AN ESSAY ON THE FOUNDATIONS OF GEOMETRY By Bertrand A. W. Russell Fellow Of Trinity College, Cambridge 1897 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 � 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 � priori, for Epistemology, when without it knowledge would be impossible 2 4. The subjective and the � priori belong respectively to Psychology and to Epistemology. The latter alone will be investigated in this essay 3 5. My test of the � priori will be purely logical: what knowledge is necessary for experience? 3 6. But since the necessary is hypothetical, we must include, in the � 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 [viii] 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.