An essay on the foundations of geometry

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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