in
contact with the plane being’s plane. He sees two points, the null _wh_
and the null _y_. The line between these lies in the cutting plane.
Hence, since the three cutting lines meet and enclose a portion of
the cube between them, he has determined the figure he sought. It is
a triangle with orange, pink, and light yellow sides, all equal, and
enclosing an ochre area.
Let us now determine in what figure the space, determined by the four
points, null _r_, null _y_, null _wh_, null _b_, cuts the tesseract. We
can see three of these points in the primary position of the tesseract
resting against our solid sheet by the ochre cube. These three points
determine a plane which lies in the space we are considering, and this
plane cuts the ochre cube in a triangle, the interior of which is
ochre (fig. 119 will serve for this view), with pink, light yellow and
orange sides, and null points. Going in the fourth direction, in one
sense, from this plane we pass into the tesseract, in the other sense
we pass away from it. The whole area inside the triangle is common to
the cutting plane we see, and a boundary of the tesseract. Hence we
conclude that the triangle drawn is common to the tesseract and the
cutting space.
Now let the ochre cube turn out and the brown cube come in. The dotted
lines show the position the ochre cube has left (fig. 120).
[Illustration: Fig. 120.]
Here we see three out of the four points through which the cutting
plane passes, null _r_, null _y_, and null _b_. The plane they
determine lies in the cutting space, and this plane cuts out of the
brown cube a triangle with orange, purple and green sides, and null
points. The orange line of this figure is the same as the orange line
in the last figure.
Now let the light purple cube swing into our space, towards us, fig.
121.
[Illustration: Fig. 121.]
The cutting space which passes through the four points, null _r_, _y_,
_wh_, _b_, passes through the null _r_, _wh_, _b_, and therefore the
plane these determine lies in the cutting space.
This triangle lies before us. It has a light purple interior and pink,
light blue, and purple edges with null points.
This, since it is all of the plane that is common to it, and this
bounding of the tesseract, gives us one of the bounding faces of our
sectional figure. The pink line in it is the same as the pink line we
found in the first figure—that of the ochre cube.
Finally, let the tesseract swing about the light yellow plane, so that
the light green cube comes into our space. It will point downwards.
The three points, _n.y_, _n.wh_, _n.b_, are in the cutting space, and
the triangle they determine is common to the tesseract and the cutting
space. Hence this boundary is a triangle having a light yellow line,
which is the same as the light yellow line of the first figure, a light
blue line and a green line.
[Illustration: Fig. 122.]
We have now traced the cutting space between every set of three that
can be made out of the four points in which it cuts the tesseract, and
have got four faces which all join on to each other by lines.
[Illustration: Fig. 123.]
The triangles are shown in fig. 123 as they join on to the triangle
in the ochre cube. But they join on each to the other in an exactly
similar manner; their edges are all identical two and two. They form a
closed figure, a tetrahedron, enclosing a light brown portion which is
the portion of the cutting space which lies inside the tesseract.
We cannot expect to see this light brown portion, any more than a plane
being could expect to see the inside of a cube if an angle of it were
pushed through his plane. All he can do is to come upon the boundaries
of it in a different way to that in which he would if it passed
straight through his plane.
Thus in this solid section; the whole interior lies perfectly open in
the fourth dimension. Go round it as we may we are simply looking at
the boundaries of the tesseract which penetrates through our solid
sheet. If the tesseract were not to pass across so far, the triangle
would be smaller; if it were to pass farther, we should have a
different figure, the outlines of which can be determined in a similar
manner.
The preceding method is open to the objection that it depends rather on
our inferring what must be, than our seeing what is. Let us therefore
consider our sectional space as consisting of a number of planes, each
very close to the last, and observe what is to be found in each plane.
The corresponding method in the case of two dimensions is as
follows:—The plane being can see that line of the sectional plane
through null _y_, null _wh_, null _r_, which lies in the orange plane.
Let him now suppose the cube and the section plane to pass half way
through his plane. Replacing the red and yellow axes are lines parallel
to them, sections of the pink and light yellow faces.
[Illustration: Fig. 124.]
Where will the section plane cut these parallels to the red and yellow
axes?
Let him suppose the cube, in the position of the drawing, fig. 124,
turned so that the pink face lies against his plane. He can see the
line from the null _r_ point to the null _wh_ point, and can see
(compare fig. 119) that it cuts AB a parallel to his red axis, drawn
at a point half way along the white line, in a point B, half way up. I
shall speak of the axis as having the length of an edge of the cube.
Similarly, by letting the cube turn so that the light yellow square
swings against his plane, he can see (compare fig. 119) that a parallel
to his yellow axis drawn from a point half-way along the white axis, is
cut at half its length by the trace of the section plane in the light
yellow face.
Hence when the cube had passed half-way through he would have—instead
of the orange line with null points, which he had at first—an ochre
line of half its length, with pink and light yellow points. Thus, as
the cube passed slowly through his plane, he would have a succession
of lines gradually diminishing in length and forming an equilateral
triangle. The whole interior would be ochre, the line from which it
started would be orange. The succession of points at the ends of
the succeeding lines would form pink and light yellow lines and the
final point would be null. Thus looking at the successive lines in
the section plane as it and the cube passed across his plane he would
determine the figure cut out bit by bit.
Coming now to the section of the tesseract, let us imagine that the
tesseract and its cutting _space_ pass slowly across our space; we can
examine portions of it, and their relation to portions of the cutting
space. Take the section space which passes through the four points,
null _r_, _wh_, _y_, _b_; we can see in the ochre cube (fig. 119) the
plane belonging to this section space, which passes through the three
extremities of the red, white, yellow axes.
Now let the tesseract pass half way through our space. Instead of our
original axes we have parallels to them, purple, light blue, and green,
each of the same length as the first axes, for the section of the
tesseract is of exactly the same shape as its ochre cube.
But the sectional space seen at this stage of the transference would
not cut the section of the tesseract in a plane disposed as at first.
To see where the sectional space would cut these parallels to the
original axes let the tesseract swing so that, the orange face
remaining stationary, the blue line comes in to the left.
Here (fig. 125) we have the null _r_, _y_, _b_ points, and of the
sectional space all we see is the plane through these three points in
it.
[Illustration: Fig. 125.]
In this figure we can draw the parallels to the red and yellow axes and
see that, if they started at a point half way along the blue axis, they
would each be cut at a point so as to be half of their previous length.
Swinging the tesseract into our space about the pink face of the ochre
cube we likewise find that the parallel to the white axis is cut at
half its length by the sectional space.
Hence in a section made when the tesseract had passed half across our
space the parallels to the red, white, yellow axes, which are now in
our space, are cut by the section space, each of them half way along,
and for this stage of the traversing motion we should have fig. 126.
The section made of this cube by the plane in which the sectional space
cuts it, is an equilateral triangle with purple, l. blue, green points,
and l. purple, brown, l. green lines.
[Illustration: Fig. 126.]
Thus the original ochre triangle, with null points and pink, orange,
light yellow lines, would be succeeded by a triangle coloured in manner
just described.
This triangle would initially be only a very little smaller than the
original triangle, it would gradually diminish, until it ended in a
point, a null point. Each of its edges would be of the same length.
Thus the successive sections of the successive planes into which we
analyse the cutting space would be a tetrahedron of the description
shown (fig. 123), and the whole interior of the tetrahedron would be
light brown.
[Illustration: Fig. 127. Front view. The rear faces.]
In fig. 127 the tetrahedron is represented by means of its faces as
two triangles which meet in the p. line, and two rear triangles which
join on to them, the diagonal of the pink face being supposed to run
vertically upward.
We have now reached a natural termination. The reader may pursue
the subject in further detail, but will find no essential novelty.
I conclude with an indication as to the manner in which figures
previously given may be used in determining sections by the method
developed above.
Applying this method to the tesseract, as represented in Chapter IX.,
sections made by a space cutting the axes equidistantly at any distance
can be drawn, and also the sections of tesseracts arranged in a block.
If we draw a plane, cutting all four axes at a point six units distance
from null, we have a slanting space. This space cuts the red, white,
yellow axes in the points LMN (fig. 128), and so in the region of our
space before we go off into the fourth dimension, we have the plane
represented by LMN extended. This is what is common to the slanting
space and our space.
[Illustration: Fig. 128.]
This plane cuts the ochre cube in the triangle EFG.
Comparing this with (fig. 72) _oh_, we see that the hexagon there drawn
is part of the triangle EFG.
Let us now imagine the tesseract and the slanting space both together
to pass transverse to our space, a distance of one unit, we have in
1_h_ a section of the tesseract, whose axes are parallels to the
previous axes. The slanting space cuts them at a distance of five units
along each. Drawing the plane through these points in 1_h_ it will be
found to cut the cubical section of the tesseract in the hexagonal
figure drawn. In 2_h_ (fig. 72) the slanting space cuts the parallels
to the axes at a distance of four along each, and the hexagonal figure
is the section of this section of the tesseract by it. Finally when
3_h_ comes in the slanting space cuts the axes at a distance of three
along each, and the section is a triangle, of which the hexagon drawn
is a truncated portion. After this the tesseract, which extends only
three units in each of the four dimensions, has completely passed
transverse of our space, and there is no more of it to be cut. Hence,
putting the plane sections together in the right relations, we have
the section determined by the particular slanting space: namely an
octahedron.
CHAPTER XIV.[6]
A RECAPITULATION AND EXTENSION OF THE PHYSICAL ARGUMENT
There are two directions of inquiry in which the research for the
physical reality of a fourth dimension can be prosecuted. One is the
investigation of the infinitely great, the other is the investigation
of the infinitely small.
[6] The contents of this chapter are taken from a paper read before
the Philosophical Society of Washington. The mathematical portion
of the paper has appeared in part in the Proceedings of the Royal
Irish Academy under the title, “Cayley’s formulæ of orthogonal
transformation,” Nov. 29th, 1903.
By the measurement of the angles of vast triangles, whose sides are the
distances between the stars, astronomers have sought to determine if
there is any deviation from the values given by geometrical deduction.
If the angles of a celestial triangle do not together equal two right
angles, there would be an evidence for the physical reality of a fourth
dimension.
This conclusion deserves a word of explanation. If space is really
four-dimensional, certain conclusions follow which must be brought
clearly into evidence if we are to frame the questions definitely which
we put to Nature. To account for our limitation let us assume a solid
material sheet against which we move. This sheet must stretch alongside
every object in every direction in which it visibly moves. Every
material body must slip or slide along this sheet, not deviating from
contact with it in any motion which we can observe.
The necessity for this assumption is clearly apparent, if we consider
the analogous case of a suppositionary plane world. If there were
any creatures whose experiences were confined to a plane, we must
account for their limitation. If they were free to move in every space
direction, they would have a three-dimensional motion; hence they must
be physically limited, and the only way in which we can conceive such
a limitation to exist is by means of a material surface against which
they slide. The existence of this surface could only be known to them
indirectly. It does not lie in any direction from them in which the
kinds of motion they know of leads them. If it were perfectly smooth
and always in contact with every material object, there would be no
difference in their relations to it which would direct their attention
to it.
But if this surface were curved—if it were, say, in the form of a vast
sphere—the triangles they drew would really be triangles of a sphere,
and when these triangles are large enough the angles diverge from
the magnitudes they would have for the same lengths of sides if the
surface were plane. Hence by the measurement of triangles of very great
magnitude a plane being might detect a difference from the laws of a
plane world in his physical world, and so be led to the conclusion that
there was in reality another dimension to space—a third dimension—as
well as the two which his ordinary experience made him familiar with.
Now, astronomers have thought it worth while to examine the
measurements of vast triangles drawn from one celestial body to another
with a view to determine if there is anything like a curvature in our
space—that is to say, they have tried astronomical measurements to
find out if the vast solid sheet against which, on the supposition of
a fourth dimension, everything slides is curved or not. These results
have been negative. The solid sheet, if it exists, is not curved or,
being curved, has not a sufficient curvature to cause any observable
deviation from the theoretical value of the angles calculated.
Hence the examination of the infinitely great leads to no decisive
criterion. If it did we should have to decide between the present
theory and that of metageometry.
Coming now to the prosecution of the inquiry in the direction of
the infinitely small, we have to state the question thus: Our laws
of movement are derived from the examination of bodies which move
in three-dimensional space. All our conceptions are founded on the
supposition of a space which is represented analytically by three
independent axes and variations along them—that is, it is a space in
which there are three independent movements. Any motion possible in it
can be compounded out of these three movements, which we may call: up,
right, away.
To examine the actions of the very small portions of matter with the
view of ascertaining if there is any evidence in the phenomena for
the supposition of a fourth dimension of space, we must commence by
clearly defining what the laws of mechanics would be on the supposition
of a fourth dimension. It is of no use asking if the phenomena of the
smallest particles of matter are like—we do not know what. We must
have a definite conception of what the laws of motion would be on the
supposition of the fourth dimension, and then inquire if the phenomena
of the activity of the smaller particles of matter resemble the
conceptions which we have elaborated.
Now, the task of forming these conceptions is by no means one to be
lightly dismissed. Movement in space has many features which differ
entirely from movement on a plane; and when we set about to form the
conception of motion in four dimensions, we find that there is at least
as great a step as from the plane to three-dimensional space.
I do not say that the step is difficult, but I want to point out
that it must be taken. When we have formed the conception of
four-dimensional motion, we can ask a rational question of Nature.
Before we have elaborated our conceptions we are asking if an unknown
is like an unknown—a futile inquiry.
As a matter of fact, four-dimensional movements are in every way simple
and more easy to calculate than three-dimensional movements, for
four-dimensional movements are simply two sets of plane movements put
together.
Without the formation of an experience of four-dimensional bodies,
their shapes and motions, the subject can be but formal—logically
conclusive, not intuitively evident. It is to this logical apprehension
that I must appeal.
It is perfectly simple to form an experiential familiarity with the
facts of four-dimensional movement. The method is analogous to that
which a plane being would have to adopt to form an experiential
familiarity with three-dimensional movements, and may be briefly summed
up as the formation of a compound sense by means of which duration is
regarded as equivalent to extension.
Consider a being confined to a plane. A square enclosed by four lines
will be to him a solid, the interior of which can only be examined by
breaking through the lines. If such a square were to pass transverse to
his plane, it would immediately disappear. It would vanish, going in no
direction to which he could point.
If, now, a cube be placed in contact with his plane, its surface of
contact would appear like the square which we have just mentioned.
But if it were to pass transverse to his plane, breaking through it,
it would appear as a lasting square. The three-dimensional matter will
give a lasting appearance in circumstances under which two-dimensional
matter will at once disappear.
Similarly, a four-dimensional cube, or, as we may call it, a tesseract,
which is generated from a cube by a movement of every part of the cube
in a fourth direction at right angles to each of the three visible
directions in the cube, if it moved transverse to our space, would
appear as a lasting cube.
A cube of three-dimensional matter, since it extends to no distance at
all in the fourth dimension, would instantly disappear, if subjected
to a motion transverse to our space. It would disappear and be gone,
without it being possible to point to any direction in which it had
moved.
All attempts to visualise a fourth dimension are futile. It must be
connected with a time experience in three space.
The most difficult notion for a plane being to acquire would be that of
rotation about a line. Consider a plane being facing a square. If he
were told that rotation about a line were possible, he would move his
square this way and that. A square in a plane can rotate about a point,
but to rotate about a line would seem to the plane being perfectly
impossible. How could those parts of his square which were on one side
of an edge come to the other side without the edge moving? He could
understand their reflection in the edge. He could form an idea of the
looking-glass image of his square lying on the opposite side of the
line of an edge, but by no motion that he knows of can he make the
actual square assume that position. The result of the rotation would be
like reflection in the edge, but it would be a physical impossibility
to produce it in the plane.
The demonstration of rotation about a line must be to him purely
formal. If he conceived the notion of a cube stretching out in an
unknown direction away from his plane, then he can see the base of
it, his square in the plane, rotating round a point. He can likewise
apprehend that every parallel section taken at successive intervals in
the unknown direction rotates in like manner round a point. Thus he
would come to conclude that the whole body rotates round a line—the
line consisting of the succession of points round which the plane
sections rotate. Thus, given three axes, _x_, _y_, _z_, if _x_ rotates
to take the place of _y_, and _y_ turns so as to point to negative
_x_, then the third axis remaining unaffected by this turning is the
axis about which the rotation takes place. This, then, would have to be
his criterion of the axis of a rotation—that which remains unchanged
when a rotation of every plane section of a body takes place.
There is another way in which a plane being can think about
three-dimensional movements; and, as it affords the type by which we
can most conveniently think about four-dimensional movements, it will
be no loss of time to consider it in detail.
[Illustration: Fig. 1 (129).]
We can represent the plane being and his object by figures cut out of
paper, which slip on a smooth surface. The thickness of these bodies
must be taken as so minute that their extension in the third dimension
escapes the observation of the plane being, and he thinks about them
as if they were mathematical plane figures in a plane instead of being
material bodies capable of moving on a plane surface. Let A_x_, A_y_
be two axes and ABCD a square. As far as movements in the plane are
concerned, the square can rotate about a point A, for example. It
cannot rotate about a side, such as AC.
But if the plane being is aware of the existence of a third dimension
he can study the movements possible in the ample space, taking his
figure portion by portion.
His plane can only hold two axes. But, since it can hold two, he is
able to represent a turning into the third dimension if he neglects one
of his axes and represents the third axis as lying in his plane. He can
make a drawing in his plane of what stands up perpendicularly from his
plane. Let A_z_ be the axis, which stands perpendicular to his plane at
A. He can draw in his plane two lines to represent the two axes, A_x_
and A_z_. Let Fig. 2 be this drawing. Here the _z_ axis has taken the
place of the _y_ axis, and the plane of A_x_ A_z_ is represented in his
plane. In this figure all that exists of the square ABCD will be the
line AB.
[Illustration: Fig. 2 (130).]
The square extends from this line in the _y_ direction, but more of
that direction is represented in Fig. 2. The plane being can study the
turning of the line AB in this diagram. It is simply a case of plane
turning around the point A. The line AB occupies intermediate portions
like AB_{1} and after half a revolution will lie on A_x_ produced
through A.
Now, in the same way, the plane being can take another point, A´, and
another line, A´B´, in his square. He can make the drawing of the two
directions at A´, one along A´B´, the other perpendicular to his plane.
He will obtain a figure precisely similar to Fig. 2, and will see that,
as AB can turn around A, so A´C´ around A.
In this turning AB and A´B´ would not interfere with each other, as
they would if they moved in the plane around the separate points A and
A´.
Hence the plane being would conclude that a rotation round a line was
possible. He could see his square as it began to make this turning. He
could see it half way round when it came to lie on the opposite side of
the line AC. But in intermediate portions he could not see it, for it
runs out of the plane.
Coming now to the question of a four-dimensional body, let us conceive
of it as a series of cubic sections, the first in our space, the rest
at intervals, stretching away from our space in the unknown direction.
We must not think of a four-dimensional body as formed by moving a
three-dimensional body in any direction which we can see.
Refer for a moment to Fig. 3. The point A, moving to the right, traces
out the line AC. The line AC, moving away in a new direction, traces
out the square ACEG at the base of the cube. The square AEGC, moving
in a new direction, will trace out the cube ACEGBDHF. The vertical
direction of this last motion is not identical with any motion possible
in the plane of the base of the cube. It is an entirely new direction,
at right angles to every line that can be drawn in the base. To trace
out a tesseract the cube must move in a new direction—a direction at
right angles to any and every line that can be drawn in the space of
the cube.
The cubic sections of the tesseract are related to the cube we see, as
the square sections of the cube are related to the square of its base
which a plane being sees.
Let us imagine the cube in our space, which is the base of a tesseract,
to turn about one of its edges. The rotation will carry the whole body
with it, and each of the cubic sections will rotate. The axis we see
in our space will remain unchanged, and likewise the series of axes
parallel to it about which each of the parallel cubic sections rotates.
The assemblage of all of these is a plane.
Hence in four dimensions a body rotates about a plane. There is no such
thing as rotation round an axis.
We may regard the rotation from a different point of view. Consider
four independent axes each at right angles to all the others, drawn in
a four-dimensional body. Of these four axes we can see any three. The
fourth extends normal to our space.
Rotation is the turning of one axis into a second, and the second
turning to take the place of the negative of the first. It involves
two axes. Thus, in this rotation of a four-dimensional body, two axes
change and two remain at rest. Four-dimensional rotation is therefore a
turning about a plane.
As in the case of a plane being, the result of rotation about a
line would appear as the production of a looking-glass image of the
original object on the other side of the line, so to us the result
of a four-dimensional rotation would appear like the production of a
looking-glass image of a body on the other side of a plane. The plane
would be the axis of the rotation, and the path of the body between its
two appearances would be unimaginable in three-dimensional space.
[Illustration: Fig. 3 (131).]
Let us now apply the method by which a plane being could examine
the nature of rotation about a line in our examination of rotation
about a plane. Fig. 3 represents a cube in our space, the three axes
_x_, _y_, _z_ denoting its three dimensions. Let _w_ represent the
fourth dimension. Now, since in our space we can represent any three
dimensions, we can, if we choose, make a representation of what is
in the space determined by the three axes _x_, _z_, _w_. This is a
three-dimensional space determined by two of the axes we have drawn,
_x_ and _z_, and in place of _y_ the fourth axis, _w_. We cannot,
keeping _x_ and _z_, have both _y_ and _w_ in our space; so we will
let _y_ go and draw _w_ in its place. What will be our view of the cube?
Evidently we shall have simply the square that is in the plane of _xz_,
the square ACDB. The rest of the cube stretches in the _y_ direction,
and, as we have none of the space so determined, we have only the face
of the cube. This is represented in fig. 4.
[Illustration: Fig. 4 (132).]
Now, suppose the whole cube to be turned from the _x_ to the _w_
direction. Conformably with our method, we will not take the whole of
the cube into consideration at once, but will begin with the face ABCD.
Let this face begin to turn. Fig. 5 represents one of the positions it
will occupy; the line AB remains on the _z_ axis. The rest of the face
extends between the _x_ and the _w_ direction.
[Illustration: Fig. 5 (133).]
Now, since we can take any three axes, let us look at what lies in the
space of _zyw_, and examine the turning there. We must now let the _z_
axis disappear and let the _w_ axis run in the direction in which the
_z_ ran.
Making this representation, what do we see of the cube? Obviously we
see only the lower face. The rest of the cube lies in the space of
_xyz_. In the space of _xyz_ we have merely the base of the cube lying
in the plane of _xy_, as shown in fig. 6.
[Illustration: Fig. 6 (134).]
Now let the _x_ to _w_ turning take place. The square ACEG will turn
about the line AE. This edge will remain along the _y_ axis and will be
stationary, however far the square turns.
Thus, if the cube be turned by an _x_ to _w_ turning, both the edge AB
and the edge AC remain stationary; hence the whole face ABEF in the
_yz_ plane remains fixed. The turning has taken place about the face
ABEF.
[Illustration: Fig. 7 (135).]
Suppose this turning to continue till AC runs to the left from
A. The cube will occupy the position shown in fig. 8. This is
the looking-glass image of the cube in fig. 3. By no rotation in
three-dimensional space can the cube be brought from the position in
fig. 3 to that shown in fig. 8.
[Illustration: Fig. 8 (136).]
We can think of this turning as a turning of the face ABCD about AB,
and a turning of each section parallel to ABCD round the vertical line
in which it intersects the face ABEF, the space in which the turning
takes place being a different one from that in which the cube lies.
One of the conditions, then, of our inquiry in the direction of the
infinitely small is that we form the conception of a rotation about
a plane. The production of a body in a state in which it presents
the appearance of a looking-glass image of its former state is the
criterion for a four-dimensional rotation.
There is some evidence for the occurrence of such transformations
of bodies in the change of bodies from those which produce a
right-handed polarisation of light to those which produce a left-handed
polarisation; but this is not a point to which any very great
importance can be attached.
Still, in this connection, let me quote a remark from Prof. John G.
McKendrick’s address on Physiology before the British Association
at Glasgow. Discussing the possibility of the hereditary production
of characteristics through the material structure of the ovum, he
estimates that in it there exist 12,000,000,000 biophors, or ultimate
particles of living matter, a sufficient number to account for
hereditary transmission, and observes: “Thus it is conceivable that
vital activities may also be determined by the kind of motion that
takes place in the molecules of that which we speak of as living
matter. It may be different in kind from some of the motions known to
physicists, and it is conceivable that life may be the transmission
to dead matter, the molecules of which have already a special kind of
motion, of a form of motion _sui generis_.”
Now, in the realm of organic beings symmetrical structures—those with a
right and left symmetry—are everywhere in evidence. Granted that four
dimensions exist, the simplest turning produces the image form, and by
a folding-over structures could be produced, duplicated right and left,
just as is the case of symmetry in a plane.
Thus one very general characteristic of the forms of organisms could
be accounted for by the supposition that a four-dimensional motion was
involved in the process of life.
But whether four-dimensional motions correspond in other respects to
the physiologist’s demand for a special kind of motion, or not, I
do not know. Our business is with the evidence for their existence
in physics. For this purpose it is necessary to examine into the
significance of rotation round a plane in the case of extensible and of
fluid matter.
Let us dwell a moment longer on the rotation of a rigid body. Looking
at the cube in fig. 3, which turns about the face of ABFE, we see that
any line in the face can take the place of the vertical and horizontal
lines we have examined. Take the diagonal line AF and the section
through it to GH. The portions of matter which were on one side of AF
in this section in fig. 3 are on the opposite side of it in fig. 8.
They have gone round the line AF. Thus the rotation round a face can be
considered as a number of rotations of sections round parallel lines in
it.
The turning about two different lines is impossible in
three-dimensional space. To take another illustration, suppose A and
B are two parallel lines in the _xy_ plane, and let CD and EF be two
rods crossing them. Now, in the space of _xyz_ if the rods turn round
the lines A and B in the same direction they will make two independent
circles.
When the end F is going down the end C will be coming up. They will
meet and conflict.
[Illustration: Fig. 9 (137).]
But if we rotate the rods about the plane of AB by the _z_ to _w_
rotation these movements will not conflict. Suppose all the figure
removed with the exception of the plane _xz_, and from this plane draw
the axis of _w_, so that we are looking at the space of _xzw_.
Here, fig. 10, we cannot see the lines A and B. We see the points G and
H, in which A and B intercept the _x_ axis, but we cannot see the lines
themselves, for they run in the _y_ direction, and that is not in our
drawing.
Now, if the rods move with the _z_ to _w_ rotation they will turn in
parallel planes, keeping their relative positions. The point D, for
instance, will describe a circle. At one time it will be above the line
A, at another time below it. Hence it rotates round A.
[Illustration: Fig. 10 (138).]
Not only two rods but any number of rods crossing the plane will move
round it harmoniously. We can think of this rotation by supposing the
rods standing up from one line to move round that line and remembering
that it is not inconsistent with this rotation for the rods standing up
along another line also to move round it, the relative positions of all
the rods being preserved. Now, if the rods are thick together, they may
represent a disk of matter, and we see that a disk of matter can rotate
round a central plane.
Rotation round a plane is exactly analogous to rotation round an axis
in three dimensions. If we want a rod to turn round, the ends must be
free; so if we want a disk of matter to turn round its central plane
by a four-dimensional turning, all the contour must be free. The whole
contour corresponds to the ends of the rod. Each point of the contour
can be looked on as the extremity of an axis in the body, round each
point of which there is a rotation of the matter in the disk.
If the one end of a rod be clamped, we can twist the rod, but not turn
it round; so if any part of the contour of a disk is clamped we can
impart a twist to the disk, but not turn it round its central plane. In
the case of extensible materials a long, thin rod will twist round its
axis, even when the axis is curved, as, for instance, in the case of a
ring of India rubber.
In an analogous manner, in four dimensions we can have rotation round
a curved plane, if I may use the expression. A sphere can be turned
inside out in four dimensions.
[Illustration: Fig. 11 (139).]
Let fig. 11 represent a spherical surface, on each side of which a
layer of matter exists. The thickness of the matter is represented by
the rods CD and EF, extending equally without and within.
[Illustration: Fig. 12 (140).]
Now, take the section of the sphere by the _yz_ plane we have a
circle—fig. 12. Now, let the _w_ axis be drawn in place of the _x_ axis
so that we have the space of _yzw_ represented. In this space all that
there will be seen of the sphere is the circle drawn.
Here we see that there is no obstacle to prevent the rods turning
round. If the matter is so elastic that it will give enough for the
particles at E and C to be separated as they are at F and D, they
can rotate round to the position D and F, and a similar motion is
possible for all other particles. There is no matter or obstacle to
prevent them from moving out in the _w_ direction, and then on round
the circumference as an axis. Now, what will hold for one section will
hold for all, as the fourth dimension is at right angles to all the
sections which can be made of the sphere.
We have supposed the matter of which the sphere is composed to be
three-dimensional. If the matter had a small thickness in the fourth
dimension, there would be a slight thickness in fig. 12 above the
plane of the paper—a thickness equal to the thickness of the matter
in the fourth dimension. The rods would have to be replaced by thin
slabs. But this would make no difference as to the possibility of the
rotation. This motion is discussed by Newcomb in the first volume of
the _American Journal of Mathematics_.
Let us now consider, not a merely extensible body, but a liquid one. A
mass of rotating liquid, a whirl, eddy, or vortex, has many remarkable
properties. On first consideration we should expect the rotating mass
of liquid immediately to spread off and lose itself in the surrounding
liquid. The water flies off a wheel whirled round, and we should expect
the rotating liquid to be dispersed. But see the eddies in a river
strangely persistent. The rings that occur in puffs of smoke and last
so long are whirls or vortices curved round so that their opposite ends
join together. A cyclone will travel over great distances.
Helmholtz was the first to investigate the properties of vortices.
He studied them as they would occur in a perfect fluid—that is, one
without friction of one moving portion or another. In such a medium
vortices would be indestructible. They would go on for ever, altering
their shape, but consisting always of the same portion of the fluid.
But a straight vortex could not exist surrounded entirely by the fluid.
The ends of a vortex must reach to some boundary inside or outside the
fluid.
A vortex which is bent round so that its opposite ends join is capable
of existing, but no vortex has a free end in the fluid. The fluid
round the vortex is always in motion, and one produces a definite
movement in another.
Lord Kelvin has proposed the hypothesis that portions of a fluid
segregated in vortices account for the origin of matter. The properties
of the ether in respect of its capacity of propagating disturbances
can be explained by the assumption of vortices in it instead of by a
property of rigidity. It is difficult to conceive, however, of any
arrangement of the vortex rings and endless vortex filaments in the
ether.
Now, the further consideration of four-dimensional rotations shows the
existence of a kind of vortex which would make an ether filled with a
homogeneous vortex motion easily thinkable.
To understand the nature of this vortex, we must go on and take a
step by which we accept the full significance of the four-dimensional
hypothesis. Granted four-dimensional axes, we have seen that a rotation
of one into another leaves two unaltered, and these two form the axial
plane about which the rotation takes place. But what about these two?
Do they necessarily remain motionless? There is nothing to prevent a
rotation of these two, one into the other, taking place concurrently
with the first rotation. This possibility of a double rotation deserves
the most careful attention, for it is the kind of movement which is
distinctly typical of four dimensions.
Rotation round a plane is analogous to rotation round an axis. But in
three-dimensional space there is no motion analogous to the double
rotation, in which, while axis 1 changes into axis 2, axis 3 changes
into axis 4.
Consider a four-dimensional body, with four independent axes, _x_,
_y_, _z_, _w_. A point in it can move in only one direction at a given
moment. If the body has a velocity of rotation by which the _x_ axis
changes into the _y_ axis and all parallel sections move in a similar
manner, then the point will describe a circle. If, now, in addition
to the rotation by which the _x_ axis changes into the _y_ axis the
body has a rotation by which the _z_ axis turns into the _w_ axis, the
point in question will have a double motion in consequence of the two
turnings. The motions will compound, and the point will describe a
circle, but not the same circle which it would describe in virtue of
either rotation separately.
We know that if a body in three-dimensional space is given two
movements of rotation they will combine into a single movement of
rotation round a definite axis. It is in no different condition
from that in which it is subjected to one movement of rotation. The
direction of the axis changes; that is all. The same is not true about
a four-dimensional body. The two rotations, _x_ to _y_ and _z_ to _w_,
are independent. A body subject to the two is in a totally different
condition to that which it is in when subject to one only. When subject
to a rotation such as that of _x_ to _y_, a whole plane in the body,
as we have seen, is stationary. When subject to the double rotation
no part of the body is stationary except the point common to the two
planes of rotation.
If the two rotations are equal in velocity, every point in the body
describes a circle. All points equally distant from the stationary
point describe circles of equal size.
We can represent a four-dimensional sphere by means of two diagrams,
in one of which we take the three axes, _x_, _y_, _z_; in the
other the axes _x_, _w_, and _z_. In fig. 13 we have the view of a
four-dimensional sphere in the space of _xyz_. Fig. 13 shows all that
we can see of the four sphere in the space of _xyz_, for it represents
all the points in that space, which are at an equal distance from the
centre.
Let us now take the _xz_ section, and let the axis of _w_ take the
place of the _y_ axis. Here, in fig. 14, we have the space of _xzw_.
In this space we have to take all the points which are at the same
distance from the centre, consequently we have another sphere. If we
had a three-dimensional sphere, as has been shown before, we should
have merely a circle in the _xzw_ space, the _xz_ circle seen in the
space of _xzw_. But now, taking the view in the space of _xzw_, we have
a sphere in that space also. In a similar manner, whichever set of
three axes we take, we obtain a sphere.
[Illustration: _Showing axes xyz_
Fig. 13 (141).]
[Illustration: _Showing axes xwz_
Fig. 14 (142).]
In fig. 13, let us imagine the rotation in the direction _xy_ to be
taking place. The point _x_ will turn to _y_, and _p_ to _p´_. The axis
_zz´_ remains stationary, and this axis is all of the plane _zw_ which
we can see in the space section exhibited in the figure.
In fig. 14, imagine the rotation from _z_ to _w_ to be taking place.
The _w_ axis now occupies the position previously occupied by the _y_
axis. This does not mean that the _w_ axis can coincide with the _y_
axis. It indicates that we are looking at the four-dimensional sphere
from a different point of view. Any three-space view will show us three
axes, and in fig. 14 we are looking at _xzw_.
The only part that is identical in the two diagrams is the circle of
the _x_ and _z_ axes, which axes are contained in both diagrams. Thus
the plane _zxz´_ is the same in both, and the point _p_ represents the
same point in both diagrams. Now, in fig. 14 let the _zw_ rotation
take place, the _z_ axis will turn toward the point _w_ of the _w_
axis, and the point _p_ will move in a circle about the point _x_.
Thus in fig. 13 the point _p_ moves in a circle parallel to the _xy_
plane; in fig. 14 it moves in a circle parallel to the _zw_ plane,
indicated by the arrow.
Now, suppose both of these independent rotations compounded, the point
_p_ will move in a circle, but this circle will coincide with neither
of the circles in which either one of the rotations will take it. The
circle the point _p_ will move in will depend on its position on the
surface of the four sphere.
In this double rotation, possible in four-dimensional space, there
is a kind of movement totally unlike any with which we are familiar
in three-dimensional space. It is a requisite preliminary to the
discussion of the behaviour of the small particles of matter,
with a view to determining whether they show the characteristics
of four-dimensional movements, to become familiar with the main
characteristics of this double rotation. And here I must rely on a
formal and logical assent rather than on the intuitive apprehension,
which can only be obtained by a more detailed study.
In the first place this double rotation consists in two varieties or
kinds, which we will call the A and B kinds. Consider four axes, _x_,
_y_, _z_, _w_. The rotation of _x_ to _y_ can be accompanied with the
rotation of _z_ to _w_. Call this the A kind.
But also the rotation of _x_ to _y_ can be accompanied by the rotation,
of not _z_ to _w_, but _w_ to _z_. Call this the B kind.
They differ in only one of the component rotations. One is not the
negative of the other. It is the semi-negative. The opposite of an
_x_ to _y_, _z_ to _w_ rotation would be _y_ to _x_, _w_ to _z_. The
semi-negative is _x_ to _y_ and _w_ to _z_.
If four dimensions exist and we cannot perceive them, because the
extension of matter is so small in the fourth dimension that all
movements are withheld from direct observation except those which are
three-dimensional, we should not observe these double rotations, but
only the effects of them in three-dimensional movements of the type
with which we are familiar.
If matter in its small particles is four-dimensional, we should expect
this double rotation to be a universal characteristic of the atoms
and molecules, for no portion of matter is at rest. The consequences
of this corpuscular motion can be perceived, but only under the form
of ordinary rotation or displacement. Thus, if the theory of four
dimensions is true, we have in the corpuscles of matter a whole world
of movement, which we can never study directly, but only by means of
inference.
The rotation A, as I have defined it, consists of two equal
rotations—one about the plane of _zw_, the other about the plane
of _xy_. It is evident that these rotations are not necessarily
equal. A body may be moving with a double rotation, in which these
two independent components are not equal; but in such a case we can
consider the body to be moving with a composite rotation—a rotation of
the A or B kind and, in addition, a rotation about a plane.
If we combine an A and a B movement, we obtain a rotation about a
plane; for, the first being _x_ to _y_ and _z_ to _w_, and the second
being _x_ to _y_ and _w_ to _z_, when they are put together the _z_
to _w_ and _w_ to _z_ rotations neutralise each other, and we obtain
an _x_ to _y_ rotation only, which is a rotation about the plane of
_zw_. Similarly, if we take a B rotation, _y_ to _x_ and _z_ to _w_,
we get, on combining this with the A rotation, a rotation of _z_ to
_w_ about the _xy_ plane. In this case the plane of rotation is in the
three-dimensional space of _xyz_, and we have—what has been described
before—a twisting about a plane in our space.
Consider now a portion of a perfect liquid having an A motion. It
can be proved that it possesses the properties of a vortex. It
forms a permanent individuality—a separated-out portion of the
liquid—accompanied by a motion of the surrounding liquid. It has
properties analogous to those of a vortex filament. But it is not
necessary for its existence that its ends should reach the boundary of
the liquid. It is self-contained and, unless disturbed, is circular in
every section.
[Illustration: Fig. 15 (143).]
If we suppose the ether to have its properties of transmitting
vibration given it by such vortices, we must inquire how they lie
together in four-dimensional space. Placing a circular disk on a plane
and surrounding it by six others, we find that if the central one is
given a motion of rotation, it imparts to the others a rotation which
is antagonistic in every two adjacent ones. If A goes round, as shown
by the arrow, B and C will be moving in opposite ways, and each tends
to destroy the motion of the other.
Now, if we suppose spheres to be arranged in a corresponding manner
in three-dimensional space, they will be grouped in figures which are
for three-dimensional space what hexagons are for plane space. If a
number of spheres of soft clay be pressed together, so as to fill up
the interstices, each will assume the form of a fourteen-sided figure
called a tetrakaidecagon.
Now, assuming space to be filled with such tetrakaidecagons, and
placing a sphere in each, it will be found that one sphere is touched
by eight others. The remaining six spheres of the fourteen which
surround the central one will not touch it, but will touch three of
those in contact with it. Hence, if the central sphere rotates, it
will not necessarily drive those around it so that their motions will
be antagonistic to each other, but the velocities will not arrange
themselves in a systematic manner.
In four-dimensional space the figure which forms the next term of the
series hexagon, tetrakaidecagon, is a thirty-sided figure. It has for
its faces ten solid tetrakaidecagons and twenty hexagonal prisms. Such
figures will exactly fill four-dimensional space, five of them meeting
at every point. If, now, in each of these figures we suppose a solid
four-dimensional sphere to be placed, any one sphere is surrounded by
thirty others. Of these it touches ten, and, if it rotates, it drives
the rest by means of these. Now, if we imagine the central sphere to be
given an A or a B rotation, it will turn the whole mass of sphere round
in a systematic manner. Suppose four-dimensional space to be filled
with such spheres, each rotating with a double rotation, the whole mass
would form one consistent system of motion, in which each one drove
every other one, with no friction or lagging behind.
Every sphere would have the same kind of rotation. In three-dimensional
space, if one body drives another round the second body rotates
with the opposite kind of rotation; but in four-dimensional space
these four-dimensional spheres would each have the double negative
of the rotation of the one next it, and we have seen that the
double negative of an A or B rotation is still an A or B rotation.
Thus four-dimensional space could be filled with a system of
self-preservative living energy. If we imagine the four-dimensional
spheres to be of liquid and not of solid matter, then, even if the
liquid were not quite perfect and there were a slight retarding effect
of one vortex on another, the system would still maintain itself.
In this hypothesis we must look on the ether as possessing energy,
and its transmission of vibrations, not as the conveying of a motion
imparted from without, but as a modification of its own motion.
We are now in possession of some of the conceptions of four-dimensional
mechanics, and will turn aside from the line of their development
to inquire if there is any evidence of their applicability to the
processes of nature.
Is there any mode of motion in the region of the minute which, giving
three-dimensional movements for its effect, still in itself escapes the
grasp of our mechanical theories? I would point to electricity. Through
the labours of Faraday and Maxwell we are convinced that the phenomena
of electricity are of the nature of the stress and strain of a medium;
but there is still a gap to be bridged over in their explanation—the
laws of elasticity, which Maxwell assumes, are not those of ordinary
matter. And, to take another instance: a magnetic pole in the
neighbourhood of a current tends to move. Maxwell has shown that the
pressures on it are analogous to the velocities in a liquid which would
exist if a vortex took the place of the electric current: but we cannot
point out the definite mechanical explanation of these pressures. There
must be some mode of motion of a body or of the medium in virtue of
which a body is said to be electrified.
Take the ions which convey charges of electricity 500 times greater in
proportion to their mass than are carried by the molecules of hydrogen
in electrolysis. In respect of what motion can these ions be said to
be electrified? It can be shown that the energy they possess is not
energy of rotation. Think of a short rod rotating. If it is turned
over it is found to be rotating in the opposite direction. Now, if
rotation in one direction corresponds to positive electricity, rotation
in the opposite direction corresponds to negative electricity, and the
smallest electrified particles would have their charges reversed by
being turned over—an absurd supposition.
If we fix on a mode of motion as a definition of electricity, we must
have two varieties of it, one for positive and one for negative; and a
body possessing the one kind must not become possessed of the other by
any change in its position.
All three-dimensional motions are compounded of rotations and
translations, and none of them satisfy this first condition for serving
as a definition of electricity.
But consider the double rotation of the A and B kinds. A body rotating
with the A motion cannot have its motion transformed into the B kind
by being turned over in any way. Suppose a body has the rotation _x_
to _y_ and _z_ to _w_. Turning it about the _xy_ plane, we reverse the
direction of the motion _x_ to _y_. But we also reverse the _z_ to _w_
motion, for the point at the extremity of the positive _z_ axis is
now at the extremity of the negative _z_ axis, and since we have not
interfered with its motion it goes in the direction of position _w_.
Hence we have _y_ to _x_ and _w_ to _z_, which is the same as _x_ to
_y_ and _z_ to _w_. Thus both components are reversed, and there is the
A motion over again. The B kind is the semi-negative, with only one
component reversed.
Hence a system of molecules with the A motion would not destroy it in
one another, and would impart it to a body in contact with them. Thus A
and B motions possess the first requisite which must be demanded in any
mode of motion representative of electricity.
Let us trace out the consequences of defining positive electricity as
an A motion and negative electricity as a B motion. The combination of
positive and negative electricity produces a current. Imagine a vortex
in the ether of the A kind and unite with this one of the B kind. An
A motion and B motion produce rotation round a plane, which is in the
ether a vortex round an axial surface. It is a vortex of the kind we
represent as a part of a sphere turning inside out. Now such a vortex
must have its rim on a boundary of the ether—on a body in the ether.
Let us suppose that a conductor is a body which has the property of
serving as the terminal abutment of such a vortex. Then the conception
we must form of a closed current is of a vortex sheet having its edge
along the circuit of the conducting wire. The whole wire will then be
like the centres on which a spindle turns in three-dimensional space,
and any interruption of the continuity of the wire will produce a
tension in place of a continuous revolution.
As the direction of the rotation of the vortex is from a three-space
direction into the fourth dimension and back again, there will be no
direction of flow to the current; but it will have two sides, according
to whether _z_ goes to _w_ or _z_ goes to negative _w_.
We can draw any line from one part of the circuit to another; then the
ether along that line is rotating round its points.
This geometric image corresponds to the definition of an electric
circuit. It is known that the action does not lie in the wire, but in
the medium, and it is known that there is no direction of flow in the
wire.
No explanation has been offered in three-dimensional mechanics of how
an action can be impressed throughout a region and yet necessarily
run itself out along a closed boundary, as is the case in an electric
current. But this phenomenon corresponds exactly to the definition of a
four-dimensional vortex.
If we take a very long magnet, so long that one of its poles is
practically isolated, and put this pole in the vicinity of an electric
circuit, we find that it moves.
Now, assuming for the sake of simplicity that the wire which determines
the current is in the form of a circle, if we take a number of small
magnets and place them all pointing in the same direction normal to
the plane of the circle, so that they fill it and the wire binds them
round, we find that this sheet of magnets has the same effect on
the magnetic pole that the current has. The sheet of magnets may be
curved, but the edge of it must coincide with the wire. The collection
of magnets is then equivalent to the vortex sheet, and an elementary
magnet to a part of it. Thus, we must think of a magnet as conditioning
a rotation in the ether round the plane which bisects at right angles
the line joining its poles.
If a current is started in a circuit, we must imagine vortices like
bowls turning themselves inside out, starting from the contour. In
reaching a parallel circuit, if the vortex sheet were interrupted and
joined momentarily to the second circuit by a free rim, the axis plane
would lie between the two circuits, and a point on the second circuit
opposite a point on the first would correspond to a point opposite
to it on the first; hence we should expect a current in the opposite
direction in the second circuit. Thus the phenomena of induction are
not inconsistent with the hypothesis of a vortex about an axial plane.
In four-dimensional space, in which all four dimensions were
commensurable, the intensity of the action transmitted by the medium
would vary inversely as the cube of the distance. Now, the action of
a current on a magnetic pole varies inversely as the square of the
distance; hence, over measurable distances the extension of the ether
in the fourth dimension cannot be assumed as other than small in
comparison with those distances.
If we suppose the ether to be filled with vortices in the shape of
four-dimensional spheres rotating with the A motion, the B motion would
correspond to electricity in the one-fluid theory. There would thus
be a possibility of electricity existing in two forms, statically,
by itself, and, combined with the universal motion, in the form of a
current.
To arrive at a definite conclusion it will be necessary to investigate
the resultant pressures which accompany the collocation of solid
vortices with surface ones.
To recapitulate:
The movements and mechanics of four-dimensional space are definite and
intelligible. A vortex with a surface as its axis affords a geometric
image of a closed circuit, and there are rotations which by their
polarity afford a possible definition of statical electricity.[7]
[7] These double rotations of the A and B kinds I should like to call
Hamiltons and co-Hamiltons, for it is a singular fact that in his
“Quaternions” Sir Wm. Rowan Hamilton has given the theory of either
the A or the B kind. They follow the laws of his symbols, I, J, K.
Hamiltons and co-Hamiltons seem to be natural units of geometrical
expression. In the paper in the “Proceedings of the Royal Irish
Academy,” Nov. 1903, already alluded to, I have shown something of the
remarkable facility which is gained in dealing with the composition of
three- and four-dimensional rotations by an alteration in Hamilton’s
notation, which enables his system to be applied to both the A and B
kinds of rotations.
The objection which has been often made to Hamilton’s system, namely,
that it is only under special conditions of application that his
processes give geometrically interpretable results, can be removed, if
we assume that he was really dealing with a four-dimensional motion,
and alter his notation to bring this circumstance into explicit
recognition.
APPENDIX I
THE MODELS
In Chapter XI. a description has been given which will enable any
one to make a set of models illustrative of the tesseract and its
properties. The set here supposed to be employed consists of:—
1. Three sets of twenty-seven cubes each.
2. Twenty-seven slabs.
3. Twelve cubes with points, lines, faces, distinguished by colours,
which will be called the catalogue cubes.
The preparation of the twelve catalogue cubes involves the expenditure
of a considerable amount of time. It is advantageous to use them, but
they can be replaced by the drawing of the views of the tesseract or by
a reference to figs. 103, 104, 105, 106 of the text.
The slabs are coloured like the twenty-seven cubes of the first cubic
block in fig. 101, the one with red, white, yellow axes.
The colours of the three sets of twenty-seven cubes are those of the
cubes shown in fig. 101.
The slabs are used to form the representation of a cube in a plane, and
can well be dispensed with by any one who is accustomed to deal with
solid figures. But the whole theory depends on a careful observation of
how the cube would be represented by these slabs.
In the first step, that of forming a clear idea how a plane being
would represent three-dimensional space, only one of the catalogue
cubes and one of the three blocks is needed.
APPLICATION TO THE STEP FROM PLANE TO SOLID.
Look at fig. 1 of the views of the tesseract, or, what comes to the
same thing, take catalogue cube No. 1 and place it before you with the
red line running up, the white line running to the right, the yellow
line running away. The three dimensions of space are then marked out
by these lines or axes. Now take a piece of cardboard, or a book, and
place it so that it forms a wall extending up and down not opposite to
you, but running away parallel to the wall of the room on your left
hand.
Placing the catalogue cube against this wall we see that it comes into
contact with it by the red and yellow lines, and by the included orange
face.
In the plane being’s world the aspect he has of the cube would be a
square surrounded by red and yellow lines with grey points.
Now, keeping the red line fixed, turn the cube about it so that the
yellow line goes out to the right, and the white line comes into
contact with the plane.
In this case a different aspect is presented to the plane being, a
square, namely, surrounded by red and white lines and grey points. You
should particularly notice that when the yellow line goes out, at right
angles to the plane, and the white comes in, the latter does not run in
the same sense that the yellow did.
From the fixed grey point at the base of the red line the yellow line
ran away from you. The white line now runs towards you. This turning
at right angles makes the line which was out of the plane before, come
into it in an opposite sense to that in which the line ran which has
just left the plane. If the cube does not break through the plane this
is always the rule.
Again turn the cube back to the normal position with red running up,
white to the right, and yellow away, and try another turning.
You can keep the yellow line fixed, and turn the cube about it. In this
case the red line going out to the right the white line will come in
pointing downwards.
You will be obliged to elevate the cube from the table in order to
carry out this turning. It is always necessary when a vertical axis
goes out of a space to imagine a movable support which will allow the
line which ran out before to come in below.
Having looked at the three ways of turning the cube so as to present
different faces to the plane, examine what would be the appearance if
a square hole were cut in the piece of cardboard, and the cube were to
pass through it. A hole can be actually cut, and it will be seen that
in the normal position, with red axis running up, yellow away, and
white to the right, the square first perceived by the plane being—the
one contained by red and yellow lines—would be replaced by another
square of which the line towards you is pink—the section line of the
pink face. The line above is light yellow, below is light yellow and on
the opposite side away from you is pink.
In the same way the cube can be pushed through a square opening in the
plane from any of the positions which you have already turned it into.
In each case the plane being will perceive a different set of contour
lines.
Having observed these facts about the catalogue cube, turn now to the
first block of twenty-seven cubes.
You notice that the colour scheme on the catalogue cube and that of
this set of blocks is the same.
Place them before you, a grey or null cube on the table, above it a
red cube, and on the top a null cube again. Then away from you place a
yellow cube, and beyond it a null cube. Then to the right place a white
cube and beyond it another null. Then complete the block, according to
the scheme of the catalogue cube, putting in the centre of all an ochre
cube.
You have now a cube like that which is described in the text. For the
sake of simplicity, in some cases, this cubic block can be reduced to
one of eight cubes, by leaving out the terminations in each direction.
Thus, instead of null, red, null, three cubes, you can take null, red,
two cubes, and so on.
It is useful, however, to practise the representation in a plane of a
block of twenty-seven cubes. For this purpose take the slabs, and build
them up against the piece of cardboard, or the book in such a way as to
represent the different aspects of the cube.
Proceed as follows:—
First, cube in normal position.
Place nine slabs against the cardboard to represent the nine cubes
in the wall of the red and yellow axes, facing the cardboard; these
represent the aspect of the cube as it touches the plane.
Now push these along the cardboard and make a different set of nine
slabs to represent the appearance which the cube would present to a
plane being, if it were to pass half way through the plane.
There would be a white slab, above it a pink one, above that another
white one, and six others, representing what would be the nature of a
section across the middle of the block of cubes. The section can be
thought of as a thin slice cut out by two parallel cuts across the
cube. Having arranged these nine slabs, push them along the plane, and
make another set of nine to represent what would be the appearance of
the cube when it had almost completely gone through. This set of nine
will be the same as the first set of nine.
Now we have in the plane three sets of nine slabs each, which represent
three sections of the twenty-seven block.
They are put alongside one another. We see that it does not matter in
what order the sets of nine are put. As the cube passes through the
plane they represent appearances which follow the one after the other.
If they were what they represented, they could not exist in the same
plane together.
This is a rather important point, namely, to notice that they should
not co-exist on the plane, and that the order in which they are placed
is indifferent. When we represent a four-dimensional body our solid
cubes are to us in the same position that the slabs are to the plane
being. You should also notice that each of these slabs represents only
the very thinnest slice of a cube. The set of nine slabs first set up
represents the side surface of the block. It is, as it were, a kind
of tray—a beginning from which the solid cube goes off. The slabs
as we use them have thickness, but this thickness is a necessity of
construction. They are to be thought of as merely of the thickness of a
line.
If now the block of cubes passed through the plane at the rate of an
inch a minute the appearance to a plane being would be represented by:—
1. The first set of nine slabs lasting for one minute.
2. The second set of nine slabs lasting for one minute.
3. The third set of nine slabs lasting for one minute.
Now the appearances which the cube would present to the plane being
in other positions can be shown by means of these slabs. The use of
such slabs would be the means by which a plane being could acquire a
familiarity with our cube. Turn the catalogue cube (or imagine the
coloured figure turned) so that the red line runs up, the yellow line
out to the right, and the white line towards you. Then turn the block
of cubes to occupy a similar position.
The block has now a different wall in contact with the plane. Its
appearance to a plane being will not be the same as before. He has,
however, enough slabs to represent this new set of appearances. But he
must remodel his former arrangement of them.
He must take a null, a red, and a null slab from the first of his sets
of slabs, then a white, a pink, and a white from the second, and then a
null, a red, and a null from the third set of slabs.
He takes the first column from the first set, the first column from the
second set, and the first column from the third set.
To represent the half-way-through appearance, which is as if a very
thin slice were cut out half way through the block, he must take the
second column of each of his sets of slabs, and to represent the final
appearance, the third column of each set.
Now turn the catalogue cube back to the normal position, and also the
block of cubes.
There is another turning—a turning about the yellow line, in which the
white axis comes below the support.
You cannot break through the surface of the table, so you must imagine
the old support to be raised. Then the top of the block of cubes in its
new position is at the level at which the base of it was before.
Now representing the appearance on the plane, we must draw a horizontal
line to represent the old base. The line should be drawn three inches
high on the cardboard.
Below this the representative slabs can be arranged.
It is easy to see what they are. The old arrangements have to be
broken up, and the layers taken in order, the first layer of each for
the representation of the aspect of the block as it touches the plane.
Then the second layers will represent the appearance half way through,
and the third layers will represent the final appearance.
It is evident that the slabs individually do not represent the same
portion of the cube in these different presentations.
In the first case each slab represents a section or a face
perpendicular to the white axis, in the second case a face or a section
which runs perpendicularly to the yellow axis, and in the third case a
section or a face perpendicular to the red axis.
But by means of these nine slabs the plane being can represent the
whole of the cubic block. He can touch and handle each portion of the
cubic block, there is no part of it which he cannot observe. Taking it
bit by bit, two axes at a time, he can examine the whole of it.
OUR REPRESENTATION OF A BLOCK OF TESSERACTS.
Look at the views of the tesseract 1, 2, 3, or take the catalogue cubes
1, 2, 3, and place them in front of you, in any order, say running from
left to right, placing 1 in the normal position, the red axis running
up, the white to the right, and yellow away.
Now notice that in catalogue cube 2 the colours of each region are
derived from those of the corresponding region of cube 1 by the
addition of blue. Thus null + blue = blue, and the corners of number 2
are blue. Again, red + blue = purple, and the vertical lines of 2 are
purple. Blue + yellow = green, and the line which runs away is coloured
green.
By means of these observations you may be sure that catalogue cube 2
is rightly placed. Catalogue cube 3 is just like number 1.
Having these cubes in what we may call their normal position, proceed
to build up the three sets of blocks.
This is easily done in accordance with the colour scheme on the
catalogue cubes.
The first block we already know. Build up the second block, beginning
with a blue corner cube, placing a purple on it, and so on.
Having these three blocks we have the means of representing the
appearances of a group of eighty-one tesseracts.
Let us consider a moment what the analogy in the case of the plane
being is.
He has his three sets of nine slabs each. We have our three sets of
twenty-seven cubes each.
Our cubes are like his slabs. As his slabs are not the things which
they represent to him, so our cubes are not the things they represent
to us.
The plane being’s slabs are to him the faces of cubes.
Our cubes then are the faces of tesseracts, the cubes by which they are
in contact with our space.
As each set of slabs in the case of the plane being might be considered
as a sort of tray from which the solid contents of the cubes came out,
so our three blocks of cubes may be considered as three-space trays,
each of which is the beginning of an inch of the solid contents of the
four-dimensional solids starting from them.
We want now to use the names null, red, white, etc., for tesseracts.
The cubes we use are only tesseract faces. Let us denote that fact
by calling the cube of null colour, null face; or, shortly, null f.,
meaning that it is the face of a tesseract.
To determine which face it is let us look at the catalogue cube 1 or
the first of the views of the tesseract, which can be used instead of
the models. It has three axes, red, white, yellow, in our space. Hence
the cube determined by these axes is the face of the tesseract which we
now have before us. It is the ochre face. It is enough, however, simply
to say null f., red f. for the cubes which we use.
To impress this in your mind, imagine that tesseracts do actually run
from each cube. Then, when you move the cubes about, you move the
tesseracts about with them. You move the face but the tesseract follows
with it, as the cube follows when its face is shifted in a plane.
The cube null in the normal position is the cube which has in it the
red, yellow, white axes. It is the face having these, but wanting the
blue. In this way you can define which face it is you are handling. I
will write an “f.” after the name of each tesseract just as the plane
being might call each of his slabs null slab, yellow slab, etc., to
denote that they were representations.
We have then in the first block of twenty-seven cubes, the
following—null f., red f., null f., going up; white f., null f., lying
to the right, and so on. Starting from the null point and travelling
up one inch we are in the null region, the same for the away and the
right-hand directions. And if we were to travel in the fourth dimension
for an inch we should still be in a null region. The tesseract
stretches equally all four ways. Hence the appearance we have in this
first block would do equally well if the tesseract block were to move
across our space for a certain distance. For anything less than an inch
of their transverse motion we should still have the same appearance.
You must notice, however, that we should not have null face after the
motion had begun.
When the tesseract, null for instance, had moved ever so little we
should not have a face of null but a section of null in our space.
Hence, when we think of the motion across our space we must call our
cubes tesseract sections. Thus on null passing across we should see
first null f., then null s., and then, finally, null f. again.
Imagine now the whole first block of twenty-seven tesseracts to have
moved tranverse to our space a distance of one inch. Then the second
set of tesseracts, which originally were an inch distant from our
space, would be ready to come in.
Their colours are shown in the second block of twenty-seven cubes which
you have before you. These represent the tesseract faces of the set of
tesseracts that lay before an inch away from our space. They are ready
now to come in, and we can observe their colours. In the place which
null f. occupied before we have blue f., in place of red f. we have
purple f., and so on. Each tesseract is coloured like the one whose
place it takes in this motion with the addition of blue.
Now if the tesseract block goes on moving at the rate of an inch a
minute, this next set of tesseracts will occupy a minute in passing
across. We shall see, to take the null one for instance, first of all
null face, then null section, then null face again.
At the end of the second minute the second set of tesseracts has gone
through, and the third set comes in. This, as you see, is coloured just
like the first. Altogether, these three sets extend three inches in the
fourth dimension, making the tesseract block of equal magnitude in all
dimensions.
We have now before us a complete catalogue of all the tesseracts in our
group. We have seen them all, and we shall refer to this arrangement
of the blocks as the “normal position.” We have seen as much of each
tesseract at a time as could be done in a three-dimensional space. Each
part of each tesseract has been in our space, and we could have touched
it.
The fourth dimension appeared to us as the duration of the block.
If a bit of our matter were to be subjected to the same motion it
would be instantly removed out of our space. Being thin in the fourth
dimension it is at once taken out of our space by a motion in the
fourth dimension.
But the tesseract block we represent having length in the fourth
dimension remains steadily before our eyes for three minutes, when it
is subjected to this transverse motion.
We have now to form representations of the other views of the same
tesseract group which are possible in our space.
Let us then turn the block of tesseracts so that another face of it
comes into contact with our space, and then by observing what we have,
and what changes come when the block traverses our space, we shall have
another view of it. The dimension which appeared as duration before
will become extension in one of our known dimensions, and a dimension
which coincided with one of our space dimensions will appear as
duration.
Leaving catalogue cube 1 in the normal position, remove the other two,
or suppose them removed. We have in space the red, the yellow, and the
white axes. Let the white axis go out into the unknown, and occupy the
position the blue axis holds. Then the blue axis, which runs in that
direction now will come into space. But it will not come in pointing
in the same way that the white axis does now. It will point in the
opposite sense. It will come in running to the left instead of running
to the right as the white axis does now.
When this turning takes place every part of the cube 1 will disappear
except the left-hand face—the orange face.
And the new cube that appears in our space will run to the left from
this orange face, having axes, red, yellow, blue.
Take models 4, 5, 6. Place 4, or suppose No. 4 of the tesseract views
placed, with its orange face coincident with the orange face of 1, red
line to red line, and yellow line to yellow line, with the blue line
pointing to the left. Then remove cube 1 and we have the tesseract face
which comes in when the white axis runs in the positive unknown, and
the blue axis comes into our space.
Now place catalogue cube 5 in some position, it does not matter which,
say to the left; and place it so that there is a correspondence of
colour corresponding to the colour of the line that runs out of space.
The line that runs out of space is white, hence, every part of this
cube 5 should differ from the corresponding part of 4 by an alteration
in the direction of white.
Thus we have white points in 5 corresponding to the null points in
4. We have a pink line corresponding to a red line, a light yellow
line corresponding to a yellow line, an ochre face corresponding to
an orange face. This cube section is completely named in Chapter XI.
Finally cube 6 is a replica of 1.
These catalogue cubes will enable us to set up our models of the block
of tesseracts.
First of all for the set of tesseracts, which beginning in our space
reach out one inch in the unknown, we have the pattern of catalogue
cube 4.
We see that we can build up a block of twenty-seven tesseract faces
after the colour scheme of cube 4, by taking the left-hand wall of
block 1, then the left-hand wall of block 2, and finally that of block
3. We take, that is, the three first walls of our previous arrangement
to form the first cubic block of this new one.
This will represent the cubic faces by which the group of tesseracts in
its new position touches our space. We have running up, null f., red
f., null f. In the next vertical line, on the side remote from us, we
have yellow f., orange f., yellow f., and then the first colours over
again. Then the three following columns are, blue f., purple f., blue
f.; green f., brown f., green f.; blue f., purple f., blue f. The last
three columns are like the first.
These tesseracts touch our space, and none of them are by any part of
them distant more than an inch from it. What lies beyond them in the
unknown?
This can be told by looking at catalogue cube 5. According to its
scheme of colour we see that the second wall of each of our old
arrangements must be taken. Putting them together we have, as the
corner, white f. above it, pink f. above it, white f. The column next
to this remote from us is as follows:—light yellow f., ochre f., light
yellow f., and beyond this a column like the first. Then for the middle
of the block, light blue f., above it light purple, then light blue.
The centre column has, at the bottom, light green f., light brown f.
in the centre and at the top light green f. The last wall is like the
first.
The third block is made by taking the third walls of our previous
arrangement, which we called the normal one.
You may ask what faces and what sections our cubes represent. To answer
this question look at what axes you have in our space. You have red,
yellow, blue. Now these determine brown. The colours red, yellow, blue
are supposed by us when mixed to produce a brown colour. And that cube
which is determined by the red, yellow, blue axes we call the brown
cube.
When the tesseract block in its new position begins to move across our
space each tesseract in it gives a section in our space. This section
is transverse to the white axis, which now runs in the unknown.
As the tesseract in its present position passes across our space, we
should see first of all the first of the blocks of cubic faces we have
put up—these would last for a minute, then would come the second block
and then the third. At first we should have a cube of tesseract faces,
each of which would be brown. Directly the movement began, we should
have tesseract sections transverse to the white line.
There are two more analogous positions in which the block of tesseracts
can be placed. To find the third position, restore the blocks to the
normal arrangement.
Let us make the yellow axis go out into the positive unknown, and let
the blue axis, consequently, come in running towards us. The yellow ran
away, so the blue will come in running towards us.
Put catalogue cube 1 in its normal position. Take catalogue cube 7
and place it so that its pink face coincides with the pink face of
cube 1, making also its red axis coincide with the red axis of 1 and
its white with the white. Moreover, make cube 7 come towards us from
cube 1. Looking at it we see in our space, red, white, and blue axes.
The yellow runs out. Place catalogue cube 8 in the neighbourhood
of 7—observe that every region in 8 has a change in the direction
of yellow from the corresponding region in 7. This is because it
represents what you come to now in going in the unknown, when the
yellow axis runs out of our space. Finally catalogue cube 9, which is
like number 7, shows the colours of the third set of tesseracts. Now
evidently, starting from the normal position, to make up our three
blocks of tesseract faces we have to take the near wall from the first
block, the near wall from the second, and then the near wall from the
third block. This gives us the cubic block formed by the faces of the
twenty-seven tesseracts which are now immediately touching our space.
Following the colour scheme of catalogue cube 8, we make the next set
of twenty-seven tesseract faces, representing the tesseracts, each of
which begins one inch off from our space, by putting the second walls
of our previous arrangement together, and the representation of the
third set of tesseracts is the cubic block formed of the remaining
three walls.
Since we have red, white, blue axes in our space to begin with, the
cubes we see at first are light purple tesseract faces, and after the
transverse motion begins we have cubic sections transverse to the
yellow line.
Restore the blocks to the normal position, there remains the case in
which the red axis turns out of space. In this case the blue axis will
come in downwards, opposite to the sense in which the red axis ran.
In this case take catalogue cubes 10, 11, 12. Lift up catalogue cube 1
and put 10 underneath it, imagining that it goes down from the previous
position of 1.
We have to keep in space the white and the yellow axes, and let the red
go out, the blue come in.
Now, you will find on cube 10 a light yellow face; this should coincide
with the base of 1, and the white and yellow lines on the two cubes
should coincide. Then the blue axis running down you have the catalogue
cube correctly placed, and it forms a guide for putting up the first
representative block.
Catalogue cube 11 will represent what lies in the fourth dimension—now
the red line runs in the fourth dimension. Thus the change from 10 to
11 should be towards red, corresponding to a null point is a red point,
to a white line is a pink line, to a yellow line an orange line, and so
on.
Catalogue cube 12 is like 10. Hence we see that to build up our blocks
of tesseract faces we must take the bottom layer of the first block,
hold that up in the air, underneath it place the bottom layer of the
second block, and finally underneath this last the bottom layer of the
last of our normal blocks.
Similarly we make the second representative group by taking the middle
courses of our three blocks. The last is made by taking the three
topmost layers. The three axes in our space before the transverse
motion begins are blue, white, yellow, so we have light green tesseract
faces, and after the motion begins sections transverse to the red light.
These three blocks represent the appearances as the tesseract group in
its new position passes across our space. The cubes of contact in this
case are those determinal by the three axes in our space, namely, the
white, the yellow, the blue. Hence they are light green.
It follows from this that light green is the interior cube of the first
block of representative cubic faces.
Practice in the manipulations described, with a realization in each
case of the face or section which is in our space, is one of the best
means of a thorough comprehension of the subject.
We have to learn how to get any part of these four-dimensional figures
into space, so that we can look at them. We must first learn to swing a
tesseract, and a group of tesseracts about in any way.
When these operations have been repeated and the method of arrangement
of the set of blocks has become familiar, it is a good plan to rotate
the axes of the normal cube 1 about a diagonal, and then repeat the
whole series of turnings.
Thus, in the normal position, red goes up, white to the right, yellow
away. Make white go up, yellow to the right, and red away. Learn the
cube in this position by putting up the set of blocks of the normal
cube, over and over again till it becomes as familiar to you as in the
normal position. Then when this is learned, and the corresponding
changes in the arrangements of the tesseract groups are made, another
change should be made: let, in the normal cube, yellow go up, red to
the right, and white away.
Learn the normal block of cubes in this new position by arranging them
and re-arranging them till you know without thought where each one
goes. Then carry out all the tesseract arrangements and turnings.
If you want to understand the subject, but do not see your way clearly,
if it does not seem natural and easy to you, practise these turnings.
Practise, first of all, the turning of a block of cubes round, so that
you know it in every position as well as in the normal one. Practise by
gradually putting up the set of cubes in their new arrangements. Then
put up the tesseract blocks in their arrangements. This will give you
a working conception of higher space, you will gain the feeling of it,
whether you take up the mathematical treatment of it or not.
APPENDIX II
A LANGUAGE OF SPACE
The mere naming the parts of the figures we consider involves a certain
amount of time and attention. This time and attention leads to no
result, for with each new figure the nomenclature applied is completely
changed, every letter or symbol is used in a different significance.
Surely it must be possible in some way to utilise the labour thus at
present wasted!
Why should we not make a language for space itself, so that every
position we want to refer to would have its own name? Then every time
we named a figure in order to demonstrate its properties we should be
exercising ourselves in the vocabulary of place.
If we use a definite system of names, and always refer to the same
space position by the same name, we create as it were a multitude of
little hands, each prepared to grasp a special point, position, or
element, and hold it for us in its proper relations.
We make, to use another analogy, a kind of mental paper, which has
somewhat of the properties of a sensitive plate, in that it will
register, without effort, complex, visual, or tactual impressions.
But of far more importance than the applications of a space language to
the plane and to solid space is the facilitation it brings with it to
the study of four-dimensional shapes.
I have delayed introducing a space language because all the systems I
made turned out, after giving them a fair trial, to be intolerable. I
have now come upon one which seems to present features of permanence,
and I will here give an outline of it, so that it can be applied to the
subject of the text, and in order that it may be subjected to criticism.
The principle on which the language is constructed is to sacrifice
every other consideration for brevity.
It is indeed curious that we are able to talk and converse on every
subject of thought except the fundamental one of space. The only way of
speaking about the spatial configurations that underlie every subject
of discursive thought is a co-ordinate system of numbers. This is so
awkward and incommodious that it is never used. In thinking also, in
realising shapes, we do not use it; we confine ourselves to a direct
visualisation.
Now, the use of words corresponds to the storing up of our experience
in a definite brain structure. A child, in the endless tactual, visual,
mental manipulations it makes for itself, is best left to itself, but
in the course of instruction the introduction of space names would
make the teachers work more cumulative, and the child’s knowledge more
social.
Their full use can only be appreciated, if they are introduced early
in the course of education; but in a minor degree any one can convince
himself of their utility, especially in our immediate subject of
handling four-dimensional shapes. The sum total of the results obtained
in the preceding pages can be compendiously and accurately expressed in
nine words of the Space Language.
In one of Plato’s dialogues Socrates makes an experiment on a slave boy
standing by. He makes certain perceptions of space awake in the mind
of Meno’s slave by directing his close attention on some simple facts
of geometry.
By means of a few words and some simple forms we can repeat Plato’s
experiment on new ground.
Do we by directing our close attention on the facts of four dimensions
awaken a latent faculty in ourselves? The old experiment of Plato’s, it
seems to me, has come down to us as novel as on the day he incepted it,
and its significance not better understood through all the discussion
of which it has been the subject.
Imagine a voiceless people living in a region where everything had
a velvety surface, and who were thus deprived of all opportunity of
experiencing what sound is. They could observe the slow pulsations
of the air caused by their movements, and arguing from analogy, they
would no doubt infer that more rapid vibrations were possible. From
the theoretical side they could determine all about these more rapid
vibrations. They merely differ, they would say, from slower ones,
by the number that occur in a given time; there is a merely formal
difference.
But suppose they were to take the trouble, go to the pains of producing
these more rapid vibrations, then a totally new sensation would fall
on their rudimentary ears. Probably at first they would only be dimly
conscious of Sound, but even from the first they would become aware
that a merely formal difference, a mere difference in point of number
in this particular respect, made a great difference practically, as
related to them. And to us the difference between three and four
dimensions is merely formal, numerical. We can tell formally all about
four dimensions, calculate the relations that would exist. But that
the difference is merely formal does not prove that it is a futile and
empty task, to present to ourselves as closely as we can the phenomena
of four dimensions. In our formal knowledge of it, the whole question
of its actual relation to us, as we are, is left in abeyance.
Possibly a new apprehension of nature may come to us through the
practical, as distinguished from the mathematical and formal, study
of four dimensions. As a child handles and examines the objects with
which he comes in contact, so we can mentally handle and examine
four-dimensional objects. The point to be determined is this. Do we
find something cognate and natural to our faculties, or are we merely
building up an artificial presentation of a scheme only formally
possible, conceivable, but which has no real connection with any
existing or possible experience?
This, it seems to me, is a question which can only be settled by
actually trying. This practical attempt is the logical and direct
continuation of the experiment Plato devised in the “Meno.”
Why do we think true? Why, by our processes of thought, can we predict
what will happen, and correctly conjecture the constitution of the
things around us? This is a problem which every modern philosopher has
considered, and of which Descartes, Leibnitz, Kant, to name a few,
have given memorable solutions. Plato was the first to suggest it.
And as he had the unique position of being the first devisor of the
problem, so his solution is the most unique. Later philosophers have
talked about consciousness and its laws, sensations, categories. But
Plato never used such words. Consciousness apart from a conscious being
meant nothing to him. His was always an objective search. He made man’s
intuitions the basis of a new kind of natural history.
In a few simple words Plato puts us in an attitude with regard to
psychic phenomena—the mind—the ego—“what we are,” which is analogous
to the attitude scientific men of the present day have with regard
to the phenomena of outward nature. Behind this first apprehension
of ours of nature, there is an infinite depth to be learned and
known. Plato said that behind the phenomena of mind that Meno’s slave
boy exhibited, there was a vast, an infinite perspective. And his
singularity, his originality, comes out most strongly marked in this,
that the perspective, the complex phenomena beyond were, according to
him, phenomena of personal experience. A footprint in the sand means a
man to a being that has the conception of a man. But to a creature that
has no such conception, it means a curious mark, somehow resulting from
the concatenation of ordinary occurrences. Such a being would attempt
merely to explain how causes known to him could so coincide as to
produce such a result; he would not recognise its significance.
Plato introduced the conception which made a new kind of natural
history possible. He said that Meno’s slave boy thought true about
things he had never learned, because his “soul” had experience. I
know this will sound absurd to some people, and it flies straight in
the face of the maxim, that explanation consists in showing how an
effect depends on simple causes. But what a mistaken maxim that is!
Can any single instance be shown of a simple cause? Take the behaviour
of spheres for instance; say those ivory spheres, billiard balls,
for example. We can explain their behaviour by supposing they are
homogeneous elastic solids. We can give formulæ which will account for
their movements in every variety. But are they homogeneous elastic
solids? No, certainly not. They are complex in physical and molecular
structure, and atoms and ions beyond open an endless vista. Our simple
explanation is false, false as it can be. The balls act as if they
were homogeneous elastic spheres. There is a statistical simplicity in
the resultant of very complex conditions, which makes that artificial
conception useful. But its usefulness must not blind us to the fact
that it is artificial. If we really look deep into nature, we find a
much greater complexity than we at first suspect. And so behind this
simple “I,” this myself, is there not a parallel complexity? Plato’s
“soul” would be quite acceptable to a large class of thinkers, if by
“soul” and the complexity he attributes to it, he meant the product of
a long course of evolutionary changes, whereby simple forms of living
matter endowed with rudimentary sensation had gradually developed into
fully conscious beings.
But Plato does not mean by “soul” a being of such a kind. His soul is
a being whose faculties are clogged by its bodily environment, or at
least hampered by the difficulty of directing its bodily frame—a being
which is essentially higher than the account it gives of itself through
its organs. At the same time Plato’s soul is not incorporeal. It is a
real being with a real experience. The question of whether Plato had
the conception of non-spatial existence has been much discussed. The
verdict is, I believe, that even his “ideas” were conceived by him as
beings in space, or, as we should say, real. Plato’s attitude is that
of Science, inasmuch as he thinks of a world in Space. But, granting
this, it cannot be denied that there is a fundamental divergence
between Plato’s conception and the evolutionary theory, and also an
absolute divergence between his conception and the genetic account of
the origin of the human faculties. The functions and capacities of
Plato’s “soul” are not derived by the interaction of the body and its
environment.
Plato was engaged on a variety of problems, and his religious and
ethical thoughts were so keen and fertile that the experimental
investigation of his soul appears involved with many other motives.
In one passage Plato will combine matter of thought of all kinds and
from all sources, overlapping, interrunning. And in no case is he more
involved and rich than in this question of the soul. In fact, I wish
there were two words, one denoting that being, corporeal and real, but
with higher faculties than we manifest in our bodily actions, which is
to be taken as the subject of experimental investigation; and the other
word denoting “soul” in the sense in which it is made the recipient and
the promise of so much that men desire. It is the soul in the former
sense that I wish to investigate, and in a limited sphere only. I wish
to find out, in continuation of the experiment in the Meno, what the
“soul” in us thinks about extension, experimenting on the grounds laid
down by Plato. He made, to state the matter briefly, the hypothesis
with regard to the thinking power of a being in us, a “soul.” This
soul is not accessible to observation by sight or touch, but it can be
observed by its functions; it is the object of a new kind of natural
history, the materials for constructing which lie in what it is natural
to us to think. With Plato “thought” was a very wide-reaching term, but
still I would claim in his general plan of procedure a place for the
particular question of extension.
The problem comes to be, “What is it natural to us to think about
matter _qua_ extended?”
First of all, I find that the ordinary intuition of any simple object
is extremely imperfect. Take a block of differently marked cubes, for
instance, and become acquainted with them in their positions. You may
think you know them quite well, but when you turn them round—rotate
the block round a diagonal, for instance—you will find that you have
lost track of the individuals in their new positions. You can mentally
construct the block in its new position, by a rule, by taking the
remembered sequences, but you don’t know it intuitively. By observation
of a block of cubes in various positions, and very expeditiously
by a use of Space names applied to the cubes in their different
presentations, it is possible to get an intuitive knowledge of the
block of cubes, which is not disturbed by any displacement. Now, with
regard to this intuition, we moderns would say that I had formed it by
my tactual visual experiences (aided by hereditary pre-disposition).
Plato would say that the soul had been stimulated to recognise an
instance of shape which it knew. Plato would consider the operation
of learning merely as a stimulus; we as completely accounting for
the result. The latter is the more common-sense view. But, on the
other hand, it presupposes the generation of experience from physical
changes. The world of sentient experience, according to the modern
view, is closed and limited; only the physical world is ample and large
and of ever-to-be-discovered complexity. Plato’s world of soul, on the
other hand, is at least as large and ample as the world of things.
Let us now try a crucial experiment. Can I form an intuition of a
four-dimensional object? Such an object is not given in the physical
range of my sense contacts. All I can do is to present to myself the
sequences of solids, which would mean the presentation to me under my
conditions of a four-dimensional object. All I can do is to visualise
and tactualise different series of solids which are alternative sets of
sectional views of a four-dimensional shape.
If now, on presenting these sequences, I find a power in me of
intuitively passing from one of these sets of sequences to another, of,
being given one, intuitively constructing another, not using a rule,
but directly apprehending it, then I have found a new fact about my
soul, that it has a four-dimensional experience; I have observed it by
a function it has.
I do not like to speak positively, for I might occasion a loss of time
on the part of others, if, as may very well be, I am mistaken. But for
my own part, I think there are indications of such an intuition; from
the results of my experiments, I adopt the hypothesis that that which
thinks in us has an ample experience, of which the intuitions we use in
dealing with the world of real objects are a part; of which experience,
the intuition of four-dimensional forms and motions is also a part. The
process we are engaged in intellectually is the reading the obscure
signals of our nerves into a world of reality, by means of intuitions
derived from the inner experience.
The image I form is as follows. Imagine the captain of a modern
battle-ship directing its course. He has his charts before him; he
is in communication with his associates and subordinates; can convey
his messages and commands to every part of the ship, and receive
information from the conning-tower and the engine-room. Now suppose the
captain immersed in the problem of the navigation of his ship over the
ocean, to have so absorbed himself in the problem of the direction of
his craft over the plane surface of the sea that he forgets himself.
All that occupies his attention is the kind of movement that his ship
makes. The operations by which that movement is produced have sunk
below the threshold of his consciousness, his own actions, by which
he pushes the buttons, gives the orders, are so familiar as to be
automatic, his mind is on the motion of the ship as a whole. In such a
case we can imagine that he identifies himself with his ship; all that
enters his conscious thought is the direction of its movement over the
plane surface of the ocean.
Such is the relation, as I imagine it, of the soul to the body. A
relation which we can imagine as existing momentarily in the case
of the captain is the normal one in the case of the soul with its
craft. As the captain is capable of a kind of movement, an amplitude
of motion, which does not enter into his thoughts with regard to the
directing the ship over the plane surface of the ocean, so the soul is
capable of a kind of movement, has an amplitude of motion, which is
not used in its task of directing the body in the three-dimensional
region in which the body’s activity lies. If for any reason it became
necessary for the captain to consider three-dimensional motions with
regard to his ship, it would not be difficult for him to gain the
materials for thinking about such motions; all he has to do is to
call his own intimate experience into play. As far as the navigation
of the ship, however, is concerned, he is not obliged to call on
such experience. The ship as a whole simply moves on a surface. The
problem of three-dimensional movement does not ordinarily concern its
steering. And thus with regard to ourselves all those movements and
activities which characterise our bodily organs are three-dimensional;
we never need to consider the ampler movements. But we do more than
use the movements of our body to effect our aims by direct means; we
have now come to the pass when we act indirectly on nature, when we
call processes into play which lie beyond the reach of any explanation
we can give by the kind of thought which has been sufficient for the
steering of our craft as a whole. When we come to the problem of what
goes on in the minute, and apply ourselves to the mechanism of the
minute, we find our habitual conceptions inadequate.
The captain in us must wake up to his own intimate nature, realise
those functions of movement which are his own, and in virtue of his
knowledge of them apprehend how to deal with the problems he has come
to.
Think of the history of man. When has there been a time, in which his
thoughts of form and movement were not exclusively of such varieties as
were adapted for his bodily performance? We have never had a demand to
conceive what our own most intimate powers are. But, just as little as
by immersing himself in the steering of his ship over the plane surface
of the ocean, a captain can lose the faculty of thinking about what he
actually does, so little can the soul lose its own nature. It can be
roused to an intuition that is not derived from the experience which
the senses give. All that is necessary is to present some few of those
appearances which, while inconsistent with three-dimensional matter,
are yet consistent with our formal knowledge of four-dimensional
matter, in order for the soul to wake up and not begin to learn, but of
its own intimate feeling fill up the gaps in the presentiment, grasp
the full orb of possibilities from the isolated points presented to
it. In relation to this question of our perceptions, let me suggest
another illustration, not taking it too seriously, only propounding it
to exhibit the possibilities in a broad and general way.
In the heavens, amongst the multitude of stars, there are some which,
when the telescope is directed on them, seem not to be single stars,
but to be split up into two. Regarding these twin stars through a
spectroscope, an astronomer sees in each a spectrum of bands of colour
and black lines. Comparing these spectrums with one another, he finds
that there is a slight relative shifting of the dark lines, and from
that shifting he knows that the stars are rotating round one another,
and can tell their relative velocity with regard to the earth. By
means of his terrestrial physics he reads this signal of the skies.
This shifting of lines, the mere slight variation of a black line in a
spectrum, is very unlike that which the astronomer knows it means. But
it is probably much more like what it means than the signals which the
nerves deliver are like the phenomena of the outer world.
No picture of an object is conveyed through the nerves. No picture of
motion, in the sense in which we postulate its existence, is conveyed
through the nerves. The actual deliverances of which our consciousness
takes account are probably identical for eye and ear, sight and touch.
If for a moment I take the whole earth together and regard it as a
sentient being, I find that the problem of its apprehension is a very
complex one, and involves a long series of personal and physical
events. Similarly the problem of our apprehension is a very complex
one. I only use this illustration to exhibit my meaning. It has this
especial merit, that, as the process of conscious apprehension takes
place in our case in the minute, so, with regard to this earth being,
the corresponding process takes place in what is relatively to it very
minute.
Now, Plato’s view of a soul leads us to the hypothesis that that
which we designate as an act of apprehension may be a very complex
event, both physically and personally. He does not seek to explain
what an intuition is; he makes it a basis from whence he sets out on
a voyage of discovery. Knowledge means knowledge; he puts conscious
being to account for conscious being. He makes an hypothesis of the
kind that is so fertile in physical science—an hypothesis making no
claim to finality, which marks out a vista of possible determination
behind determination, like the hypothesis of space itself, the type of
serviceable hypotheses.
And, above all, Plato’s hypothesis is conducive to experiment. He
gives the perspective in which real objects can be determined; and,
in our present enquiry, we are making the simplest of all possible
experiments—we are enquiring what it is natural to the soul to think of
matter as extended.
Aristotle says we always use a “phantasm” in thinking, a phantasm of
our corporeal senses a visualisation or a tactualisation. But we can
so modify that visualisation or tactualisation that it represents
something not known by the senses. Do we by that representation wake
up an intuition of the soul? Can we by the presentation of these
hypothetical forms, that are the subject of our present discussion,
wake ourselves up to higher intuitions? And can we explain the world
around by a motion that we only know by our souls?
Apart from all speculation, however, it seems to me that the interest
of these four-dimensional shapes and motions is sufficient reason for
studying them, and that they are the way by which we can grow into a
fuller apprehension of the world as a concrete whole.
SPACE NAMES.
If the words written in the squares drawn in fig. 1 are used as the
names of the squares in the positions in which they are placed, it is
evident that a combination of these names will denote a figure composed
of the designated squares. It is found to be most convenient to take as
the initial square that marked with an asterisk, so that the directions
of progression are towards the observer and to his right. The
directions of progression, however, are arbitrary, and can be chosen at
will.
[Illustration: Fig. 1.]
Thus _et_, _at_, _it_, _an_, _al_ will denote a figure in the form of a
cross composed of five squares.
Here, by means of the double sequence, _e_, _a_, _i_ and _n_, _t_, _l_,
it is possible to name a limited collection of space elements.
The system can obviously be extended by using letter sequences of more
members.
But, without introducing such a complexity, the principles of a space
language can be exhibited, and a nomenclature obtained adequate to all
the considerations of the preceding pages.
1. _Extension._
Call the large squares in fig. 2 by the name written in them. It is
evident that each can be divided as shown in fig. 1. Then the small
square marked 1 will be “en” in “En,” or “Enen.” The square marked 2
will be “et” in “En” or “Enet,” while the square marked 4 will be “en”
in “Et” or “Eten.” Thus the square 5 will be called “Ilil.”
[Illustration: Fig. 2.]
This principle of extension can be applied in any number of dimensions.
2. _Application to Three-Dimensional Space._
To name a three-dimensional collocation of cubes take the upward
direction first, secondly the direction towards the observer, thirdly
the direction to his right hand.
[Illustration]
These form a word in which the first letter gives the place of the cube
upwards, the second letter its place towards the observer, the third
letter its place to the right.
We have thus the following scheme, which represents the set of cubes of
column 1, fig. 101, page 165.
We begin with the remote lowest cube at the left hand, where the
asterisk is placed (this proves to be by far the most convenient origin
to take for the normal system).
Thus “nen” is a “null” cube, “ten” a red cube on it, and “len” a “null”
cube above “ten.”
By using a more extended sequence of consonants and vowels a larger set
of cubes can be named.
To name a four-dimensional block of tesseracts it is simply necessary
to prefix an “e,” an “a,” or an “i” to the cube names.
Thus the tesseract blocks schematically represented on page 165, fig.
101 are named as follows:—
[Illustration: 1 2 3]
2. DERIVATION OF POINT, LINE, FACE, ETC., NAMES.
[Illustration]
The principle of derivation can be shown as follows: Taking the square
of squares the number of squares in it can be enlarged and the whole
kept the same size.
[Illustration]
Compare fig. 79, p. 138, for instance, or the bottom layer of fig. 84.
Now use an initial “s” to denote the result of carrying this process on
to a great extent, and we obtain the limit names, that is the point,
line, area names for a square. “Sat” is the whole interior. The corners
are “sen,” “sel,” “sin,” “sil,” while the lines are “san,” “sal,”
“set,” “sit.”
[Illustration]
I find that by the use of the initial “s” these names come to be
practically entirely disconnected with the systematic names for the
square from which they are derived. They are easy to learn, and when
learned can be used readily with the axes running in any direction.
To derive the limit names for a four-dimensional rectangular figure,
like the tesseract, is a simple extension of this process. These point,
line, etc., names include those which apply to a cube, as will be
evident on inspection of the first cube of the diagrams which follow.
All that is necessary is to place an “s” before each of the names given
for a tesseract block. We then obtain apellatives which, like the
colour names on page 174, fig. 103, apply to all the points, lines,
faces, solids, and to the hyper-solid of the tesseract. These names
have the advantage over the colour marks that each point, line, etc.,
has its own individual name.
In the diagrams I give the names corresponding to the positions shown
in the coloured plate or described on p. 174. By comparing cubes 1, 2,
3 with the first row of cubes in the coloured plate, the systematic
names of each of the points, lines, faces, etc., can be determined. The
asterisk shows the origin from which the names run.
These point, line, face, etc., names should be used in connection with
the corresponding colours. The names should call up coloured images of
the parts named in their right connection.
[Illustration]
It is found that a certain abbreviation adds vividness of distinction
to these names. If the final “en” be dropped wherever it occurs the
system is improved. Thus instead of “senen,” “seten,” “selen,” it is
preferable to abbreviate to “sen,” “set,” “sel,” and also use “san,”
“sin” for “sanen,” “sinen.”
[Illustration]
[Illustration]
We can now name any section. Take _e.g._ the line in the first cube
from senin to senel, we should call the line running from senin to
senel, senin senat senel, a line light yellow in colour with null
points.
[Illustration]
Here senat is the name for all of the line except its ends. Using
“senat” in this way does not mean that the line is the whole of senat,
but what there is of it is senat. It is a part of the senat region.
Thus also the triangle, which has its three vertices in senin, senel,
selen, is named thus:
Area: setat.
Sides: setan, senat, setet.
Vertices: senin, senel, sel.
The tetrahedron section of the tesseract can be thought of as a series
of plane sections in the successive sections of the tesseract shown in
fig. 114, p. 191. In b_{0} the section is the one written above. In
b_{1} the section is made by a plane which cuts the three edges from
sanen intermediate of their lengths and thus will be:
Area: satat.
Sides: satan, sanat, satet.
Vertices: sanan, sanet, sat.
The sections in b_{2}, b_{3} will be like the section in b_{1} but
smaller.
Finally in b_{4} the section plane simply passes through the corner
named sin.
Hence, putting these sections together in their right relation, from
the face setat, surrounded by the lines and points mentioned above,
there run:
3 faces: satan, sanat, satet
3 lines: sanan, sanet, sat
and these faces and lines run to the point sin. Thus the tetrahedron is
completely named.
The octahedron section of the tesseract, which can be traced from fig.
72, p. 129 by extending the lines there drawn, is named:
Front triangle selin, selat, selel, setal, senil, setit, selin with
area setat.
The sections between the front and rear triangle, of which one is shown
in 1b, another in 2b, are thus named, points and lines, salan, salat,
salet, satet, satel, satal, sanal, sanat, sanit, satit, satin, satan,
salan.
The rear triangle found in 3b by producing lines is sil, sitet, sinel,
sinat, sinin, sitan, sil.
The assemblage of sections constitute the solid body of the octahedron
satat with triangular faces. The one from the line selat to the point
sil, for instance, is named selin, selat, selel, salet, salat, salan,
sil. The whole interior is salat.
Shapes can easily be cut out of cardboard which, when folded together,
form not only the tetrahedron and the octahedron, but also samples of
all the sections of the tesseract taken as it passes cornerwise through
our space. To name and visualise with appropriate colours a series of
these sections is an admirable exercise for obtaining familiarity with
the subject.
EXTENSION AND CONNECTION WITH NUMBERS.
By extending the letter sequence it is of course possible to name a
larger field. By using the limit names the corners of each square can
be named.
Thus “en sen,” “an sen,” etc., will be the names of the points nearest
the origin in “en” and in “an.”
A field of points of which each one is indefinitely small is given by
the names written below.
[Illustration]
The squares are shown in dotted lines, the names denote the points.
These points are not mathematical points, but really minute areas.
Instead of starting with a set of squares and naming them, we can start
with a set of points.
By an easily remembered convention we can give names to such a region
of points.
Let the space names with a final “e” added denote the mathematical
points at the corner of each square nearest the origin. We have then
for the set of mathematical points indicated. This system is really
completely independent of the area system and is connected with it
merely for the purpose of facilitating the memory processes. The word
“ene” is pronounced like “eny,” with just sufficient attention to the
final vowel to distinguish it from the word “en.”
[Illustration]
Now, connecting the numbers 0, 1, 2 with the sequence e, a, i, and
also with the sequence n, t, l, we have a set of points named as with
numbers in a co-ordinate system. Thus “ene” is (0, 0) “ate” is (1,
1) “ite” is (2, 1). To pass to the area system the rule is that the
name of the square is formed from the name of its point nearest to the
origin by dropping the final e.
By using a notation analogous to the decimal system a larger field of
points can be named. It remains to assign a letter sequence to the
numbers from positive 0 to positive 9, and from negative 0 to negative
9, to obtain a system which can be used to denote both the usual
co-ordinate system of mapping and a system of named squares. The names
denoting the points all end with e. Those that denote squares end with
a consonant.
There are many considerations which must be attended to in extending
the sequences to be used, such as uniqueness in the meaning of the
words formed, ease of pronunciation, avoidance of awkward combinations.
I drop “s” altogether from the consonant series and short “u” from
the vowel series. It is convenient to have unsignificant letters at
disposal. A double consonant like “st” for instance can be referred to
without giving it a local significance by calling it “ust.” I increase
the number of vowels by considering a sound like “ra” to be a vowel,
using, that is, the letter “r” as forming a compound vowel.
The series is as follows:—
CONSONANTS.
0 1 2 3 4 5 6 7 8 9
positive n t l p f sh k ch nt st
negative z d th b v m g j nd sp
VOWELS.
0 1 2 3 4 5 6 7 8 9
positive e a i ee ae ai ar ra ri ree
negative er o oo io oe iu or ro roo rio
_Pronunciation._—e as in men; a as in man; i as in in; ee as in
between; ae as ay in may; ai as i in mine; ar as in art; er as ear in
earth; o as in on; oo as oo in soon; io as in clarion; oe as oa in oat;
iu pronounced like yew.
To name a point such as (23, 41) it is considered as (3, 1) on from
(20, 40) and is called “ifeete.” It is the initial point of the square
ifeet of the area system.
The preceding amplification of a space language has been introduced
merely for the sake of completeness. As has already been said nine
words and their combinations, applied to a few simple models suffice
for the purposes of our present enquiry.
_Printed by Hazell, Watson & Viney, Ld., London and Aylesbury._
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