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 CHAPTER II. CRITICAL ACCOUNT OF SOME PREVIOUS PHILOSOPHICAL THEORIES OF GEOMETRY. 51. A criticism of representative modern theories need not begin before Kant 54 52. Kant's doctrine must be taken, in an argument about Geometry, on its purely logical side 55 53. Kant contends that since Geometry is apodeictic, space must be _à priori_ and subjective, while since space is _à priori_ and subjective, Geometry must be apodeictic 55 54. Metageometry has upset the first line of argument, not the second 56 55. The second may be attacked by criticizing either the distinction of synthetic and analytic judgments, or the first two arguments of the metaphysical deduction of space 57 56. Modern Logic regards every judgment as both synthetic and analytic, 57 57. But leaves the _à priori_, as that which is presupposed in the possibility of experience 59 58. Kant's first two arguments as to space suffice to prove _some_ form of externality, but not necessarily Euclidean space, a necessary condition of experience 60 59. Among the successors of Kant, Herbart alone advanced the theory of Geometry, by influencing Riemann 62 60. Riemann regarded space as a particular kind of manifold, i.e. wholly quantitatively 63