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 [x] 53. Kant contends that since Geometry is apodeictic, space must be � priori and subjective, while since space is � 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 � 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 61. He therefore unduly neglected the qualitative adjectives of space 64 62. His philosophy rests on a vicious disjunction 65 63. His definition of a manifold is obscure, 66 64. And his definition of measurement applies only to space 67 65. Though mathematically invaluable, his view of space as a manifold is philosophically misleading 69 66. Helmholtz attacked Kant both on the mathematical and on the psychological side; 70 67. But his criterion of apriority is changeable and often invalid; 71 68. His proof that non-Euclidean spaces are imaginable is inconclusive; 72 69. And his assertion of the dependence of measurement on rigid bodies, which may be taken in three senses, 74 70. Is wholly false if it means that the axiom of Congruence actually asserts the existence of rigid bodies, 75 71. Is untrue if it means that the necessary reference of geometrical propositions to matter renders pure Geometry empirical, 76 72. And is inadequate to his conclusion if it means, what is true, that actual measurement involves approximately rigid bodies 78 73. Geometry deals with an abstract matter, whose physical properties are disregarded; and Physics must presuppose Geometry 80 74. Erdmann accepted the conclusions of Riemann and Helmholtz, 81 [xi] 75. And regarded the axioms as necessarily successive steps in classifying space as a species of manifold 82 76. His deduction involves four fallacious assumptions, namely: 82 77. That conceptions must be abstracted from a series of instances; 83 78. That all definition is classification; 83 79. That conceptions of magnitude can be applied to space as a whole; 84 80.