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