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Title: The Fourth Dimension

Author: Charles Howard Hinton

Release date: January 13, 2022 [eBook #67153]
Most recently updated: December 23, 2024

Language: English

Original publication: United Kingdom: Swan Sonnenschein & Co. Limited,
1906

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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.

THE

FOURTH DIMENSION

BY

C. HOWARD HINTON, M.A.
AUTHOR OF “SCIENTIFIC ROMANCES”
“A NEW ERA OF THOUGHT,” ETC., ETC.

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
APPENDIX 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
Fig. 1.
be arranged in a series along the line, as at a, b, c, etc.,
according to their degrees of brightness.
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
Fig. 2. 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.
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
Fig. 3.
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.
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.
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
Fig. 4. 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

Fig. 5.
movement compounded by these axial
movements.
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.
The lines az and ax determine a plane. If he
Fig. 6. 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.
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
Fig. 7. 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.
The plane being can trace out a
movement in the third dimension by
Fig. 8.
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.
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
Fig. 9.
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.
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.

Fig. 10.

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.
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
Fig. 11. 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.

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.
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
Fig. 13.
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.
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
Fig. 14. 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.
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
Fig. 15.
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.
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.
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.
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.
Fig. 17.
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.
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
Fig. 18. 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.

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.

Eternal principles, Appearances in the mind,
as beauty. as whiteness, equality

Then as A is to B, so is A1 to B1
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.
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.
Fig. 19.
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.
Thus the square acde (fig. 21) contains one point,
Fig. 20. 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
Fig. 21. in question is equal to the sum of Fig. 22.
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.

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.
This deformation is not shear
alone, but shear accompanied by
rotation.
Fig. 24.
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.
In pure shear a body is compressed and extended in two directions at
right angles to each other, so that its volume remains unchanged.

Fig. 26.
Fig. 25.
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.
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
Fig. 27. slanting
placed Fig. 28.
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.
Shear square on hypothenuse—
7 internal 7
4 at corners 1
8
Fig. 29.
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?
Fig. 29 bis.
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—
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
Fig. 30. 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.
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
Fig. 31. 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.
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
Fig. 32.
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.
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.
Fig. 33.
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
Fig. 34. observation, and is therefore, except for a process
of reasoning, inconceivable to him.
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
Fig. 35. sections, each like the square which forms its
base, all rigidly connected together.
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.
In this case all that we see of the plane
Fig. 36.
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 ax ay.
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 ab1, come after half a
revolution to lie as ax 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.
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
Fig. 37.
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
Fig. 38. 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.
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.
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. 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
Fig. 40. 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
Fig. 41.
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.
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
Fig. 42. 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.
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.
Fig. 43.
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
Fig. 44. external portion, because the
Axis of x running towards the observer. internal parts are hidden by the
sphere.
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.
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
Fig. 45.
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,
Fig. 46. 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.
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.

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
Fig. 47. 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
Fig. 48. sides are 5 and 5 (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
Fig. 49.
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.
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.

Figure 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.
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
Fig. 51.
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
Fig. 52.
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.
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.
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.
Fig. 53.
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
Fig. 54. 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 mp pm mp pm
Minor sm sm ms ms

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.

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.

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 ferioque [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.

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.

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 1a in
which the dual aspect is represented by the combination of symbols. And
similarly let b be 2b, c be 3c, 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 2a, 2b, 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.
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,
Fig. 59. 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.
Thus, for instance, the one marked by an
asterisk can be called 1c, 2b, 3c, 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
Fig. 60.
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.
Consider the points named in the figure 1c, 2a, 3c; 1c, 2c, 3a; 1a, 2c, 3c,
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.
Fig. 61. Before the 1c 2a 1c 2c 1a 2c
change 3c 3a 3c }
After the 1c 2b 1c 2c 1b 2c Conjugates.
change 3c 3b 3c

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
Fig. 62.
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?
Such a set can be found. Consider the set represented in fig. 62, and
written down in the first of the two lines—
Self- 1a 2b 3c 1b 2a 3c 1c 2a 3b 1c 2b 3a 1b 2c 3a 1a 2c 3b
{
conjugate 1c 2b 3a 1b 2c 3a 1a 2c 3b 1a 2b 3c 1b 2a 3c 1c 2a 3b

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.

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
Fig. 63. 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 1i and oj, and so on as shown in fig. 63b. 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 1j line because the points on it are at a distance,
1 in the j direction.
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
Fig. 63b. 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 Oi, 1j.
Fig. 64.

Fig. 65.

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.
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 2i, 0j, 0k.
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 2i, 1j, 1k, for instance, 1 is
repeated. The point we obtain is that marked 210,
Fig. 66. 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
Fig. 67. 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.

Fig. 68.

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.
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
Fig. 70.
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
0i meant the whole line from which the distances in the i direction was
measured, and as in space 0i means the whole plane from which distances
in the i direction are measured, so now 0h 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.

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, 3i, 2j, 1k, 0h. 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 3i we must also take 1j, 2k, 0h. This point is shown by the second
star in the cube 0h.

Fig. 72.

In the first cube, since all the points are 0h 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.

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 0k
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
Fig. 74.
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 0h, a 1h, a 2h, or a 3h 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.
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.

Fig. 75.

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.
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 0h, a 1h, a
2h, etc., cube. Putting then the 0h, 1h, 2h, 3h, 4h 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 0l 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 0h, a 1h, a 2h, a 3h, or a 4h 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 0l points.
We cannot have 0 with any other letter. Then, keeping in the first figure,
which is that of the 0l positions, take first of all that selection which always
contains 1h. We suppose, therefore, that the cube is a 1h 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 0l 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 2h points. We then have a
0l, 2h region, and have the sets ijk and 431 left over. We must then pick out
in accordance with our rule all such points as 4i, 3j, 1k.
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.

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.
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
Fig. 77.
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
Fig. 78. 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.
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.

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.
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
Fig. 80. 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.
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
Fig. 81.
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.

Fig. 82.

Fig. 83.

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.
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,
Fig. 84. 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.
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.

Fig. 85.
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
Fig. 86. name for the solidity of a cube.
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.

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
Fig. 88. 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.
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.

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.
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
Fig. 90. 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.

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.
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
Fig. 92. 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.

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.

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.

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 an, af, aa, ab.
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
Fig. 96.
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 al and ar. 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.

The plane being could not see the square a with
the squares an, af, 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 an, a far af,
Fig. 97. a above aa, a below ab, a left al, a right ar,
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 an, af, 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 ak, and ends in a final position an (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 an, af, 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,
Fig. 98.
then travelling from the same null point for one
inch in that direction, we must come to the body
lying beyond the null region.
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.
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
Fig. 99. eight
A plane being’s representation of a block of eight cubes by two sets of four cubes, by
squares.
eight cubes
placed in
some arbitrary position relative to our first eight cubes.

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.

Fig. 101.
1 2 3
Each cube is the beginning of the first Each cube is the beginning of the Each cube is the beginning of the third
tesseract going in the fourth dimension. second tesseract. tesseract.

Fig. 102.[4]
1 2 3 4
A cube of 64 cubes, A cube of 64 cubes, A cube of 64 cubes,
each 1 in. × 1 in. × 1 in. each 1 in. × 1 in. × 1 in. each 1 in. × 1 in. × 1 in.
A cube of 64 cubes each
the beginning of the beginning of the beginning of
1. in × 1 in., the
tesseracts 1 in. from our tesseracts 2 in. from our tesseracts 3 in. from our
beginning of a tesseract.
space in the 4th space in the 4th space in the 4th
dimension. dimension. dimension.

[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.

Fig. 103.

The tesseract, red, white, yellow axes in space. In the lower line the three
rear faces are shown, the interior being removed.]

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.

Fig. 105.
The tesseract, with red, white, blue axes in space. Opposite faces are coloured identically.

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.

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 b0, b1, b2, b3, b4, 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 b0 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 b0 cube. In the final cube b4, the colouring of the
original b0 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 b1, b2, b3, 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 b1, b2, b3, and

the farther face of this cube in the blue direction is shown in b4—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 b1, b2, b3. In b4 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.

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 b1,
b2, b3. 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 b1, b2, b3. 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 wh0, wh1,
wh2, wh3, wh4, 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 wh1, wh2, wh3, wh4, fig. 108.
It is the same cube that is represented in the sections b1, b2, b3: 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 b1, b2, b3, b4, on the opposite side to the l purple squares. Also
the light yellow face at the base of the cube b0, 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
wh0 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 b0, b1, b2, b3, b4.

The case is, however, a little different with the brown cube. This cube we
have altogether in space in the section wh0, fig. 108, while it exists as a
series of squares, the left-hand ones, in the sections b0, b1, b2, b3, b4. 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 b1, b2, b3,
are the very same brown sections that would be obtained by cutting the
brown cube, wh0, across at the right distances along the blue line, as shown
in fig. 108. But as these sections are placed in the brown cube, wh0, they
come behind one another in the blue direction. Now, in the sections wh1,
wh2, wh3, 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 wh0. 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 y0, y1, y2, y3, y4, 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.

Fig. 109.

The interior of the first cube, y0, 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, y0, y1, y2, etc., can be considered as
derived from sections of the b0 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 b0, 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 y1 when we go on in the yellow direction. Thus,
the first section, y1, 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.

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 r0, r1, r2, r3, r4, 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 r0, when the
section r1 comes in, by looking at the cube b0, 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 r1. Now we can determine the
nomenclature of all the regions of r1 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 r0, 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 r0 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, b1, b2,
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 r1
replaces the section of r0 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.

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.

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.y1, fig. 112. Between figure C and C.y1 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.
Now, the whole of this red, yellow, blue, or
Fig. 113.
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.

Fig. 114.

Thus drawing our initial cube and the successive sections, calling them
b0, b1, b2, b3, b4, 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.
Drawin
g this
cube we
have fig.
116.
Fig. 115. Now
Fig. 116.
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 b0, b1, b2, b3, b4, 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, 41, 42, 43, 44.

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.

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.
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
Fig. 119.
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).

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.
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.
Fig. 121.
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
Fig. 122. first figure, a light blue line and a green
line.
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.
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
Fig. 123. 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
Fig. 124.
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.
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.
In this figure we can draw the
parallels to the red and yellow axes
Fig. 125.
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.

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
Fig. 126.
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.

Front view. The rear faces.
Fig. 127.

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.
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.

Fig. 128. 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
1h 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 1h it will be found to cut the cubical
section of the tesseract in the hexagonal figure drawn. In 2h (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 3h 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

[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.

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.
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.
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
Fig. 1 (129).
plane surface. Let ax, ay 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 az be the
axis, which stands perpendicular to his plane at a. He can draw in his plane
two lines to represent the two axes, ax and az. Let Fig. 2 be this drawing.
Here the z axis has taken the place of the y axis, and the plane of ax az is
represented in his plane. In this figure all that exists of the square abcd will
be the line ab.
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 ab1 and after
Fig. 2 (130).
half a revolution will lie on ax 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.
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
Fig. 3 (131).
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 (132). fig. 4.
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.
Now, since we can take any three axes, let us
look at what lies in the space of zyw, and examine
Fig. 5 (133).
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.
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.
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.
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.
Fig. 8 (136).
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.
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
Fig. 9 (137).
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.
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.
Fig. 10 (138). 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.
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.
Now, take the section of the sphere
Fig. 11 (139).
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
Fig. 12 (140). 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.
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
Showing axes xyz Showing axes xwz
Fig. 13 (141).
indicates Fig. 14 (142).
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.
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,
Fig. 15 (143). 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.
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
Fig. 1.
considerations of the preceding pages.
1. Extension.
Call the large squares in 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.”
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.
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:—

2. Derivation of Point, Line, Face, etc., Names.
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.

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.”
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.
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.”

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.
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 b0 the section is the one written above. In b1 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 b2, b3 will be like the section in b1 but smaller.

Finally in b4 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.

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.”
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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