TRANSCRIBER'S NOTE Italic text is denoted by _underscores_. 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 corrected after careful comparison with other occurrences within the text and consultation of external sources. More detail can be found at the end of the book. 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