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 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. And that if conceptions of magnitude could be so applied, all the adjectives of space would result from their application 86 81. Erdmann regards Geometry alone as incapable of deciding on the truth of the axiom of Congruence, 86 82. Which he affirms to be empirically proved by Mechanics. 88 83. The variety and inadequacy of Erdmann's tests of apriority 89 84. Invalidate his final conclusions on the theory of Geometry 90 85. Lotze has discussed two questions in the theory of Geometry: 93