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Full text of "The fourth dimension"

SATURDAY 2 APR 1955 



SATURDAY 2 



1955 









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 GUABDIAN. 

" The fourth dimension is a subject which has /tad 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 tlie 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 ivill find abundance of exercise in 
Mr. C. H t Hinton's Fourth Dimension." WESTMINSTER REVIEW. 



FIRST EDITION, April 1904; SECOND EDITION, May 1906; 
THIRD EDITION, January 1912. 



THE 



FOURTH DIMENSION 



BY 



C, HOWARD HINTON, M.A. 

AUTHOR OF " SCIENTIFIC ROMANCES " 
"A NEW ERA OF THOUGHT," ETC., ETC. 




LONDON 

GEORGE ALLEN & CO., LTD. 

EUSKIN HOUSE 
44, 45 RATHBONE PLACE 

1912 



PREFACE TO FIRST EDITION 

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 mathe- 
matical 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 



VI PBEFACE 

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 THK HISTORY OF 

FOUR SPACE 41 

Lobatchewsky, Bolyai, and Gauss 
Metageometry 

VI. THE HIGHER WORLD . . 61 

VII. THE EVIDENCES 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-DlMEXSIONAL FlGURE . .122 

XI. NOMENCLATURE AND ANALOGIES 136 



Vlll CONTENTS 

CHAP. PAGE 

XII. THE SIMPLEST FOUR-DIMENSIONAL SOLID . . 157 

XTII. REMARKS ON THE FIGURES . . . 178 

XIV. A RECAPITULATION AND EXTENSION OF THE 

PHYSICAL ARGUMENT 203 

APPENDIX I. THE MODELS 231 

II. A LANGUAGE OF SPACE 248 



THE FOURTH DIMENSION 



CHAPTER I 

POUR 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 com- 
plexity 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 

1 



2 THE FOURTH DIMENSION 

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 them- 
selves 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 



FOUR-UIMENSIONAL SPACE 3 

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 appre- 
hended 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 



4 THE FOURTH DIMENSION 

another begins is a point, and the straight line itself can 
be generated by the motion of the point. 

One portion of a plane is bounded from another by a 
straight line, and the plane itself can be generated by 
the straight line moving in a direction not contained 
in itself. 

Again, two portions of solid space are limited with 
regard to each other by a plane ; and the plane, moving 
in a direction not contained in itself, can generate solid 
space. 

Thus, going on, we may say that space is that which 
limits two portions of higher space from each other, and 
that our space will generate the higher space by moving 
in a direction not contained in itself. 

Another indication of the nature of four-dimensional 
space can be gained by considering the problem of the 
arrangement of objects. 

If I have a number of swords of varying degrees of 
brightness, I can represent them in respect of this quality 
by points arranged along a straight line. 

If I place a sword at A, fig. 1, and regard it as having 
a certain brightness, then the other swords 

can be arranged in a series along the 

line, as at A, B, c, etc., according to 
their degrees of brightness. 

If now I take account of another quality, say length, 
thev can be arranged in a plane. Starting from A, B, c, I 
can find points to represent different 
E degrees of length along such lines as 
I AF, BD, CE, drawn from A and B and C. 

' ' Points on these lines represent different 

* * 2i 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. 




FOUK-DIMKNSIONAL SPACE 5 

Bringing in a third quality, say sharpness, I can draw, 
as in fig. 3, any number of upright 
lines. Let distances along these 
upright lines represent degrees of 
sharpness, thus the points F and G 
will represent swords of certain 
definite degrees of the three qualities 
mentioned, and the whole of space will serve to represent 
all conceivable degrees of these three qualities. 

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 desciiption, 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 



THE ANALOGY OF A PLANE WOELD 7 

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. Le't 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 
Fig. 4. he can move along the surface of his earth, 
he will have the sense of a direction parallel to its surface, 
which we may call forwards and backwards. 

He will have no sense of right and left that is, of the 
direction which we recognise as extending out from the 
plane to our right and left. 

The distinction of right and left is the one that we 
must suppose to be absent, in order to project ourselves, 
into the condition of a plane being. 



8 THE FOURTH DIMENSION 

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 in- 
vestigate 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 



THE ANALOGY OF A PLANE WORLD 



( J 



to any other, by means of two straight lines drawn at 
right angles to each other. 

Let AX and AY be two such axes. He can accomplish 
the translation from A to B by going along AX to C, and 
then from c along CB parallel to AY. 

The same result can of course be obtained 
by moving to D along AY and then parallel 
to AX from D to B, or of course by any 
diagonal movement compounded by these 
axial movements. 

By means of movements parallel to 
these two axes he can proceed (except for 



C 



rig. 5. 



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 

^"^^ Z direction represented by A z. 

Fig. 6. The lines AZ and AX determine a 

plane. If he could be taken off his plane, and trans- 
ferred 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 consider the square to represent an object in the plane 



Fig. 7. 



10 THE FOURTH DIMENSION 

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 preceive, 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 con- 
viction 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 dis- 
tinction 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 
.'olid. 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 



THE ANALOGY OF A PLANE WORLD 11 

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 



12 



THE FOURTH DIMENSION 




Fig. 8. 



his plane, each lying in the third dimension a little 

further off from his plane than 
the preceding one. These sec- 
tions 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 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. 

{. I One material particle takes the place of another, 

but it does not come from anywhere or go 
anywhere in the space which the plane being 
knows. 

The analogous manner of representing a higher solid in 
our case, is to conceive it as composed of a number of 




TUE ANALOGY OF A PLANE WORLD 



13 



sections, each lying a little farther 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 







B 



C 

Fig. 1U 



O intervals in the unknown 

dimension of a higher 
cube. Arranged thus their coherence in the higher figure 
is destroyed, they are mere representations. 

A motion in the fourth dimension from A through B, c, 
etc., would be continuous, but we can only represent it as 
the occupation of the positions A, B, c, etc., in succession. 
We can exhibit the results of the motion at different 
stages, but no more. 

In this representation we have left out the distance 
between one section and another ; we have considered the 
higher body merely as a series of sections, and so left out 
its contents. The only way to exhibit its contents is to 
call in the aid of the conception of motion. 

If a higher cube passes transverse to our space, it will 
appear as a cube isolated in space, the part 
that has not come into our space and the part 
that has passed through will not be visible. 
The gradual passing through our space would 
appear as the change of the matter of the cube 




Fig. 11. 



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. 



14 



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 com- 
posed 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. 

Take the face A and conceive it to exist as simply a 





Fig. 12. 

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 com- 
posed 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 ex- 
perience 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 con- 
ceived 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 



15 



16 THE FOURTH DIMENSION 

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 sup- 
positions 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 el her, we should 
have a strong presumption in favour of the reality of 
the fourth dimension. 

By proceeding in the direction of finer r.nd finer sub- 
division, 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 



THE SIGNIFICANCE OF A FOUR-DIMENSIONAL EXISTENCE 17 

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 

2 



18 THE FOURTH DIMENSION 

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 phenome- 
non 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 SIGNIFICANCE OF A FOUR-DIMENSIONAL EXISTENCE 19 

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,1must 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 pecuHar 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 



20 THE FOURTH DIMENSION 

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 



THE SIGNIFICANCE OF A FOUR-DIMENSIONAL EXISTENCE 21 

being, despite his bodily environment, has come to recog- 
nise 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 differ- 
ence 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 accom- 
panied 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 explana- 
tion founded on aim, purpose, volition, always presupposes 



22 THE FOURTH DIMENSION 

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 pro- 
cesses 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 
hiftory 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 be- 
tween 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 

23 



24 



THE FOURTH DIMENSION 



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 verti- 
cally 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 meet- 
ing of the film and the stick there 
will be an interruption of the film. 
If we suppose the film to have a pro- 





Fig. 13. 



perty, 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. 









THE FIRST CHAPTER IN THE HISTORY OF FOUR SPACE 25 

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 
.x^*^ ^J the film to move vertically 

f upwards, the whole spiral will 

Xf**"*"""*^ be represented in the film of 

/^*** "^ the consecutive positions of the 

point of intersection. In the 
film the permanent existence 
of the spiral is experienced as 
a time series the record of 
traversing the spiral is a point 
Fi J 4 moving in a circle. If now 

we suppose a consciousness con- 
nected 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 distinguish- 
able 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. 



26 THE FOURTH DIMENSION 

Thus we can imagine a plane world, in which all the 
variety of motion is the phenomenon of structures con- 
sisting 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. 
Kepresenting 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 uncom- 
promising 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, #s 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 uriacting. 
Slight modifications and deviations from these move- 
ments 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 FIRST CHAPTER IN THE HISTORY OF FOUR SPACE 27 

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 repre- 
sentative 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, 
andidiffering 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. 



28 THE FOUIITH DIMENSION 

Transferring our conceptions to those of an existence in 
a higher dimensionality traversed by a space of con- 
sciousness, 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 sym- 
bolical. 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 con- 
ceptions 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 concep- 
tions 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 con- 
ception of distance necessarily involved. The difference 



THK FIRST CHAPTER IN THK HISTORY OF FOUR SPACE 29 

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, 66, cb, db, and so on, and forming 

a scheme : 

da db dc dd 

ca cb cc cd 

ba bb be bd 

aa ab ao 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 
Fig. 15. , . 

bears his name. 

The Pythagorean property of an extended material 
system can be exhibited in a manner which will be of 
use to us afterwards, and which therefore I will employ 
now instead of using the kind of figure which he himself 
employed. 

Consider a two-fold field of points arranged in regular 
rows. Such a field will be presupposed in the following 
argument. 
It is evident that in fig. 1 6 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 
Fig. 16. . ,, 

in another way. 

Fig. 16 (1) shows four points determining a square. 
But four squares also meet in a point, fig. 16 (2). 
Hence a point at the corner of a square belongs equally 
to four squares. 




30 THE FOU11TH DIMENSION 

Thus we may say that the point value of the square 
shown is one point, for if we take the square in fig. 1G (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. 
. U-.1 . . Looking now at the sides of this figure 

... we see that there is a unit square on each 
i'ig. i7. O f 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 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 





THE FIRST CHAPTER IN THE HISTORY OF FOUR SPACE 31 

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 appre- 
hension. 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. Kock 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 



- 



32 THE FOUiiTH DIMENSION 

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 



TfiE FIRST CHAPTER Itf THE HISTORY OF tOtJR SPACE S3 

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 incom- 
prehensible changes of a harsh experience scatters and 
destroys. 

Plato believed in the reality of ideas. He meets us 
fairly and squarely. Divide a line into two parts, he 
says ; one to represent the real objects in the world, the 
other to represent the transitory appearances, such as the 
image in still water, the glitter of the sun on a bright 
surface, the shadows on the clouds. 

A B 

1 . 

Real things: Appearances: 

e.g., the sun. e.y., 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 repre- 
senting our true knowledge, which is of eternal principles, 
such "as beauty, goodness. 

A 1 B 1 

Eternal principles, Appearances in the mind, 

as beauty as whiteness, equality 

Then as A is to B, so is A 1 to B 1 . 

That is, the soul can proceed, going away from real 

3 



34 TSE FOURTH DIMENSION 

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 argu- 
ment 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. 



THE FIRST CHAPTER IN THE HISTORY OF FOUR SPACE 35 

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 pro- 
cessions, 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 return- 
ing 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 



36 THE FOURTH DIMENSION 

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 im- 
possible 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 tran- 
scending 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 exist- 
ence, 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 



THE FIRST CHAPTER IN THE HISTORY OF FOUR SPACE 37 

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 Jimension- 
ality 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 quarry men, 
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 



38 THE FOUKTH DIMENSION 

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 mathe- 
matician with his analytical weapons, so in the formulation 
o v 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 accus- 
tomed 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, 



THE FIRST CHAPTER IN THE HISTORY OF FOUR SPACE 39 

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 every- 
thing. 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, some- 
thing 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, {f we pass over the intervening time 



40 , THE FOURTH DIMENSION 

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 con- 
tributions to science. His theory of parallels is 



42 THE FOURTH DIMENSION. 

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 loth, 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 
Gottingen. 

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 : 
is it that we can gee him from here as well as from 



THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE 43 

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, 



44 THE FOURTH DIMENSION 

" I am unbroken in my determination to publish a 
work on Parallels, as soon as I have put rny 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 con- 
vinced 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 wor^, 



THE SECOND CHAPTER IN THE HISTORY Of FOUR SPACE 45 

" 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 inten- 
tion was to let nothing be known in my lifetime about 
my own work, of which, for the rest, but little is com- 
mitted 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 com- 
pletely 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 in- 
explicable 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 appre- 
ciate his greatness. 



46 THE FOURTH DIMENSION 

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 mathe- 
matics. 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 communi- 
cations 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. 



THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE 47 

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 deter- 
..... mine a square. 

.big. iy. Now it is evident that as four points 

determine a square, so four squares meet in a point. 

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 
. . . four others equally : belongs a quarter of it 
Fig. 20. to each square. 
Thus the square ACDE (fig. 21) contains one point, and 



D 

'E^C'- 


D 

E* 1 


c 


SX. 


^. J 




A ' B 


' A ' 


B 



Fig. 21. Fig. 22. 

has four points at the four corners. Since one-fourth of 
each of these four belongs to the square, the four together 
count as one point, and the point value of the square is 
two points the one inside and the four at the corner 
make two points belonging to it exclusively. 

Now the area of this square is two unit squares, as can 
be seen by drawing two diagonals in fig. 22. 

We also notice that the square in question is equal to 
the sum of the squares on the sides AB, BC, of the right- 
angled triangle ABC. Thus we recognise the proposition 
that the square on the hypothenuse is equal to the sum 
of the squares on the two sides of a right-angled triangle. 

Now suppose we set ourselves the question of deter- 
mining the whereabouts in the ordered system of points, 



48 



FOURTH DIMENSION 



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. 




Ing. 23. 

In fig. 23, in the square on AB, there are 
9 points interior . . . 

4 at the corners 

4 sides with 3 on each side, considered as 
1 on each side, because belonging 
equally to two squares . . , 



The total 
on BC. 



is 16. There are 9 points in the square 



TBE feECOtfb CHAPTER IN THE HISTORY OF FOUR SPACE 49 

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 



y over the preceding one, 



a b and the whole assumes 

the shape b in fig. 24. 

This deformation is not shear alone, but shear accom- 
panied by rotation. 

Shear can be considered as produced in another way. 
Take the square ABCD (fig. 25), and suppose that it 
is pulled out from along one of its diagonals both ways, 
and proportionately compressed along the other diagonal. 
It will assume the shape in fig. 26. 

4 



50 



THE FOURTH DIMENSION 



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. 

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 




Fig. 27. 



Fig. 28. 



represents a square sheared. It is not the figure into 
which the square in fig. 27 would turn, but the result of 
shear on some square not drawn. It is a simple slanting 
placed figure, taken now as we took a simple slanting 
placed square before. Now, since bodies in this world of 
shear offer no internal resistance to shearing, and keep 
their volume when sheared, an inhabitant accustomed to 
them would not consider that they altered their shape 
under shear. He would call ACDE as much a square as 
the square in fig. 27. We will call such figures shear 
squares. Counting the dots in ACDE, we find 

2 inside = 2 
4 at corners = 1 

or a total of 3. 

Now, the square on the side AB has 4 points, that on BC 
has 1 point. Here the shear square on the hypothenuse 
has not 5 points but 3 ; it is not the sum of the squares on 
the sides, but the difference. 



52 




THE FOURTH DIMENSION 

This relation always holds. Look at 
fig. 29. 

Shear square on hypothenuse 



7 internal 
4 at corners 



Fig. 29. 



Square on one side which the reader can draw for 

himself 




4 internal 
8 on sides . 
4 at corners 



Fig. 29 bis. 



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, 91 = 8. 
Thus in a world of shear 
the square on the hypothen- 
use would be equal to the 
difference of the squares on 
the sides of a right-angled 
triangle. 

In fig. 29 bis another shear square is drawn on which 
the above relation can be tested. 

What now would be the position a line on turning by 
shear would take up ? 

We must settle this in the same way as previously with 
our turning. 

Since a body shear, d 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. 



THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE 53 



We can solve the problem in a particular case 

In the figure ACDE 
(fig. 30) there are 

15 inside . . 15 
4 at corners r 1 

a total of 16. 

Now in the square ABGF, 
there are 16 

9 inside . . 9 
12 on sides . . 6 
4 at corners . 1 

16 

Hence the square on AB 
would, by the shear turn- 
ing, become the shear square 

ACDE. 




Fig. 30. 



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 th*e 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 



54 THE FOURTH DIMENSION 

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 ques- 
tion 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. 



THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE 55 

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 illus- 
tration 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, 






56 THE FOURTH DIMENSION 

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 
a<sf>rtir.r. t^f "Rnlvai and Lobatchewsky. 

^ ft Take a line AB and a point c. We 

say and see and know that through c 

^ Q can only be drawn one line parallel 

i'ig. 3i. to AB. 

But Bolyai said : " I will draw two." Let CD be parallel 




THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE 57 

to AB, that is, not meet AB however far produced, and let 

lines beyond CD also not meet 
AB; let there be a certain 
region between CD and CE, 
in which no line drawn meets 
AB. CE and CD produced 

lackwards 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 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 ^oul with 
the body, the struggle for mastery, had come to a cul- 
mination. 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 re- 
sulted in this theory so free from the bonds of sense ? No 
such supposition appears to be possible. 

Practically, however, metageometry had a great in- 
fluence 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 



58 THE FOURTH DIMENSION 

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 measure- 
ment 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 A BCD be the equator of a globe, and AP, BF, 
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 
Fig. 33. definition of parallels. Yet they 

meet in P. Hence a being living on a spherical surface, 
and unconscious of its curvature, would find that parallel 
lines would meet. He would also find that the angles 
in a triangle were greater than two right angles. In 
the triangle PAB, for instance, the angles at A and B 
are right angles, so the three angles of the triangle 
PAB are greater than two right angles. 

Now in one of the systems of metageometry (for after 
Lobatchewsky had shown the way it was found that other 
systems were possible besides his) the angles of a triangle 
are greater than two right angles. 

Thus a being on a sphere would form conclusions about 
his space which are the same as he would form if he lived 
on a plane, the matter in which had such properties as 
are presupposed by one of these systems of geometry. 
Beltrami also discovered a certain surface on which there 
could be drawn more than one "straight" line through a 




THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE 59 

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 ex- 
perience 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 dimen- 
sionality 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 accus- 
tomed 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 inevitable- 
ness 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 



60 THE FOURTH DIMENSION 

product of complex material conditions and cannot be 
measured by hard and fast rules. Its ultimate signi- 
ficance 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. 



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 qf 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 

61 



FOURTH DIMENSION 

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 cor- 
rector 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 



63 

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 repre- 
sent a turning from a known into the unknown dimension 
as a turning from one to another of his known dimensions. 



64 THE FOURTH DIMENSION 

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 observa- 

tion, 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 appprehend. 
He can see the first and last positions occupied in a half 
revolution about the line AC. The result of such a half revo- 
lution is to place the square ABCD on the left hand instead 
of on the right hand of the line AC. It would correspond 
to a pulling of the whole body ABCD through the line AC, 
or to the production of a solid body which was the exact 
reflection of it in the line AC. It would be as if the square 
ABCD turned into its image, the line AB acting as a mirror. 
Such a reversal of the positions of the parts of the square 
would be impossible in his space. The occurrence of it 
would be a proof of the existence of a higher dimensionality. 
Let him now, adopting the conception of a three- 
dimensional body as a series of 
sections lying, each removed a little 
farther than the preceding one, in 
direction at right angles to his 

plane, regard a cube, fig. 36, as a 
series of sections, each like the 



* square which forms its base, all 
Fig. 35. rigidly connected together. 



filGttBR WOELt) 



65 



tf DOW 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 pre- 
vented 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 con- 
sisting 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 





t 


f 


H 




\ 


\ 


\ 




8 
G 


D 


\ 







A C 

Fig. 36. 

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. 



(>6 1HE FOURta DIMENSION 



In this case all that we see of the pjane 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 repre- 
sentation 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 repre- 
sent 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 



THE HIGHER WORLD 67 

to him that, by a turning such as he knows, by a rotation 
about a point, the line AB can turn round A, and occu- 
pying all the intermediate positions, such as ABi, 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 con- 
sistency 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, a;, y, z, represent iv 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. 



B 



B 



Let us suppose that we let the y axis drop, and that 
we represent the w axis as occupy- 
ing 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 fo 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 a? to w 
direction about this line. In fig. 38 
it is shown on its way, and it can 
evidently continue this rotation till 
A * it lies on the other side of the z 

axis in the plane of xz. 

We can also take a section parallel to the face ABCD, 
and then letting drop all of our space except the plane of 
that section, introduce the w axis, running in the old y 
direction. This section can be represented by the same 
drawing, fig. 38, and we see that it can rotate about the 
line on its left until it swings half way round and runs in 
the opposite direction to that which it ran in before. 
These turnings of the different sections are not incon- 
sistent, and taken all together they will bring the cubt- 
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 th 
fourth axis take its place. 

In fig. 36 suppose the z axis to go. We have then 



THE HIGHER WORLD 



w 



simply the plane of xy and the square base of the 
cube ACEG, fig. 39, is all that could 
be seen of it. Let now the w axis 
take the place of the z axis and 
we have, in fig. 39 again, a repre- 
sentation of the space of xyiv, in 
A C which all that exists of the cube is 

its square base. Now, by a turning 

of x to w, this base can rotate around the line AE, it is 
shown on its way in fig. 40, and 
finally it will, after half a revolution, 
lie on the other side of the y axis. 
In a similar way we may rotate 
sections parallel to the base of the 
xw rotation, and each of them comes 
to run in the opposite direction from 
that which they occupied at first. 

Thus again the cube comes from the position of fig. 36. 

to that of fig. 41. In this x 
to w turning, we see that it 
takes place by the rotations of 
sections parallel to the front 
face about lines parallel to AB, 
or else we may consider it as 
consisting of the rotation of 
sections parallel to the base 
about lines parallel to AE. It 




H 



C A * 

2-posificn I -position 

Fig. 41. 

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- 
(Ijrnensjonal space, so we can think of a rotation about a 



70 



THE FOURTH DIMENSION 



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 per- 
pendicular 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 con- 
sidering 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 
If we 




8 



* 

Fig. 42. 

above a.n4 below the plane pf xy, from these lines. 



THE HIGHER WORLD 7] 

turn these rods in our space about the lines A and B, as 
the upper end of one, F, is going down, the lower end of 
the other, c, will be coming up. They will meet and 
conflict. But it is quite possible for these two rods 
each of them to turn about the two lines without altering 
their relative distances. 

To see this suppose the y axis to go, and let the w axis 
take its place. We shall see the lines A and B no longer, 
for they run in the y direction from the points G and H. 
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 form- 
ing part of a rigid body. 

Now we have but to suppose 
a central plane with rods crossing 
it at every point, like CD and EF cross the plane of xy, 
to have an image of a mass of matter extending equal 
distances on each side of a diametral plane. As two of 
these rods can rotate round, so can all, and the whole 
mass of matter can rotate round its diametral plane. 

This rotation round a plane corresponds, in four 
dimensions, to the rotation round an axis in three 
dimensions. Rotation of a body round a plane is the 
analogue of rotation of a rod round an axis. 

In a plane we have rotation round a point, in three- 
space rotation round an axis line, in four-space rotation 
round an axis plane. 

The four-dimensional being's shaft by which he trans- 
mits power is a disk rotating round its central plane 




72 THE FOURTH DIMENSION 

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 



THE HIGHER WORLD 73 

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 fi,nd a region which, though without mobility of the 



74 THE FOURTH DIMENSION 

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-dimen- 
sional movements down to their last detail. Then, knowing 



THE HIGHER WORLD. 75 

the mode of action which would be characteristic of the 
mioutest particles, if they were free, we can draw con- 
clusions 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 con- 
cerned, there would be a difference as to which way the 
motion took place. This would be shown in the dissimi- 
larity 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 multi- 
tude of forms which show a certain relation to a plane, 
a relation of symmetry, which indicates more than an acci- 
dental 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 sym- 
metrical about a line as the earth is symmetrical about an 
axis can easily be explained. But where there is symmetry 
a.bout a plane no simple physical motion, such as we 



EVIDENCES FOR A FotJRtS DIMENSION 

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 produc- 
tion 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 life-like repre- 
sentation 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 antenna 
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 concep- 
tions 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 our- 
selves, 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. 



78 THE FOURTH DIMENSION 

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 dimen- 
sions 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 dimen- 
sions 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 in- 
validate 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 sym- 
metry exists, it is only in the very minute particles 
of bodies that that motion is to be found, for there is 



THE EVIDENCES FOR A FOURTH DIMENSION 79 

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 observa- 
tions 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 move- 
ments 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 ? 



80 



tHE FOtJRtti 



Let us consider a sphere of our three-dimensional 

matter having a definite 
thickness. To represent 
this thickness let us sup- 
pose that from every point 
of the sphere in fig. 44 rods 
project both ways, in and 
out, like D and F. We can 
only see the external por- 
tion, because the internal 
parts are hidden by the 
sphere. 

In this sphere the axis 
of x is supposed to come 
towards the observer, the 




Fig. 44. 

Axis ofx running 
the observer. 



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 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 exten- 
sible to allow the particles c and E to become as widely 
separated as they would be in the positions D and F, then 




45< 



THE EVIDENCES FOR A FOUfcTfi DIMENSION 81 

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-dimen- 
sional 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 perpen- 
dicular 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 iv 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 

6 



82 tHE FOUKTS DIMENSION 

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 con- 
sists 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 ai: 



THE EVIDENCES FOR A FOURTH DIMENSION 83 

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 



84 THE FOURTH DIMENSION 

exactly that which we should expect if it were a four-dimen- 
sional 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 rota- 
tion 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 pheno- 
menon 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 con- 
structions 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 

85 



86 THE FOURTH DIMENSION 

of drawing conclusions and the use of higher space 
figures.* 

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. 

* 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 USE OF FOUR DIMENSIONS IN THOUGHT 87 

To take an instance chosen on account of its ready 

availability. Let us take 
two right-angled triangles of 
a given hypothenuse, but 
having sides of different 
lengths (fig. 46). These 
triangles are shapes which have a certain relation to each 
other. Let us exhibit their relation as a figure. 

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-ordin- 
ating lines we can represent a 
double set of magnitudes ; one set 
j as distances to the right of the ver- 

Fig. 47. tical 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 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 ricrht of the vertical level and 5 above the 
.5,5 horizontal level will represent the 

triangle whose sides are 5 and 5 




Thus we have obtained a figure 
consisting of the two points 7, 1, 
Fig. 48. and 5 ? 5^ representative of our two 

triangles. But we can go further, and, drawing an arc 



88 THE FOURTH DIMENSION 

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 indi- 
viduals of the class represented, the difference being 
measured by the length of the line between two repre- 
sentative 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 pro- 
perties 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 K. 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 
Mendeleeff'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 



THE USE OF FOUR DIMENSIONS IN THOUGHT 89 

it is worth while to bring into the full light of our atten- 
tion 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, elec- 
trical 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 dimen- 
sions. If the symmetry and completeness of our repre- 
sentatation 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 



90 



THE FOURTH DIMENSION 



us into the complexities incident on its application in 
classificatory science, let us follow Mrs. Alicia Boole Stott 
in her representation of the syllogism by its means. She 
will be interested to find that the curious gap she detected 
has a significance. 

A syllogism consists of two statements, the major and 
the minor premiss, with the conclusion that can be drawn 
from them. Thus, to take an instance, fig. 49. It is 
evident, from looking at the successive figures that, if we 
know that the region M lies altogether within the region 
p, and also know that the region s lies altogether within 
the region M, we can conclude that the region s lies 
altogether within the region P. M is P, 
major premiss; s is M, minor premiss; s 
is p, conclusion. Given the first two data 
we must conclude that s lies in p. The 
conclusion s is P involves two terms, s and 
p, which are respectively called the subject 
and the predicate, the letters s and P 
being chosen with reference to the parts 
the notions they designate play in the 
conclusion, s is the subject of the con- 
clusion, 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. 




Fig. 49. 



THE USE OF FOUR DIMENSIONS IN THOUGHT 



91 



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. 
The dotted lines in 3 and 4, fig. 50, denote that it is 

not known whether or no any objects exist, corresponding 




4. 
Mood o. 



Fig. 50. 



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 column?, 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 



Q 
o 



o s 



R PA E I 

Fig. 51. 

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 



92 



THE FOURTH DIMENSION 



M 


















L.U.. 


i 


n 



letters E, i, o, in the Same way show the mood character- 
ising the minor premiss in the rows opposite these letters. 
We have still to exhibit the conclusion. To do this we 
must consider the conclusion as a third variable, character- 
ised in its different varieties by four moods this being 
the syllogistic classification. The introduction of a third 
variable involves a change in our system of representation. 
Before we started with the regions to the right of a 
certain line as representing successively the major premiss 
in its moods ; now we must start with the regions to the 
right of a certain plane. Let LMNR 
be the plane face of a cube, fig. 52, and 
let the cube be divided into four parts 
by vertical sections parallel to LMNR. 
The variable, the major premiss, is re- 
A c ' ~ 'presented by the successive regions 
Fig. 52. 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. 



THE USE Of FOtJR DIMENSIONS IN fHOUGHT S3 

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, 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 com- 
bination 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, 



94 



THE fOtlKTH DIMENSION 




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 of the conclusion is represented 
in the top layer. Hence we see that the marking is 
correct in this respect. 

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 re- 
present merely the beginning 
of the region it stands for. 
Thus the whole cube can be 
represented by four vertical 
squares, each standing for a 
kind of vertical tray, and the 




Fig. 54. 



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 X USE OF FOUR DIMENSIONS IN THOUGH* 95 

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 pre- 
misses 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 pos- 
sibilities : 

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 represent- 
ing 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. 



96 ftfE FOUlVrti DIMENSION 



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 dimen- 
sions 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 soli(J from which we measure figure is that in 



"tHE USE Of tfOtift blMENSlONS IN THOUGflt 9* 

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 measure- 
ment 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 



98 THE FOURTH DIMENSION 

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 correspond- 
ing 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 con- 
clusion, 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. 



THE USE OF FOUR DIMENSIONS IN THOUGHT 



From this square, let it be supposed that that direc- 
tion 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 




(D 



Fig. 55. 



4321 



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. 

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, runn'ng 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 conclu- 
sion, 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 s^pond to major 



100 



'J'HE FOURTH DIMENSION 



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. 

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 




Fig. 5( : . 

by moving it in the fourth dimension. The first por- 
tion 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 dis- 
connected cubes drawn are our representation in space by 
means of disconnected sections of this higher body. 



THE USE OF FOUR DIMENSIONS IN THOUGHT 101 

Now it is only a limited number of conclusions which 
are true their truth depends on the particular combina- 
tions 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 corre- 
spond 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 con- 
cerned 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 habei [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 p 5 some s is not p. The region to be marked then 



102 THE FOURTH DIMENSION 

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 com- 
binations 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 com- 
binations 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. 

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 
Fig. 57. unmarked, which if marked would give 
symmetry. It is the one which would be denoted by the 




THE USE OF FOUR DIMENSIONS IN THOUGHT 



103 



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. 



M 









2nd figure. 








8rd figure. 



Fig. 58. 



4th figure. 



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 
jf some M is P and no S is M, we have no conclusion of 



104 THE FOURTH DIMENSION 

the form s is p in the various moods. It is quite inde- 
terminate 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 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 complete- 
ness 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 con- 
tain 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. 

}t is not that thp result is given expressed Jn another 



THE USE OF FOUR DIMENSIONS IN THOUGHT 105 

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 mediaeval 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 sav 
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 



106 THE FOURTH DIMENSION 

Americans." And this method is what is called the 
" quantification of the predicate." 

The laws of formal logic are coincident with the con- 
clusions 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 OP 
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. 

wr 



108 THE FOURTH DIMENSION 

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 observa- 
tion, 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 trans- 
ference of this relation to a region in which the con- 
stituents 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 direptjon?. 



APPLICATION TO KAJTr's THEORY OF EXPERIENCE 109 

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 ether 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 ex- 
perience such as ours is very different from experience 
in the abstract. Imagine just simply experience, suc- 
cession of states, of consciousness ! Why, there would 
be no connecting any two together, there would be no 



110 TftE FOURTH DIMENSION 

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 our- 
selves 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 exis- 
tence 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 exis- 
tence, 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. 



APPLICATION TO KANT*S THEORY OF EXPERIENCE 111 

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 par- 
tially 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 






112 THE FOURTH DIMENSION 

would correspond to a sudden and arbitrary change of rt 
into 6 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 intro- 
duced 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 la in which the dual aspect is represented by the 
combination of symbols. And similarly let b be 26, 
c be 3c, in which 2 and b represent the dual aspects 
of 6, 3 and c those of c. 

Since a can arbitrarily change into 6, or into c, and 
so on, the particular combinations written above cannot 
be kept. We have to assume the equally possible occur- 
rence 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 represention. 

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 imme- 
diately following passages. They are there carried 




APPLICATION TO KANT'S THEORY OF EXPERIENCE 113 

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, 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 
<*k be called Ic, 26, 3c, because it is 

opposite to c on 1, to 6 on 2, to 
c on 3. 

Let us now treat of the states of 
consciousness corresponding to these 
positions. Each point represents a 
composite of posits, and the mani- 
fold of consciousness corresponding 
Fig. <;o. ,, r , . ,*_.. 

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 sub- 
ject 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 

8 




114 



FOURTH DIMENSION 



c2a3c 



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 Ic, 2a, 3c ; 

Ic, 2c, 3a ; la, 2c, 3c, and 
examine the effect on them 
>32c3c w hen a change of order takes 
( place. Let us suppose, for 

instance, that a changes into 6, 
and let us call the two sets of 
points we get, the one before 
and the one after, their change 
conjugates. 




1c2c3a 
Fig. 01. 



Before the change Ic 2a Be Ic 2c 3a la 2c 3c 
After the change Ic 2b 3f Ic 2e 3b Ib 2c 3o 



\ Conjugates. 



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 repre- 
sented by those points would have a permanence. It 
would have a principle of identity. Despite the no law, 
the no order, of the ultimate constituents, it would have 
an order, it would form a system, the condition of a 
personal identity would be fulfilled. 

The question comes to this, then. Can we find a 
system of points which is self-conjugate which is such 
that when any posit on the axes becomes any other, or 



APPLICATION TO KANT'S THEORY OF EXPERIENCE 115 



when any axis becomes any other, such a set is trans- 
formed 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 




Fig. 62. 



Self- flaZbSc \b la 3c Ic 2a 3b Ic 2b 3a Ib 2c 3a Ia2o3b 
conjugate. \lc2b3a Ib 2c 3a la 2c 3b la 2b 3c Ib 2d 3c Ic 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.* Any arbitrary change of the points on the axes, 
or of the axes themselves, reproduces the same set. 

Thus, a function, by which a random, an unordered, con- 
sciousness 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 conscious- 
ness 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 con- 
sciousness capable of giving au account of itself is one 
which is characterised by this combination. All com- 
binations exist of this kind is the consciousness which 
can give an account of itself. And the very duality which 

* These figures are described more fully, and extended, in the next 
chapter. 



116 THE FOURTH DIMENSION 

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 random- 
ness, in any way. But any consciousness which is per- 
manent 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 



APPLICATION TO KANT's THEORY OP EXPERIENCE 117 

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 con- 
sciousness, 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 
"-f law and order by a thought of an invariable sequence. 
lie 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 appre- 
ciable 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 



118 THE FOURTH DIMENSION 

being, those that do survive will present such and such 
characteristics. This is the necessary beginning for ascer- 
taining 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 pheno- 
mena. It gives the general laws constitutive of any 
experience. If, on the assumption of absolute random- 
ness in the constituents, such and such would be 
characteristic of the experience, then, whatever the con- 
stituents, 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 



APPLICATION TO KANT'S THEORY OF EXPERIENCE 119 

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 per- 
tinent 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 arbitary 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 ex- 
perience 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 some- 
times 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. 



120 THE FOURTH 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 con- 
ditioned 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 con- 
sidering 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 in- 
tellectual 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 examina- 



APPLICATION TO KANT'S THEORY OF EXPERIENCE 121 

tion of the pigments. And in any method of thought it 
is the complexity of the whole that brings us to a know- 
ledge 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 repre- 
sents 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 subse- 
quently 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. 

122 



A FOUR-DIMENSIONAL FIGURE 123 

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. 



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 
J rise to a certain complexity. I will 
let the two lines i and j meet in the 
position I call o on each, and I will 
consider i as a direction starting equally 
big. 63. f rom ever y position on j, and j as 



B 



it.oj 



starting equally from every position on i. We thus 
obtain the following figure : A is both oi and oj, B is 1 i 
A _ C ai] d o}, and so on as shown in fig. 636. 
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 
Fig. 63ft. from AB in thej direction, and we can 

call them oj points and the line AB the oj line. Again, 
the line CD can be called the Ij 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 



134 



THE FOURTH DIMENSION 



with it o of the kind j ; and then, that we take o of the 
kind i and with it 1 of the kind j. 

Thus we get a pair of positions lying in the straight 
C 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, Ij. 

Coining 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 




. 64. 




Fig. 65. 

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 
\DC, we see that every position in the face ADC is a oi 
position, and the whole plane of the face may be called an 
oi plane. Thus we see that with the introduction of a 



A FOUR-DIMENSIONAL FIGURE 



125 



hew 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, uk. 
It is at the distance along i, along j, 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 dis- 
tance 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, in the 
j direction from the plane ABD, and in the k direction, 
measured from the plane ABC. Hence it is 200 written 
for 2i, Oj, Ok. 

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 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, Ij, Ik, for 
instance, 1 is repeated. The point 
we obtain is that marked 210, 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 




126 



FOURTft DIMENSION 



could be taken out of the cube and placed on a plane. 
It is a figure which will fill a 
plane by equal repetitions of itself. 
The plane being representing this 
construction in his plane would 
take three squares to represent the 
cube. Let us suppose that he 
takes the ij axes in his space and 
k represents the axis running out 
of his space, fig. 68. In each of 
the three squares shown here as drawn separately he 
could select the points given by the rule, and he would 




Fig. Gi 




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 drawing^ and discovering in them what the relation 
between these two extremities is. 



I02.X' 

"x-fc 



201 




C2I 



^B 

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 



127 




pick out by the rule the two points 201, 102 u, 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 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 assemllage is shown. 

We can represent this assemblage of points by four 
solid figures. The first giving all those positions which 



Fig. 70. 



128 



THE FOURTH DIMENSION 



are at a distance o from our space in the fourth dimen* 
sion, 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 




Fig. 71. 

all the points thus determined to be contained in solid 
models of which our drawings in fig. 71 are represen- 
tatives. Here we notice that as on the plane Oi meant 
the whole line from which the distances in the i direction 
was measured, and as in space Oi means the whole plane 
from which distances in the i direction are measured, so 
now Oh 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 cuhe represented. 



A FOUR-DIMENSIONAL FIGUEE 



129 



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, Ik, Oh. 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, 
in the h direction. 

With 3i we must also take 1^ 2k, Oh. This point 
is shown by the second star in the cube Ohi, 




In the first cube, since all the points are Oh points, 
we can only have varieties in which i, j, k, are accom- 
panied 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 

9 



130 



THE FOURTH DIMENSION 



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. 



2103 



0215 



5102 




2K 



Fig. 73. 



3K 



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 Ok 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 



A POUR-DIMENSIONAL FlGURfc 



131 



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 ques- 
tion are marked with a * in fig. 73. We see that the 
distance in question is the diagonal of a unit square. In 
like manner we find that the distance between corres- 
ponding 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 T BX^XJ 
drawing all the four 
cubes in the catalogue 
figure in one (fig. 74). 
These cubes are exact 
repetitions of one 
another, so one draw- 
ing will serve as a 
representation of the 
whole series, if we 
take care to remember 
where we are, whether 
in a Oh, a Ih, a 2h, 
or a 3h figure, when 
we pick out the points required. Fig. 74 is a represen- 
tation 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. 




Fig. 74. 



132 



THE FOURTH DIMENSION 



Hence these two hexagons fit together, forming one 
hexagon, and the line TVV is only wanted when we con- 
sider a section of the whole figure, we thus obtain the 
solid represented in the lower part of fig. 74. Equal 
repetitions of this figure, called a tetrakaidecagon, will 
fill up three-dimensional space. 

To make the corresponding four-dimensional figure we 
have to take five axes mutually at right angles with five 
points on each. A catalogue of the positions determined 
in five-dimensional space can be found thus. 

Take a cube with five points on each of its axes, the 
fifth point is at a distance of four units of length from the 
first on any one of the axes. And since the fourth dimen- 
also stretches to a distance of four we shall need to 

represent the succes- 

BO 



-ion 



4L 




sive sets of points at 
distances 0, 1, 2, 3,4, 
in the fourth dimen- 
sions, five cubes. Now 
all of these extend to 
no distance at all in 
the fifth dimension. 
To represent what 
lies in the fifth dimen 
sion we shall have to 
draw, starting from 
each of our cubes, five 
similar cubes to re- 
present 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 



OH 



Fig. 7r,. 



A FOUR-DIMENSIONAL FIGURE 133 

note when we look at it, whether we consider it as a OA, a 
\h, a 2h, etc., cube. Putting then the Oh, Ih, 2h, 3h, 4/4 
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 01 points, and has in it the i points 
from to 4, the j points from to 4, the k points from 
to 4, while we have to specify with regard to any 
selection we make from it, whether we regard it as a Oh, 
a Ih, a 2h, a 3&, or a 4 h figure. In fig. 76 each cube 
is represented by two drawings, one of the front part, the 
other of the rear part. 

Let then our five cubes be arranged before us and our 
selection be made according to the rule. Take the first 
figure in which all points are Ql points. We cannot 
have with any other letter. Then, keeping in the first 
figure, which is that of the Ql positions, take first of all 
that selection which always contains Ih. We suppose, 
therefore, that the cube is a Ih 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 Ql 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 01, 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, 3jf, Ik. 

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 some- 
thing like the 



A FOUR-DIMENSIONAL FIGURE 135 

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 basii 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-dimen- 
sional space by equal repetitions of themselves. 



CHAPTER XI 



NOMENCLATURE AND ANALOGIES PRELIM- 

INARY TO THE STUDY OF FOUR DIMEN- 

SIONAL 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 



c 




130 



NOMENCLATURE AND ANALOGIES 



to any colour with which it mixed. If there is no 
such real colour let us imagine such a colour, and 
assign to it the properties of the number zero, which 
makes no difference in any number to which it is 
added. 

Above this square place a red square. Thus we symbolise 
the going up by adding red to null. 

Away from this null square place a yellow square, and 
represent going away by adding yellow to null. 

To complete the figure we need a fourth square. 
Colour this orange, which is a mixture of red and 
yellow, and so appropriately represents a going in a 
direction compounded of up and away. We have thus 
a colour scheme which will serve to name the set of 
squares drawn. We have two axes of colours red and 

yellow and they may oc- 
cupy as in the figure the 
direction up and away, or 
they may be turned about ; 
in any case they enable us 
to name the four squares 
drawn in their relation to 
one another. 

Now take, in Fig. 78, 
nine squares, and suppose 
that at the end of the 
going in any direction the 




Fig. 78. 



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 
fvboye it, three yellows between the first null apd the 



138 



THE FOURTH DIMENSION 



null beyond it, while the oranges increase in a doublo 
way. 




Red 



Null 



Orange 



""Yellow 
Fig. 80. 



Red 
Null 



Fig. 79. 

Suppose this process of enlarging the number of the 
Null Yellow Null 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 



NOMENCLATURE AND ANALOGIES 



139 




null points. The yellow line will trace out a yellow and 
red, or orange square. 

Now, turning back to fig. 78, we see that these two 
ways of naming, the one we started with and the one we 
arrived at, can be combined. 

By its position in the group of four squares, in fig. 77, 
the null square has a relation to the yellow and to the red 
directions. We can speak therefore of the red line of the 
null square without confusion, meaning thereby the line 
AB, fig. 81, which runs up from the 
initial null point A in the figure as 
drawn. The yellow line of the null 
square is its lower horizontal line AC 
as it is situated in the figure. 

If we wish to denote the upper 
yellow line BD, fig. 81, we can speak 
of it as the yellow r line, meaning 
Flg * 8L the yellow line which is separated 

from the primary yellow line by the red movement. 

In a similar way each of the other squares has null 
points, red and yellow lines. Although the yellow square 
is all yellow, its line CD, for instance, can be referred to as 
its red line. 

This nomenclature can be extended. 
If the eight cubes drawn, in fig. 82, are put close 
together, as on the right hand of the diagram, they form 
a cube, and in them, as thus arranged, a going up is 
represented by adding red to the zero, or null colour, a 
going away by adding yellow, a going to the right by 
adding white. White is used as a colour, as a pigment, 
which produces a colour change in the pigments with which 
it is mixed. From whatever cube of the lower set we 
start, a motion up brings us to a cube showing a change 
to red, thus light yellow becomes light yellow red, or 
light orange, which is called ochre, And going tq the 



140 



THE FOURTH DIMENSION 



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, 





yellow 



Fig. 82. 



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. 



/ Null /White/ Null / 


A', ll.m/ 1 ' 1 ^' /Yellow/ 
/ / yellow / / 


/ /' , 


/ 


/ 
/ 

( 

/ 

/ 


/ 

/T3 
U 
M 

% / 

/ 
^3 

i z , 
*/ 

<u f 

f 


^ 

\ 


& 

4? 


* 


<? 


S 


^ 


/ 

X 


^ 


^ 



/ Null /White/ Null 

7 /LiVht /V n 
ellow/ . /Yellow 
_ /yellow/ 

Null /White/ Null 

/ Red /Pink / 
/ - -/ + 
/Orange /Ochre /Orange 

/Red /Pink Red 

Null 



/ Null / 



White 



Null White Null 




Third 

, QT7 

layer. 



Second 
layer. 



Fig. 88. 



We can conveniently represent a block of cubes by 
three sets of squares, representing each the base of a cube, 
the block, fig. 83, can be represented by the 



NOMENCLATURE AND ANALOGIES 141 

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 

. - 7 - 7 - 7 - 7 - 7 we obtain fig. 84, 
/n /wh/wh/wh/ n / 

5y ' - 



v Otters of the colours 

/ y- /'.y-/'-y : /.y : / y- / , 

/ y. /'y./'-y.A-y./y. / are glven mstead of 

/ n./ wh./ wh,./ wh./ n. / their f ull names. 

/ - -7 - T - -7 - -7 - -7 Here we see that 

/ - 7^ - / f- /- - / there are four null 
4, / or. / oc. / oc. / oc. / or. / 

/or./oc./oc./oc./or./ cubes as betore ' but 

/ or. / oc. / oc. / oc . /or/ the senes whlch SP"ng 

/ r. / p. / p. / p. / r. / f rom tbe initial corner 

i - 7 - 7 - 7 - 7 - 7 will tend to become 

/ T- / P- / P- / P- / ' / ,. , , 

3 /on /oc. /oc /oc. /or. / lines of CubeS ' as al ^ 

/ or./oc./oc./oc /or./ the Sets f Cubes 

/ or./ oc. / oc. /oc. / or. / parallel to them, start- 

/ */ p- / P- / P- / r - / in S from otner cor ners. 

ri p, p, p r v ThuS ' from the initial 

a line of 



g/or./oc./ oc.oc./ or. 

/ or. / oc . / oc. / oc. / or"/ red cubes a line of 
/ or. / oc. / oc. / oc. / or> / white cubes, and a line 
/ r. / p. / p. / p./ n / of yellow cubes. 

/ n /wh./ wh./whi./n. 7 If the number of the 
1 / y. /I. y / I. y./l.'y./ y. / cubes is ^rgely in- 
/ y> / ] - y-/l- y*>/ 1. y./ y- / creased, and the size 
/ y. /l-'y./i- y./l. y./ y. / of the whole cube is 
/ TI. / wh./ wh ./ wh../ n. / diminished, we get 

p. 84 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, ultimately the cube 



142 



THE FOURTH DIMENSION 



Null 



Null 



Null 



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 ulti- 
mately 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. 

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 




Fig. 85. 



NOMENCLATURE AND ANALOGIES 



143 



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 

Taking all the colours 



Fig. 86 

by taking these colours in pairs, 
together we get a colour name for the solidity of a cube. 
Let us now ask ourselves how the cube could be pre- 
sented 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 



White 





Null White Null wH. 



kce previously perceived 



Fig. 87. 

face, the plane being would observe a square surrounded 
by red and yellow lines, and having null points. See the 
dotted square, 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 



144 



THE 



DIMENSION 




is turning from its first position all of it except the red 
line leaves the plane goes absolutely out of the range 
of the plane being's apprehension. But when the yellow 
line points straight out from the plane then the pink 
face comes into contact with it. Thus the same red line 
remaining as he saw it at first, now towards him comes 
a face surrounded by white and red lines. 

If we call the direction to the right the unknown 
direction, then the line he saw before, the yellow line, 
goes out into this unknown direction, and the line which 
before went into the unknown direction, comes in. It 
comes in in the opposite direction to that in which the 
yellow line ran before ; the interior of the face now 

against the plane is pink. It is 
a property of two lines at right 
angles that, if one turns out of 
a given direction and stands at 
right angles to it, then the other 
'B 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 un- 
known dimension, the white line which before ran in the 



Fig. 83. 



NOMENCLATURE ANt> ANALOGIES 



145 



unknown dimension would come in downwards in the 
opposite sense to that in which the red line ran before. 

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 










Null-y * Null Wfcite NuIJ 
^Yellow 



Fig. 89. 

the first turning round the k axis, white runs negative j, 
yellow runs i, red runs k ; thus we have the table : 





i 


j 


li 


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 

10 



146 THE FOURTH DIMENSION 

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 



NOMENCLATIVE AND ANALOGIES 



147 



Null 





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 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, tf 

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 pre- 
sented 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 



Fig. 90. 



148 THE FOUfcTH DIMENSION 

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 appre- 
hend 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 



NOMENCLATURE AND ANALOGIES 149 

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 move- 
ment 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 thus 
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 



150 THE FOURTH DIMENSION 

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 realisa- 
tion 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. 



NOMENCLATURE AND ANALOGIES 



151 



Let the cubes in fig. 91 be the eight making a larger 
cube. 

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 





Fig. 91. 

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 eacli, as in fig. 92. The first 



152 



THE FOURTH DIMENSION 




are supposed to last a minute, and then the others to 

eome 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 repre- 
sents. Thus, at the begin- 
ning he would have null 
cube orange face, and after 
the motion had begun he 
would have null cube ochre 
section. As he could get 
the same coloured section whichever way the cube passed 
through, it would be best for him to call this section white 
section, meaning that it is transverse to the white axis. 
These colour-names, of course, are merely used as names, 
and do not imply in this case that the object is really 
coloured. Finally, after a minute, as the first cube was 
passing beyond his plane he would have null cube orange 
face again. 

The same names will hold for each of the other cubes, 
describing what face or section of them the plane being 
has before him ; and the second wall of cubes will come 
on, continue, and go out in the same manner. In the 
area he thus has he can represent any movement which 
we carry out in the cubes, as long as it does not involve 
a motion in the direction of the white axis. The relation 
of parts that succeed one another in the direction of the 
white axis is realised by him as a consecution of states. 

Now, his means of developing his space apprehension 
lies in this, that that which is represented as a time 
sequence in one position of the cubes, can become a real 
co-existence, if something that has a real co-existence 
becomes a time sequence. 



NOMENCLATURE AND ANALOGIES 



153 



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 




och, 



I. y. 



Fig. 93. 

The cu>e swung round the red line, so that the white line points 
towards us. 

point marked with a star is the same in the cubes and in 
the plane view. 

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 



-154 



THE FOURTH DIMENSION 



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 p'ane perpendicular to the yellow line. When 




A 



\ 



\ 



Fig. 94. 



The cube swung round yellow line, with red line running from left 
to right, and white line running down. 

the null cube has passed through the plane, as it is 
leaving it, we get again a pink face. 

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 



NOMENCLATURE AND ANALOGIES 



155 



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 



Mill 





Null 

r. y. wh s 






X 



XI 



Null 
r. y, wh 



X 



ite 



Null 



3 4 



have the same series of views of the red as he had of the 
null cube. 

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, w/i, 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 



156 THE FOURTH DIMENSION 

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. 



CHAFFER 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 sup- 
posing 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 

157 



158 tttE FOURT& 



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 



fttE SIMPLEST FOUfe-fciMENSiOKAL SOLID 



159 



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 ATI, A/, 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 sur- 
round 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 




Fig. 96. 



square A in its first position ; the next, that which coincides 
with the square A in its final position. Against these 
two faces cubes must be placed in order to completely 
enclose the cube A. These may be called the cubes left 
and right or Al and AT. 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, A/, 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 ATI, A far A/, A above 
Aa, A below Ab, A left Al, A right Ar, 
shown in fig. 97. If now the cube A 






Aa 


7f Af 


/ / 


Al 


/ 


/ 


Ar 


An 




D 


J 




Al 



Pig. 97. 



moves in the fourth dimension right out of space, it traces 
out a higher cube a tesseract, as it may be called. 



160 THE FOURTH DlMENSlOK 

Each of the six surrounding cubes carried on in the satne 
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, A/, 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 dimen- 
sion). Now the movement in this fourth dimension is 
not bounded by any of the cubes An, A/, 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, 



THE SIMPLEST FOUH-DlMENSIONAL SOLID 



161 



but the thought of an abstract boundary , the face of 
a cube. 

Let us now take our eight coloured cubes, which form 
ft cube in space, and ask what additions we must make 
to them to represent the simplest collection of four-dimen- 
sional 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-dimen- 
sional bodies bodies which in four-dimensional space 
correspond to cubes in three-dimensional space, and these 
bodies we call tesseracts. 

Given then the null, white, red, yellow cubes, and 
those which make up the block, we 
notice that we represent perfectly 
well the extension in three directions 
(fig. 98). From the null point of 
the null cube, travelling one inch, we 
come to the white cube ; travelling 
one inch away we come to the yellow 
cube ; travelling one inch up we come 
to the red cube. Now, if there is 
a fourth dimension, then travelling 
from the same null point for one 



Red 


Pink 


N 
\ 
\ 


u 

\ 


Null 

X 


White 


\ \ 


\fellow\ , 
\ vellov 



( Orange hidden) 
Fig. 98. 

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 

11 



162 



THE FOURTH DIMENSION 



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 re- 
gions, 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 re- 
gions, which we 

Fig. 99. should come to by 

A plane being's representation of a block going in the fourth 
of eight cubes by two sets of four squares. dimension from each 

of our eight cubes, by eight cubes placed in some arbitrary 
position relative to our first eight cubes. 




Red 


Pink 


\ 
\ 
\ 


CA 
o 

* 


Null 

X 


White 


\ \ 


\Yellow\L'g h < v 




(1) 

Orange hidden 



(2) 
Brown hidden 



Fig. 100. 



Our representation of a block of sixteen tesse acts by 
two blocks of eight cubes.* 

Hence, of the two sets of eight cubes, each one will serve 

* The eight cubes used here in 2 can be found in the second of the 
model blocks. They can be taken out and used. 



THE SIMPLEST FOUR-DIMENSIONAL SOLID 163 

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 di- 
mension, 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 



164 THE FOURTH DIMENSION 

natural supposition to make. It is also natural to siippOse 
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 hav^e 
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 SIMPLEST FOUR-DIMENSIONAL SOLID 



165 



the first block, but in their proper positions they could 
not be in space with the first set. 3 represents 27 cubes 

Fig. 101. 



Null 


White 


Null 


bellow 


Light 
yellow 


Yellow 


Null 


White 


Null 



Bhie 


Light 
blue 


Blue 


Green 


Light 
green 


Green 


Blue 


Light 
blue 


Blue 



Null 


White 


Null 


Yellow 


Light 
yellow 


Yellow 


Null 


White 


Null 



Red 


Pink 


Red 


)range 


Ochre 


Grange 


Red 


Pink 


Red 



Purple 


Light 
purple 


Purple 


Brown 


Light 
brown 


Brown 


Purple 


Light 
purple 


Purple 



Red 


Pink 


Red 


)range 


Ochre i 


) range 


Red 


Pink 


Red 



Null 


White 


Null 


Yellow 


Light 
yellow 


Yellow 


Null 

I 


White 


Null 



Eao'.i cube is the begin- 
ning of the first tesseract 
going in the fourth di- 
mension. 




Bach cube is the begin- 
ning of the second 
tessenict. 



Null 


White 


Null 


Yellow 


Light 
yellow 


Yellow 


Null 


White 


Null 



3 

Each cube is the l>e<:i li- 
ning of the thhd 
tesseract. 



(forming a larger cube) which are the beginnings of the 
tesseraets, which begin two inches in the fourth direction 
from our space and coi}tinu.e another incft, 



166 



THE FOURTH DIMENSION 



In fig. 102, we have the representation of a block of 
4x4x4x4 or 256 tesseracts. They are given in 

Fig. 102,* 




A cube of 64 cubes, 
each 1 in. x 1 in. 
x 1 in., the begin- 
ning of a tesseract. 



A cube of 04 tubes, 
each 1 in. x 1 in. 
x 1 in. the begin- 
ning of tesseracts 
1 in from our space 
on the 4th dimen- 
sion. 



A cube or 04 cubes, 
each 1 in. x 1 in. 
x 1 in., the begin- 
ning of tesseracts 
2 in. from our space 
in the 4th dimen- 
sion. 



A cube of 64 cubes, 
each 1 in. x 1 in. 
x 1 in., the begin- 
ning of tesseracts 
starting 3 in. from 
our space in the 4th 
dimension. 



four consecutive sections, each supposed to be taken one 
inch apart in the fourth dimension, and so giving four 

* The coloured plate, figs. 1, 2, 3, shovys these relations more. 



THE SIMPLEST FOUR-DIMENSIONAL SOLID 167 

blocks of cubes, 64 in each block. Here we see, com- 
paring 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 repre- 
sented 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 1. 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 



168 THE FOURTH DIMENSION 

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 repre- 
sented 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 an4 the right and left dimensions, 



THE SIMPLEST FOUR-DIMENSIONAL SOLID 169 

We get then altogether, as three-dimensional regions, 
ochre, brown, light purple, light green. 

Finally, there i;> 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 dimen- 
sions, 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 is 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, 



170 THE FOURTH DIMENSION 

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 pf its, own kind, an initial a.nd a final 



THE SIMPLEST FOUR-DIMENSIONAL SOLID 171 

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 jn this direction traces out a cub. 



172 THE FOURTH DIMENSION 

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 move- 
ment 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 inter- 
mediate 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 correspond- 
ing 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 ciibes, squares, lines, points, a,$ 
boundaries, 



THE SIMPLEST FOUR- DIMENSIONAL SOLID 173 

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 



174 



THE FOURTH DIMENSION 



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 






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ochre cube, the light purple cube, the brown cube, and 

the light green cube. :v . 

A complete view of the tesseract in its various space 



THE SIMPLEST FOUR-DIMENSIONAL SOLID 



176 




f 



presentations is given in the following figures or catalogue 
cubes, figp. 103-106. The first cube in each figure 



176 



THE FOURTH DIMENSION 



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, 



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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. 



THE SIMPLEST tOUR-DlMfcN SlONAL SOtlt) 



177 






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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 

178 



REMARKS ON THE FIGURES 



170 



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 nomen- 
clature, which is merely indicated here. 




'iw I \i I ^ A-NJ I ^J/^NJ I -*\\L 

" wh. n. bl. 1. bl.bl. bl. I. bl. bl. bl 1. bl.bl. n. w h. n. 
interior interior interior interior interior 



Ochre L.Brown L.Brown L. Brown 
Fig. 107. 



Ochre 



In this figure the tesseract is shown in five stages 
distant from our space: first, zero ; second, in. 5 third, 
f in. ; fourth, in. ; fifth, 1 in.; which are called 60, 61, 
62, 63, 64, 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 60 cube, 
as before, simply by remembering that transference along 
the 6 axis gives the addition of blue to the colour of 
(he region in the ochre, the 60 cube. In the final cube 
64, the colouring of the original 60 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 61, 62, 63, taken 
at , , | 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 



180 THE FOURTH DIMENSION 

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 &,, 6 2 , 6 3 , and the farther 
face of this cube in the blue direction is shown in 6 4 
a pink face, called pink b because it is distant from the 
pink face we began with in the blue direction. Thus 
the cube which we colour light purple appears as a lasting 
square. The square face itself, the pink face, vanishes 
instantly the tesseract begins to move, but the light 
purple cube appears as a lasting square. Here also 
duration is the equivalent of a dimension of space a 
lasting square is a cube. It is useful to connect these 
diagrams with the views given in the coloured plate. 

Take again the orange face, that determined by the 
red and yellow axes ; from it goes a brown cube in the 
blue direction, for red and yellow and blue are supposed 
to make brown. This brown cube is shown in three 
sections in the faces 6 t , 6 2 , 6 3 . In 6 4 is the opposite 
orange face of the brown cube, the face called orangp b, 



BEMARKS ON THE FIGURES 



181 



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 dimen- 
sion, and take the blue axis into our known space 
dimension. Since the white and blue axes are perpen- 
dicular 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. 



wh/j 



\vh 



wh, 




- bl. n - I. bl.wh. 1. bl.wh. 1. bl.wh. n. bl. n. 
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 dimen- 
sion, 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 per- 
fectly systematic way. The red line traces out a purple 



182 THE FOURTH DIMENSION 

square in the transference along the blue axis by which 
this cube is generated from the orange face. This 
purple square made by the motion of the red line is 
the same purple face that we saw before as a series of 
lines in the sections b lt 6 2 . ^3- Here, since both red and 
blue axes are in our space, we have no need of duration 
to represent the area they determine. In the motion 
of the tesseract across space this purple face would 
instantly disappear. 

From the orange face, which is common to the initial 
cubes in fig. 107 and fig. 108, there goes in the blue 
direction a cube coloured brown. This brown cube is 
now all in our space, because each of its three axes run 
in space directions, up, away, to the left. It is the same 
brown cube which appeared as the successive faces on the 
sections 6 lf 6 2 , 6 3 . Having all its three axes in our 
space, it is given in extension ; no part of it needs to 
be represented as a succession. The tesseract is now 
in a new position with regard to our space, and when 
it moves across our space the brown cube instantly 
disappears. 

In order to exhibit the other regions of the tesseract 
we must remember that now the white line runs in the 
unknown dimension. Where shall we put the sections 
at distances along the line ? Any arbitrary position in 
our space will do : there is no way by which we can 
represent their real position. 

However, as the brown cube comes off from the orange 
face to the left, let us put these successive sections to 
the left. We can call them wh , ivh lt wh. 2 , wh 3 , ivh, 
because they are sections along the white axis, which 
now runs in the unknown dimension. 

Eunning from the purple square in the white direction 
we find the light purplejmbe. This is represented in the 
sections wh lt wh 2 , tdi*(t0A^fig. 108. It is the same cube 



REMARKS ON THE FIGURES 183 

that is represented in the sections 6 1} 6 2 , b 3 in fig. 107 
the red and white axes are in our space, the blue out of 
it ; in the other case, the red and blue are in our space, 
the white out of it. It is evident that the face pink y, 
opposite the pink face in fig. 107, makes a cube shown 
in squares in 6 1? 6 2 , 6 3 , 6 4 , on the opposite side to the I 
purple squares. Also the light yellow face at the base 
of the cube 6 , makes a light green cube, shown as a series 
of base squares. 

The same light green cube can be found in fig. 107. 
The base square in wh 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 

6 , 61, & 2 > &3> b t . 

The case is, however, a little different with the brown 
cube. This cube we have altogether in space in the 
section ivh 6 , fig. 108, while it exists as a series of squares, 
the left-hand ones, in the sections &<>, b u b& 6 3 , 6 4 . The 
brown cube exists as a solid in our space, as shown in 
fig. 108. In the mode of representation of the tesseract 
exhibited in fig. 107, the same brown cube appears as a 
succession of squares. That is, as the tesseract moves 
across space, the brown cube would actually be to us a 
square it would be merely the lasting boundary of another 
solid. It would have no thickness at all, only extension 
in two dimensions, and its duration would show its solidity 
in three dimensions. 

It is obvious that, if there is a four-dimensional space, 
matter in three dimensions only is a mere abstraction ; all 
material objects must then have a slight four-dimensional 
thickness. In this case the above statement will undergo 
modification. The material cube which is used as the 
model of the boundary of a tesseract will have a slight 
thickness in the fourth dimension, and when the cube is 



184 THE FOURTH DIMENSION 

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 6 19 6 2 > b 3 , are the 
very same brown sections that would be obtained by 
cutting the brown cube, wh , across at the right distances 
along the blue line, as shown in fig. 108. But as these 
sections are placed in the brown cube, wh , they come 
behind one another in the blue direction. Now, in the 
sections ivh lt wh 2 , wh 3 , we are looking at these sections 
from the white direction the blue direction does not 
exist in these figures. So we see them in a direction at 
right angles to that in which they occur behind one 
another in wh . There are intermediate vi^ews, 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 



REMARKS ON THE FIGURES 



185 



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 
2/o, 2/1, 2/2> 2/3? 2/4> as they are made at distances 0, 1, 2, 3, 4, 
quarter inches along the yellow line. We suppose these 
cubes arranged in a line coming towards us not that 
that is any more natural than any other arbitrary series 
of positions, but it agrees with the plan previously adopted. 




Fig. 109. 

The interior of the first cube, 2/0, is that derived from 
pink by adding blue, or, as we call it, light purple. The 
faces of the cube are light blue, purple, pink. As drawn, 
we can only see the face nearest to us, which is not the 
one from which the cube starts but the face on the 
opposite side has the same colour name as the face 
towards us. 

The successive sections of the series, y oj y^ y 2 , etc., can 
be considered as derived from sections of the 6 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 &< fig. 107. 
It is an ochre section with lines orange and light yellow. 
This section will therefore take the place of the pink face 



186 



THE FOURTH DIMENSION 



in ^ when we go on in the yellow direction. Thus, the 
first section, y lt 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 




Fig. 110. 



come in beneath the position of our ochre cube, the one 
we have been in the habit of starting with. 

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. 



REMARKS ON THE FIGURES 187 

As the red line now runs in the fourth dimension, the 
imccessive sections can be called TO, r\, r 2 , r 3 , r 4 , these 
letters indicating that at distances 0, J, , f , 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 
ro, when the section r\ comes in, by looking at the cube 
60, fig. 107. What is distant in it one-quarter of an inch 
from the light yellow face in the red direction ? It is an 
ochre section with orange and pink lines and red points ; 
see also fig. 103. 

This square then forms the top square of r\. Now we 
can determine the nomenclature of all the regions of r\ 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 hori- 
zontal section of ro, 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 ro 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, 61, b 2 , etc. In them we see that 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 n replaces the section of r 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 
tesseract s. We can take a group of eighty-one of them 
3x3x3x3, in four dimensions, and each tesseract will 
have its name null, red, white, yellow, blue, etc., and 



188 



THE FOURTH DIMENSION 



whatever cubic view we take 
what sides of the tesseracts 
they touch each other.* 

Thus, for instance, if we 
shown below, we can ask how 

In the arrangement given 
white, red, yellow, in space, 
dimension. Hence we have 
Imagine now the tesseractic 
our space we have first of 



of them we can say exactly 
we are handling, and how 

have the sixteen tesseracts 
does null touch blue, 
in fig. Ill we have the axes 
blue running in the fourth 

the ochre cubes as bases, 
group to pass transverse to 

all null ochre cube, white 



O'J 
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os 



cj 




\c/JrX 


White 


\ 
\ 

\ 


\ \ 


Red 


Pink 


Null 

X 


White 



r. 
a 




\ -\Ligl 


it brown 
White 


\ 
\ 
\ 


\ \ 


Purple 


Light 
purple 


Blue 


Light 
blue 



direction 



yellow hidden 
A,b 



Light green 
!B 



Fig. 111. 



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 trans- 
verse 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. Ill, where we have the ochre 

* At this point the reader will find it advantageous, if he has the 
models, to go through the manipulations described in the appendix. 



REMARKS ON THE FIGURES 



180 



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. 

If we ask again how red touches light blue tesseract, 
let us rearrange our group, fig. 112, or rather turn it 





White 


\ 

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Light 
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Light 
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\P^X 


s 

1 
J 


~S x 




Light 
Brown 


(^> 

-^ 

5 




Light 
green 



White 
direction 



Light yellow 
Bidden 



Fig. 112. 



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 arrange- 
ment 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. 



190 THE FOURTH 

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.yu fig. 112. Between figure C 
and C.yi 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. 



ON THE FIGURES 



191 




Red 



Yellow 



Blue Null 



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 inch each, it 
takes the positions 11, 22, 33, 
and ends as a red line. 

Now, the whole of this red, 
yellow, blue, or brown cube ap- 
j'ig liy pears as a series of faces on the 

successive sections of the tes- 
seract 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. 



Yello 




Nufo White ' 
bo 



Fig. 114. 



Thus drawing our initial cube and the successive 
sections, calling them b , 61, 63? 63, &4, fig- 115, we have 
the red line subject to this movement appearing in the 
positions indicated. 

We will now investigate what positions in the tesseract 
another line in the pink face assumes when it is moved in 
a similar manner. 

Take a section of tha 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. 



192 



FOtfRTft 



From this section a cube goes off in the fourth dimen- 
sion, which is formed by moving each point of the section 
in the blue direction. 



Yellow 




Null White 
FiR. 115. 




Light blue White 
Fig. 11H. 



Drawing this cube we have fig. 116. 

Now this cube occurs as a series of sections in our 
original representation of the tesseract. Taking four steps 
as before this cube appears as the sections drawn in 6 , b\, 
b-2, b 3 , 64, fig. 117, and if the line 4 is subjected to a 
movement equal in the blue and yellow directions, it will 
occupy the positions designated by 4, 4^ 4 2 , 4 3 , 4 4 . 




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 repre- 
sented by the set of planes delineated in Fig. 118, pink 



REMARKS ON TfiE FIGURES 



fade 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 




Null b 



the light purple cube, this space going off from the pink 
face is fixed. 

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 

13 



194 



THE FOURTH DIMENSION 



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 deter- 
mination 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 parsing through the 
null r, null wh, and null y 
points, fig. 119. Looking at 
the orange square, which, as 
usual, we suppose to be ini- 
tially in his plane, he seen 
that the line from null r to 
null y, which is a line in the 



Nully. 




Null-wh. 



Null A 
Fig. 119. 

section plane, the plane, namely, through the three 
extremities of the edges meeting in null, cuts the orange 



fcEMAfcKS ON THE fIGUfcES 196 

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 ivh 
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 ivh 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. 



196 



FOURTH DIMENSION 



Red 






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). 

Here we see three out 
j of the four points through 
1*J which the cutting plane 
passes, null r, null y, and 
null b. The plane they 
cutting space, and this plane 



*\ 


g r - 


or. 


pur. 

A 



ellow 



Null- b. Blue Null 
Fig. 120. 

determine lies in the 
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 there- 
fore the plane these determine 
lies in the cutting space. 

This triangle lies before us. 
It has a light purple interior 
and pink, light blue, and 
purple edges with null points. 
This, since it is all of the 
plane that is common to it, and this bounding of the 
tesseract, gives us one of the bounding faces of our sec- 
tional 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 




Nullb. 
Fig. 121. 



REMARKS ON THE FIGURES 



197 



Null 




Null-wb. 



pur-n.r.pur. 



space, and the triangle they determine is common to 

the tesseract and the cut- 
ting space. Hence this 
boundary is a triangle hav- 
ing a light yellow line, 
which is the same as the 
light yellow line of the first 
figure, a light blue line and 
a green line. 

We have now traced the 
cutting space between every 

NHll ' b : 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 fines. 

The triangles are shown in fig. 123 as they join on to 
the triangle in the ochre cube. But 
they join on each to the other in an 
exactly similar manner; their edges 
are all identical two and two. They 
form a closed figure, a tetrahedron, 
enclosing a light brown portion which 
is the portion of the cutting space 
which lies inside the tesseract. 

We cannot expect to see this light brown portion, any 
more than a plane being could expect to see the inside 
of a cube if an angle of it were pushed through his 
plane. All he can do is to come upon the boundaries 
of it in a different way to that in which he would if it 
passed straight through his plane. 

Thus in this solid section ; the whole interior lies per- 
fectly open in the fourth dimension. G-O round it as 
we may we are simply looking at the boundaries of the 
tesseract which penetrates through our solid sheet. If 
the tess^rapt were not to pass across so far, tl^e triangle 




198 



THE FOURTH DIMENSION 



Null-y. 



P.: 



Null 
Fig. 124. 



Null-wh. 



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 dimen- 
sions is as follows : The plane 
being can see that line of the 
sectional plane through null y, 
null iv, null r, which lies in 
the orange plane. Let him 
now suppose the cube and the 
section plane to pass half way 
through his plane. Replacing 
the red and yellow axes are lines parallel to them, sections 
of the pink and light .yellow faces. 

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 ivh point, and can see (compare fig. 119) 
that it cuts A& 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 pf the section plane in the 
light yellow face r 



REMARKS ON THE felGURES 199 

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 suc- 
cession 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 tlie 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 trans- 
ference 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 
cpmes in to the left. 



200 



THE FOURTH DIMENSION 




Null-b. Blue 
Fi 



. _ 



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 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 




Blue L.blue bl. 
Section bg interior Light brown 
Fig. 126. 



section made of this cube by 
the plane in which the sec- 
tional space cuts it, is an 
equilateral triangle with purple, 1. blue, green points, and 
1. purple, brown, 1. 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 it ended in a point, a null point. Each of its 
edges would be of the same length. Thus the successive 



REMARKS ON THE FIGURES 



201 



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. 
Jhjs space cuts the red, white, yellow axes in the 



- : 5 - .-- 




nd the 

to our spaee, a 
11 a section of the 
to the 

"of fire 

pninti in 

the cubical wtioo of the 
predrawn. In 2* (fig. 72) the 
to the axes at a distance 
iathe wetTO 
br it. FinaOj when 3& 
fjiy jtxes at a distance 
tnsngle, of which 
the hfii&iiB drawn is a truncated portion. After ttia 
the teaKnet, whidi I'lTiin 1 ! rnfy three mnU in each of 

of ov ffiacevaad there if no more of it to be eat. Hence, 
we hare the section determined by the jtkoiar slanting 



CHAPTER XIV.* 



07 



-i- 



(fee it O 



:ir 



tlte. 



- - - - ^ - - 

dk VBfr m^BBB^t *fc 



204 THE FOURTH DIMENSION 

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 experi- 
ence 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 



RECAPITULATION AND EXTENSION 205 

out if the vast solid sheet against which, on the sup- 
position 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 sup- 
position 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 con- 
ception 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 



206 THE FOURTH DIMENSION 

on a plane; and when we set about to form the Con- 
ception 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 by 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 



RECAPITULATION AND EXTENSION 207 

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 under- 
stand 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 im- 
possibility to produce it in the plane. 

The demonstration of rotation about a line must be to 



208 THE FOURTH DIMENSION 

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 
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 
y that their extension in the third di- 

mension escapes the observation of the 
plane being, and he thinks about them 
as if they were mathematical plane 
5' figures in a plane instead of being 
material bodies capable of moving on 



A B x a plane surface. Let Ax, Ay be two 

Fig. 1 (129;. axes an d 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. 



RECAPITULATION AND EXTENSION 209 

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 neglect one of his axes and represent 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 draw- 
ing. Here the z axis has taken 



^ 8 the place of the y axis, and the 

Fig. 2 (130). 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 AB : 
and after 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'(f 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 

14 



21 TttE FOURTH DIMENSION 

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 ACEGHDIIF. 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. 



RECAPITULATION AND EXTENSION 



211 



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 rota- 
tion about a line in our examination 
of rotation about a plane. Fig. 3 
represents a cube in our space, the 
three axes x, y, z denoting its 
three dimensions. Let w represent 
the fourth dimension. Now, since 
in our space we can represent any 
three dimensions, we can, if we 



H 



B 



A C 

Fig. 3 (131). 

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 ; 



B 



212 THE FOURTH DIMENSION 

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. 

Now, suppose the whole cube to 
, Fig. 4 (132). 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 
Fig. 5 (133). the space of zyiv, and examine the 

turning there. We must now let the z axis disappear 
and let the iv axis run in the direction in which the z ran. 
Making this representation, what 
do we see of the cube ? Obviously 

\we bee only the lower face. The rest 
^ of the cube lies in the space of xyz. 

\ In the space of xyz we have merely 



A C the base of the cube lying in the 

Fig. 6 (134). plane of xy, as shown in fig. 6. 

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. 



BECAPITULAT10N AND EXTENSION 



213 



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. 

Suppose this turning to continue 
* till AC runs to the left from A. 
Fig. 7 (135). r occupy the position 



H 




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 



2"?pQSiftcn . Imposition 
Fig. 8 (13G> 



space in which the turning takes place being a different 
one from that in which the cube lies. 

One of the conditions, then, of our inquiry in the 
direction of the infinitely small is that we form the con- 
ception 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 trans- 
formations 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 



214 THE FOURTH DIMENSION 

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 struc- 
tures those with a right and left symmetry are every- 
where 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 signifi- 
cance 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 



RECAPITULATION AND EXTENSION 



215 



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 goin^ 
down the end c will be coming 
up. They will meet and con- 
flict. 

But if we rotate the rods 
about the plane of AB by the 
z to w rotation these move- 
ments will not conflict. Sup- 
pose all the figure removed 




B 



Fig. 9 (137). 



with the exception of the plane xz, and from this plane 
draw the axis of w, so that we are looking at the space 
of xzw. 

Here, fig. 10, we cannot see the lines A and B. We 
see the points G and H, in which A and B intercept 
the x axis, but we cannot see the lines themselves, for 
they run in the y direction, and that is not in our 
drawing. 

Now, if the rods move with the z to w rotation they will 



216 THE FOURTH DIMENSION 

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 har- 
moniously. We can think of 
w* 

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-dimen- 
sional turning, all the contour must be free. The whole 
contour corresponds to the ends of the rod. Each point 
of the contour can be looked on as the extremity of an 
axis in the body, round each point of which there is a 
rotation of the matter in the disk. 

If the one end of a rod be clamped, we can twist the 
rod, but not turn it round ; so if any part of the contour 
of a disk is clamped we can impart a twist to the disk, 
but not turn it round its central plane. In the case of 
extensible materials a long, thin rod will twist round its 
axis, even when the axis is curved, as, for instance, in the 
case of a ring of India rubber. 



RECAPITULATION AND EXTENSION 



217 



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 thick- 
ness of the matter is rep- 
resented by the rods CD and 
EF, extending equally with- 
' out and within. 

Now, take the section of 
the sphere by the yz plane 
we have a circle fig. 12. 
Now, let the w axis be drawn 
in place of the x axis so that 




Fig. ll (139). 



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 
Fig. 12 (140). f rom 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 




218 THE FOURTH DIMENSION 

all, as the fourth dimension is at right angles to all the 
sections which can be made of the sphere. 

We have supposed the matter of which the sphere is 
composed to be three-dimensional. If the matter had a 
small thickness in the fourth dimension, there would be 
a slight thickness in fig. 12 above the plane of the paper 
a thickness equal to the thickness of the matter in the 
fourth dimension. The rods would have to be replaced 
by thin slabs. But this would make no difference as to 
the possibility of the rotation. This motion is discussed 
by Newcomb in the first volume of the American Journal 
of Mathematics. 

Let us now consider, not a merely extensible body, but 
a liquid one. A mass of rotating liquid, a whirl, eddy, 
or vortex, has many remarkable properties. On first 
consideration we should expect the rotating mass of 
liquid immediately to spread off and lose itself in the 
surrounding liquid. The water flies off a wheel whirled 
round, and we should expect the rotating liquid to be 
dispersed. But see the eddies in a river strangely per- 
sistent. 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 inde- 
structible. 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 



RECAPITULATION AND EXTENSION 219 

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 pro- 
perty 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 signifi- 
cance 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 



220 THE FOURTH DIMENSION 

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 sub- 
jected 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 wheiv 
subject to one only. When subject to a rotation such a* 
that of x to y, a whole plane in the body, as we have 
seen, is stationary. When subject to the double rotatioh 
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 



RECAPITULATION AND EXTENSION 



221 



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. 



p' Showing axes 
xyz 

y 



Fig. 13 (141). 





Fig. 14 (142). 



In fig. 13, let us imagine the rotation in the direction 
xy to be taking place. The point x will turn to y-, and p 
to p'. The axis zz remains stationary, and this axis is all 
of the plane zw which we can see in the space section 
exhibited in the figure. 

In fig. 14, imagine the rotation from z to w to be taking 
place. The lu axis now occupies the position previously 
occupied by the y axis. This does not mean that the 
w axis can coincide with the y axis. It indicates that we 
are looking at the four-dimensional sphere from a different 
point of view. Any three-space view will show us three 
axes, and in fig. 14 we are looking at xzw. 

The only part that is identical in the two diagrams is 
the circle of the x and z axes, which axes are contained 
in both diagrams. Thus the plane zxz' is the same in 
both, and the point p represents the same point in both 



222 ME FOURTH 

diagrams. Now, in fig. 14 let the zw rotation take place, 
the z axis will turn toward the point iv 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 com- 
pounded, 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, iv. The rotation of x to y can 
be accompanied with the rotation of z to iv. 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. 



feECAPlTtTLAflON AND EXTENSION 223 

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 corpus- 
cular 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 com- 
ponents 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 
to iv, and the second being x to y and iv to z, when (hey 
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 



224 THE FOURTH DIMENSION 

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 trans- 
mitting vibration given it by such vortices, we must 
inquire how they lie together in four-dimensional space. 
Placing a circular disk on a plane and surrounding it by 
six others, we find that if the central one is given a motion 
of rotation, it imparts to the others a rotation which is 

antagonistic in every two ad- 
jacent ones. If A goes round, 
as shown by the arrow, B and 
C will be moving in opposite 
ways, and each tends to de- 
stroy the motion of the other. 
Now, if we suppose spheres 
to be arranged in a corre- 
sponding manner in three- 
dimensional space, they will 
Fig. 15(143). , . 1 ' J . . 

be grouped m 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 
tetrakai decagon. 

Now, assuming space to be filled with such tetrakai- 
decagons, and placing a sphere in each, it will be found. 




EECAPITULATION AND EXTENSION 225 

that one sphere is touched by eight others. The re- 
maining 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 tetra- 
kaidecagons and twenty hexagonal prisms. Such figures 
will exactly fill four-dimensional space, five of them meet- 
ing 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 fpur-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 

15 



226 THE FOURTH DIMENSION 

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 pome 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 



RECAPITULATION AND EXTENSION 221 

direction. Now, if rotation in one direction corresponds to 
positive electricity, rotation in the opposite direction cor- 
responds 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. 

Ah 1 three-dimensional motions are compounded of rota- 
tions 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 
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 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 0, 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 



228 THE FOURTH DIMENSION 

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. 



RECAPITULATION AND EXTENSION 229 

If we take a very long magnet, so long that one of its 
poles is practically isolated, and pat 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. 



230 THE FOUBTH DIMENSION 

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. * 

* 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. Eowan 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. 

Jn the first step, that of forming a clear idea how a. 

28} 



232 THE FOURTH DIMENSION 

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 card- 
board, or a book, and place it so that it forms a wall 
extending up and down not opposite to you, but run- 
ning 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 



APPENDIX I 233 

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, 



234 THE FOURTH DIMENSION 

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 



APPENDIX I 235 

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 ap- 
pearances 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 



236 THE FOURTH DIMENSION 

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. Bat 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 
thiid 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. 

Jt is easy to see what they are. The old arrangements 



APPENDIX I 237 

have to be broken up, and the layers taken in order, the 
fir^t layer of each for the representation of the aspect ol 
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 presenta- 
tions. 

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 rum ing 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 -f-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 



238 THE FOUETH DIMENSION 

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 



APPENDIX I 239 

can be used instead of the models. It has three axes, 
red, white, yellow, in our space. Hence the cube deter- 
mined 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 trans- 
verse 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 



240 THE FOURTH DIMENSION 

a cross our space we must call our cubes tesseraci sections. 
Thus on null pass-ing 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-dimen- 
sional space. Each part of each tesseract has been in 
our space, and we could have touched it. 



APPENDIX I 241 

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. 

16 



242 THE FOURTH DIMENSION 

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., 



APPENDIX I 243 

orangfr 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 



244 THE FOURTH DIMENSION 

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, 



APPENDIX I 245 

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 tessaracts 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 down- 
wards, 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 dimen- 
sion. 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) 



246 THE FOURTH DIMENSION 

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 corre- 



APPENDIX I 247 

spending 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 OP SPACE 

THE mere naming the parts of the figures we con Rider 
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 

248 



APPENDIX II 249 

facilitation it brings with it to the study of four-dimen- 
sional 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 funda- 
mental 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 experi- 
ment on a slave boy standing by. He makes certain 



250 THE FOURTH DIMENSION 

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 dis- 
cussion 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 



APPENDIX H 251 

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 mathe- 
matical 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 oi 
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 



252 THE FOURTH DIMENSION 

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 per- 
spective, 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 con- 
ception 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 formulae 
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. 



APPENDIX n 253 

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 plogged 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 



254 THE FOURTH DIMENSION 

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 con- 
tinuation 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 acces- 
sible 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 ? " 

Fir. t 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 ac- 
quainted 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 remem- 
bered 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 



n 265 

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 con- 
ditions 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 appre- 
hending 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 



256 THE FOURTH DIMENSION 

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 intui- 
tions 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 informa- 
tion from the conn ing-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 con- 
sciousness, 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 



APPENDIX II 257 

directing the ship over the plane surface of the ocean, so 
the soul is capable of a kind of movement, has an ampli- 
tude 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, 

17 



258 THE FOUfcTH DIMENSION 

just as little as by immersing himself in the steering of 
his ship over the plane surface of the ocean, a captain 
can loose the faculty of thinking about what he actually 
does, so little can the soul loose 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 possi- 
bilities 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 



APPENDIX II 259 

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 per- 
sonally. 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 ex- 
periment. 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 



260 THE FOURTH DIMENSION 

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 con- 
venient to take as the initial square that 
marked with an asterisk, so that the 
Fig. i. 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 considerations 
of the preceding pages. 




APPENDIX II 



261 



I. Extension. 

Call the large squares in fig. 2 by the name written 
in them. It is evident that each 
can be divided as shown in fig. 1. 
Then s 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." 

This principle of extension can 
be applied in any number of dimensions. 



Fig. 2. 



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." 




262 



THE FOURTH DIMENSION 



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 




APPENDIX 



263 



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 
.*.. *. se i 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 rect- 
angular 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 
apellatires which, like the colour names on page 174, 
fig. 103, apply to all the points, lines, faces, solids, and to 




264 



THE FOURTH DIMENSION 



the hypersolid 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." 



5tt(n S*Uf 5ell 5aW Siltt 5<jll 5ilen. Oilet Stlel 



or&l 


\5ela^ N^/> Cx> 


\^ 5 ala > x<> or s'<L 


\^> 5ilaf N^y 


Srv 


Win 5liT\ ^ 


N,^ 


t ! s l-K CoU 
Sal in .'XiMX gjj 


X-5'lin 5iW\ 




_ 


Sel it 


? 


.t 


Salit 


3 


- Silt^ 




^, 


C/" C/> 




\ 


2^ &" 




^ 0> <s> 


Snra 
or,W 


^ 


? ^Seh't ^ 


trjn 


\, 


5-5.Hl ^ 


Sinen 
or5in 


\>, ? ^ 


\ 


Stni'n 


Seiil \ 


\imrt 


oauit \ Ociun 


5*nil" Soiil Snn.it 




Cr. 


\ 




M ? 


\ 




gN 




tf 


f *s 




Sanel JJiitl 


"^ 

.r 




C-ffiJ '^ 


*\ , ^ 

V"" \ 


X ^ 

V Santf ^V 


\ Smit'V 



Stntl 



fjvtsri.r 5t*t 



iTiferior Jj^f luTfrior Oil 



APPENDIX II 



265 



Sil Mia Siltt Saia M Slid 5al,l M 




OantL 



f o 
Sit 2, 


X 




Sifcl ^ 


\ 




Cx* 


\ 




./ 






t/: 




(* r 


1* 


<5dii- .Sen. 


H 




3anef<S*et 


^ 




S I.S 1 


e< v 


\Jana. ^ 


Sanat Xtfx 


^mr^ 

\ OiniL ^y 



Inferior 5arai 



inltrior Jat^ 



iianjcur 5;m Sill* Si m S:?ii 




Sfniil 



5itet S 
5niJ ii'tif^ 


\ 

^ 




g 


\ 

X 




5m if Jmil 


X 

\ 


\J^ 


Ny 5a n.dt 


X 4 '" 1 ' 



266 



THE FOURTH DIMENSION 



Se.t Qtle.t OcTet Sel 

Cx- N&x c !- f f^* f f 



Sel. I 




Inferior Oan*t 



Interior iat.t 



We can now name any section. Take .(7. 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 b the section 
is the one written above. In bj the section is made by a 



APPENDIX II 267 

plane which cuts the three edges from sanen intermediate 
of their lengths and thus will be : 

Area : satat. 

Sides : satan, sanat, satet. 

Vertices : sanan, sanet, sat. 

The sections in b a , b 3 will be like the section in b, but 
smaller. 

Finally in b 4 the section plane simply passes through the 
corner named sin. 

Hence, putting these sections together in their right 
relation, from the face setat, surrounded by the lines and 
points mentioned above, there run : 

3 faces : satan, sanat, satet 
3 lines : sanan, sanet, sat 

and these faces and lines run to the point sin. Thus 
the tetrahedron is completely named. 

The octahedron section of the tesseract, which can be 
traced from fig. 72, p. 129 by extending the lines there 
drawn, is named : 

Front triangle selin, selat, selel, setal, senil, setit, selin 
with area setat. 

The sections between the front and rear triangle, of 
which one is shown in Ib 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 



268 



THE FOURTH DIMENSION 



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 
octohedron, 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. 



ensen 


eisen 


! 

clsen 





4 


k 


arisen 


atsen 


alben 


insen 


itscn 


ilsen ! 






1 



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. 



269 

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 



ene 


etc 


ele ! 

i 


ane 


ate 


ale 


ine 

L... 


K- 4 

ite 


iTe" 



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, 1, 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 
to positive 9, and from negative 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. 



270 THE FOURTH DIMENSION 

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. 







1 


2 


3 


4 


5 


6 


7 


8 


9 


positive 


n 


t 


1 


P 


f 


sh 


k 


ch 


nt 


st 


negative 


z 


d 


th 


b 


V 


m 


g 


j 


nd 


sp 



VOWELS. 

012345 678 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 <k Viney, Ld,, London and Aylttbury. 



By THE SAME AUTHOR 

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Views of the Tessaract. 



No. 1. 



No. 2. 



No. 3. 




No. 4. 





No. 5. 







No. 6. 




No. 7. 



No. 8. 



No. 9. 




No. 10. 





No. 11. 





No. 12