Transcriber’s Notes
Obvious typographical errors have been silently corrected. All other
spelling and punctuation remains unchanged.
Italics are represented thus _italic_, bold thus =bold= and
superscripts thus y^{en}.
It should be noted that much of the text is a discussion centred on the
many illustrations which have not been included.
THE FOURTH DIMENSION
SOME OPINIONS OF THE PRESS
“_Mr. C. H. Hinton discusses the subject of the higher dimensionality
of space, his aim being to avoid mathematical subtleties and
technicalities, and thus enable his argument to be followed by readers
who are not sufficiently conversant with mathematics to follow these
processes of reasoning._”—NOTTS GUARDIAN.
“_The fourth dimension is a subject which has had a great fascination
for many teachers, and though one cannot pretend to have quite grasped
Mr. Hinton’s conceptions and arguments, yet it must be admitted that
he reveals the elusive idea in quite a fascinating light. Quite
apart from the main thesis of the book many chapters are of great
independent interest. Altogether an interesting, clever and ingenious
book._”—DUNDEE COURIER.
“_The book will well repay the study of men who like to exercise their
wits upon the problems of abstract thought._”—SCOTSMAN.
“_Professor Hinton has done well to attempt a treatise of moderate
size, which shall at once be clear in method and free from
technicalities of the schools._”—PALL MALL GAZETTE.
“_A very interesting book he has made of it._”—PUBLISHERS’ CIRCULAR.
“_Mr. Hinton tries to explain the theory of the fourth dimension so
that the ordinary reasoning mind can get a grasp of what metaphysical
mathematicians mean by it. If he is not altogether successful it is not
from want of clearness on his part, but because the whole theory comes
as such an absolute shock to all one’s preconceived ideas._”—BRISTOL
TIMES.
“_Mr. Hinton’s enthusiasm is only the result of an exhaustive study,
which has enabled him to set his subject before the reader with far
more than the amount of lucidity to which it is accustomed._”—PALL MALL
GAZETTE.
“_The book throughout is a very solid piece of reasoning in the domain
of higher mathematics._”—GLASGOW HERALD.
“_Those who wish to grasp the meaning of this somewhat difficult
subject would do well to read_ The Fourth Dimension. _No mathematical
knowledge is demanded of the reader, and any one, who is not afraid of
a little hard thinking, should be able to follow the argument._”—LIGHT.
“_A splendidly clear re-statement of the old problem of the fourth
dimension. All who are interested in this subject will find the
work not only fascinating, but lucid, it being written in a style
easily understandable. The illustrations make still more clear
the letterpress, and the whole is most admirably adapted to the
requirements of the novice or the student._”—TWO WORLDS.
“_Those in search of mental gymnastics will find abundance of exercise
in Mr. C. H. Hinton’s_ Fourth Dimension.”—WESTMINSTER REVIEW.
FIRST EDITION, _April 1904_; SECOND EDITION, _May 1906_.
Views of the Tessaract.
No. 1. No. 2. No. 3.
No. 4. No. 5. No. 6.
No. 7. No. 8. No. 9.
No. 10. No. 11. No. 12.
THE
FOURTH DIMENSION
BY
C. HOWARD HINTON, M.A.
AUTHOR OF “SCIENTIFIC ROMANCES”
“A NEW ERA OF THOUGHT,” ETC., ETC.
[Illustration: Colophon]
LONDON
SWAN SONNENSCHEIN & CO., LIMITED
25 HIGH STREET, BLOOMSBURY
1906
PRINTED BY
HAZELL, WATSON AND VINEY, LD.,
LONDON AND AYLESBURY.
PREFACE
I have endeavoured to present the subject of the higher dimensionality
of space in a clear manner, devoid of mathematical subtleties and
technicalities. In order to engage the interest of the reader, I have
in the earlier chapters dwelt on the perspective the hypothesis of a
fourth dimension opens, and have treated of the many connections there
are between this hypothesis and the ordinary topics of our thoughts.
A lack of mathematical knowledge will prove of no disadvantage to the
reader, for I have used no mathematical processes of reasoning. I have
taken the view that the space which we ordinarily think of, the space
of real things (which I would call permeable matter), is different from
the space treated of by mathematics. Mathematics will tell us a great
deal about space, just as the atomic theory will tell us a great deal
about the chemical combinations of bodies. But after all, a theory is
not precisely equivalent to the subject with regard to which it is
held. There is an opening, therefore, from the side of our ordinary
space perceptions for a simple, altogether rational, mechanical, and
observational way of treating this subject of higher space, and of
this opportunity I have availed myself.
The details introduced in the earlier chapters, especially in
Chapters VIII., IX., X., may perhaps be found wearisome. They are of
no essential importance in the main line of argument, and if left
till Chapters XI. and XII. have been read, will be found to afford
interesting and obvious illustrations of the properties discussed in
the later chapters.
My thanks are due to the friends who have assisted me in designing and
preparing the modifications of my previous models, and in no small
degree to the publisher of this volume, Mr. Sonnenschein, to whose
unique appreciation of the line of thought of this, as of my former
essays, their publication is owing. By the provision of a coloured
plate, in addition to the other illustrations, he has added greatly to
the convenience of the reader.
C. HOWARD HINTON.
CONTENTS
CHAP. PAGE
I. FOUR-DIMENSIONAL SPACE 1
II. THE ANALOGY OF A PLANE WORLD 6
III. THE SIGNIFICANCE OF A FOUR-DIMENSIONAL
EXISTENCE 15
IV. THE FIRST CHAPTER IN THE HISTORY OF FOUR
SPACE 23
V. THE SECOND CHAPTER IN THE HISTORY OF
FOUR SPACE 41
Lobatchewsky, Bolyai, and Gauss
Metageometry
VI. THE HIGHER WORLD 61
VII. THE EVIDENCE FOR A FOURTH DIMENSION 76
VIII. THE USE OF FOUR DIMENSIONS IN THOUGHT 85
IX. APPLICATION TO KANT’S THEORY OF EXPERIENCE 107
X. A FOUR-DIMENSIONAL FIGURE 122
XI. NOMENCLATURE AND ANALOGIES 136
XII. THE SIMPLEST FOUR-DIMENSIONAL SOLID 157
XIII. REMARKS ON THE FIGURES 178
XIV. A RECAPITULATION AND EXTENSION OF THE
PHYSICAL ARGUMENT 203
APPENDIX I.—THE MODELS 231
" II.—A LANGUAGE OF SPACE 248
THE FOURTH DIMENSION
CHAPTER I
FOUR-DIMENSIONAL SPACE
There is nothing more indefinite, and at the same time more real, than
that which we indicate when we speak of the “higher.” In our social
life we see it evidenced in a greater complexity of relations. But this
complexity is not all. There is, at the same time, a contact with, an
apprehension of, something more fundamental, more real.
With the greater development of man there comes a consciousness of
something more than all the forms in which it shows itself. There is
a readiness to give up all the visible and tangible for the sake of
those principles and values of which the visible and tangible are the
representation. The physical life of civilised man and of a mere savage
are practically the same, but the civilised man has discovered a depth
in his existence, which makes him feel that that which appears all to
the savage is a mere externality and appurtenage to his true being.
Now, this higher—how shall we apprehend it? It is generally embraced
by our religious faculties, by our idealising tendency. But the higher
existence has two sides. It has a being as well as qualities. And in
trying to realise it through our emotions we are always taking the
subjective view. Our attention is always fixed on what we feel, what
we think. Is there any way of apprehending the higher after the purely
objective method of a natural science? I think that there is.
Plato, in a wonderful allegory, speaks of some men living in such a
condition that they were practically reduced to be the denizens of
a shadow world. They were chained, and perceived but the shadows of
themselves and all real objects projected on a wall, towards which
their faces were turned. All movements to them were but movements
on the surface, all shapes but the shapes of outlines with no
substantiality.
Plato uses this illustration to portray the relation between true
being and the illusions of the sense world. He says that just as a man
liberated from his chains could learn and discover that the world was
solid and real, and could go back and tell his bound companions of this
greater higher reality, so the philosopher who has been liberated, who
has gone into the thought of the ideal world, into the world of ideas
greater and more real than the things of sense, can come and tell his
fellow men of that which is more true than the visible sun—more noble
than Athens, the visible state.
Now, I take Plato’s suggestion; but literally, not metaphorically.
He imagines a world which is lower than this world, in that shadow
figures and shadow motions are its constituents; and to it he contrasts
the real world. As the real world is to this shadow world, so is the
higher world to our world. I accept his analogy. As our world in three
dimensions is to a shadow or plane world, so is the higher world to our
three-dimensional world. That is, the higher world is four-dimensional;
the higher being is, so far as its existence is concerned apart from
its qualities, to be sought through the conception of an actual
existence spatially higher than that which we realise with our senses.
Here you will observe I necessarily leave out all that gives its
charm and interest to Plato’s writings. All those conceptions of the
beautiful and good which live immortally in his pages.
All that I keep from his great storehouse of wealth is this one thing
simply—a world spatially higher than this world, a world which can only
be approached through the stocks and stones of it, a world which must
be apprehended laboriously, patiently, through the material things of
it, the shapes, the movements, the figures of it.
We must learn to realise the shapes of objects in this world of the
higher man; we must become familiar with the movements that objects
make in his world, so that we can learn something about his daily
experience, his thoughts of material objects, his machinery.
The means for the prosecution of this enquiry are given in the
conception of space itself.
It often happens that that which we consider to be unique and unrelated
gives us, within itself, those relations by means of which we are able
to see it as related to others, determining and determined by them.
Thus, on the earth is given that phenomenon of weight by means of which
Newton brought the earth into its true relation to the sun and other
planets. Our terrestrial globe was determined in regard to other bodies
of the solar system by means of a relation which subsisted on the earth
itself.
And so space itself bears within it relations of which we can
determine it as related to other space. For within space are given the
conceptions of point and line, line and plane, which really involve the
relation of space to a higher space.
Where one segment of a straight line leaves off and another begins is
a point, and the straight line itself can be generated by the motion of
the point.
One portion of a plane is bounded from another by a straight line, and
the plane itself can be generated by the straight line moving in a
direction not contained in itself.
Again, two portions of solid space are limited with regard to each
other by a plane; and the plane, moving in a direction not contained in
itself, can generate solid space.
Thus, going on, we may say that space is that which limits two portions
of higher space from each other, and that our space will generate the
higher space by moving in a direction not contained in itself.
Another indication of the nature of four-dimensional space can be
gained by considering the problem of the arrangement of objects.
If I have a number of swords of varying degrees of brightness, I can
represent them in respect of this quality by points arranged along a
straight line.
If I place a sword at A, fig. 1, and regard it as having a certain
brightness, then the other swords can be arranged in a series along the
line, as at A, B, C, etc., according to their degrees of brightness.
[Illustration: Fig. 1.]
If now I take account of another quality, say length, they can be
arranged in a plane. Starting from A, B, C, I can find points to
represent different degrees of length along such lines as AF, BD, CE,
drawn from A and B and C. Points on these lines represent different
degrees of length with the same degree of brightness. Thus the whole
plane is occupied by points representing all conceivable varieties of
brightness and length.
[Illustration: Fig. 2.]
Bringing in a third quality, say sharpness, I can draw, as in fig. 3,
any number of upright lines. Let distances along these upright lines
represent degrees of sharpness, thus the points F and G will represent
swords of certain definite degrees of the three qualities mentioned,
and the whole of space will serve to represent all conceivable degrees
of these three qualities.
[Illustration: Fig. 3.]
If now I bring in a fourth quality, such as weight, and try to find a
means of representing it as I did the other three qualities, I find
a difficulty. Every point in space is taken up by some conceivable
combination of the three qualities already taken.
To represent four qualities in the same way as that in which I have
represented three, I should need another dimension of space.
Thus we may indicate the nature of four-dimensional space by saying
that it is a kind of space which would give positions representative
of four qualities, as three-dimensional space gives positions
representative of three qualities.