TRANSCRIBER'S NOTE
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Bold text is denoted by =equal signs=.
A superscript is denoted by ^{x}. For example, y^{4} indicates y to
the power 4.
A subscript is denoted by {x}. For example, a{ik} indicates the ik-th
element of array a.
Algebraic expressions in the original text usually had the variables in
italic. The italic underscore markup has been removed for readability,
so for example: _dx_{i}_ = λ′_d_′_x_{i}_ + λ″_d″x_{i}_
has been changed to: dx{i} = λ′d′x{i} + λ″d″x{i}
Obvious typographical errors and punctuation errors have been
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THE FOUNDATIONS OF GEOMETRY.
London: C. J. CLAY AND SONS,
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
AVE MARIA LANE.
Glasgow: 263, ARGYLE STREET.
[Illustration]
Leipzig: F. A. BROCKHAUS.
New York: THE MACMILLAN COMPANY.
Bombay: GEORGE BELL AND SONS.
AN ESSAY
ON THE
FOUNDATIONS OF GEOMETRY
BY
BERTRAND A. W. RUSSELL. M.A.
FELLOW OF TRINITY COLLEGE, CAMBRIDGE.
CAMBRIDGE:
AT THE UNIVERSITY PRESS.
1897
[_All Rights reserved._]
Cambridge:
PRINTED BY J. AND C. F. CLAY,
AT THE UNIVERSITY PRESS.
PREFACE.
The present work is based on a dissertation submitted at the
Fellowship Examination of Trinity College, Cambridge, in the year
1895. Section B of the third chapter is in the main a reprint, with
some serious alterations, of an article in _Mind_ (New Series, No.
17). The substance of the book has been given in the form of lectures
at the Johns Hopkins University, Baltimore, and at Bryn Mawr College,
Pennsylvania.
My chief obligation is to Professor Klein. Throughout the first
chapter, I have found his "Lectures on non-Euclidean Geometry"
an invaluable guide; I have accepted from him the division of
Metageometry into three periods, and have found my historical work
much lightened by his references to previous writers. In Logic, I
have learnt most from Mr Bradley, and next to him, from Sigwart and
Dr Bosanquet. On several important points, I have derived useful
suggestions from Professor James's "Principles of Psychology."
My thanks are due to Mr G. F. Stout and Mr A. N. Whitehead for
kindly reading my proofs, and helping me by many useful criticisms.
To Mr Whitehead I owe, also, the inestimable assistance of constant
criticism and suggestion throughout the course of construction,
especially as regards the philosophical importance of projective
Geometry.
HASLEMERE.
_May, 1897._
TO
JOHN McTAGGART ELLIS McTAGGART
TO WHOSE DISCOURSE AND FRIENDSHIP IS OWING
THE EXISTENCE OF THIS BOOK.
TABLE OF CONTENTS.
INTRODUCTION.
OUR PROBLEM DEFINED BY ITS RELATIONS TO LOGIC,
PSYCHOLOGY AND MATHEMATICS.
PAGE
1. The problem first received a modern form through Kant, who
connected the _à priori_ with the subjective 1
2. A mental state is subjective, for Psychology, when its immediate
cause does not lie in the outer world 2
3. A piece of knowledge is _à priori_, for Epistemology, when
without it knowledge would be impossible 2
4. The subjective and the _à priori_ belong respectively to
Psychology and to Epistemology. The latter alone will be
investigated in this essay 3
5. My test of the _à priori_ will be purely logical: what knowledge
is necessary for experience? 3
6. But since the necessary is hypothetical, we must include, in
the _à priori_, the ground of necessity 4
7. This may be the essential postulate of our science, or the
element, in the subject-matter, which is necessary to
experience; 4
8. Which, however, are both at bottom the same ground 5
9. Forecast of the work 5
CHAPTER I.
A SHORT HISTORY OF METAGEOMETRY.
10. Metageometry began by rejecting the axiom of parallels 7
11. Its history may be divided into three periods: the synthetic,
the metrical and the projective 7
12. The first period was inaugurated by Gauss, 10
13. Whose suggestions were developed independently by
Lobatchewsky 10
14. And Bolyai 11
15. The purpose of all three was to show that the axiom of parallels
could not be deduced from the others, since its denial did
not lead to contradictions 12
16. The second period had a more philosophical aim, and was
inspired chiefly by Gauss and Herbart 13
17. The first work of this period, that of Riemann, invented two
new conceptions: 14
18. The first, that of a manifold, is a class-conception, containing
space as a species, 14
19. And defined as such that its determinations form a collection
of magnitudes 15
20. The second, the measure of curvature of a manifold, grew out
of curvature in curves and surfaces 16
21. By means of Gauss's analytical formula for the curvature of
surfaces, 19
22. Which enables us to define a _constant_ measure of curvature
of a three-dimensional space without reference to a fourth
dimension 20
23. The main result of Riemann's mathematical work was to
show that, if magnitudes are independent of place, the
measure of curvature of space must be constant 21
24. Helmholtz, who was more of a philosopher than a mathematician, 22
25. Gave a new but incorrect formulation of the essential axioms, 23
26. And deduced the quadratic formula for the infinitesimal arc,
which Riemann had assumed 24
27. Beltrami gave Lobatchewsky's planimetry a Euclidean
interpretation, 25
28. Which is analogous to Cayley's theory of distance; 26
29. And dealt with _n_-dimensional spaces of constant negative
curvature 27
30. The third period abandons the metrical methods of the second,
and extrudes the notion of spatial quantity 27
31. Cayley reduced metrical properties to projective properties,
relative to a certain conic or quadric, the Absolute; 28
32. And Klein showed that the Euclidean or non-Euclidean systems
result, according to the nature of the Absolute; 29
33. Hence Euclidean _space_ appeared to give rise to all the kinds
of Geometry, and the question, which is true, appeared
reduced to one of convention 30
34. But this view is due to a confusion as to the nature of the
coordinates employed 30
35. Projective coordinates have been regarded as dependent on
distance, and thus really metrical 31
36. But this is not the case, since anharmonic ratio can be
projectively defined 32
37. Projective coordinates, being purely descriptive, can give no
information as to metrical properties, and the reduction of
metrical to projective properties is purely technical 33
38. The true connection of Cayley's measure of distance with
non-Euclidean Geometry is that suggested by Beltrami's
Saggio, and worked out by Sir R. Ball, 36
39. Which provides a Euclidean equivalent for every non-Euclidean
proposition, and so removes the possibility of contradictions
in Metageometry 38
40. Klein's elliptic Geometry has not been proved to have a
corresponding variety of space 39
41. The geometrical use of imaginaries, of which Cayley demanded
a philosophical discussion, 41
42. Has a merely technical validity, 42
43. And is capable of giving geometrical results only when it
begins and ends with real points and figures 45
44. We have now seen that projective Geometry is logically prior
to metrical Geometry, but cannot supersede it 46
45. Sophus Lie has applied projective methods to Helmholtz's
formulation of the axioms, and has shown the axiom of
Monodromy to be superfluous 46
46. Metageometry has gradually grown independent of philosophy,
but has grown continually more interesting to philosophy 50
47. Metrical Geometry has three indispensable axioms, 50
48. Which we shall find to be not results, but conditions, of
measurement, 51
49. And which are nearly equivalent to the three axioms of
projective Geometry 52
50. Both sets of axioms are necessitated, not by facts, but by
logic 52