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 86. (1) He regards the possibility of non-Euclidean spaces as suggested by the subjectivity of space, 93 87. And rejects it owing to a mathematical misunderstanding, 96 88. Having missed the most important sense of their possibility, 96 89. Which is that they fulfil the logical conditions to which any form of externality must conform 97 90. (2) He attacks the mathematical procedure of Metageometry 98 91. The attack begins with a question-begging definition of parallels 99 92. Lotze maintains that all apparent departures from Euclid could be physically explained, a view which really makes Euclid empirical 99 93. His criticism of Helmholtz's analogies rests wholly on mathematical mistakes 101 94. His proof that space must have three dimensions rests on neglect of different orders of infinity 104 95. He attacks non-Euclidean spaces on the mistaken ground that they are not homogeneous 107 96. Lotze's objections fall under four heads 108 97. Two other semi-philosophical objections may be urged, 109 98. One of which, the absence of similarity, has been made the basis of attack by Delbouf, 110 99. But does not form a valid ground of objection 111 100. Recent French speculation on the foundations of Geometry has suggested few new views 112 101. All homogeneous spaces are � priori possible, and the decision between them is empirical 114 [xii] CHAPTER III. Section A. the axioms of projective geometry. 102. Projective Geometry does not deal with magnitude, and applies to all spaces alike 117 103. It will be found wholly � priori 117 104. Its axioms have not yet been formulated philosophically 118 105. Coordinates, in projective Geometry, are not spatial magnitudes, but convenient names for points 118 106. The possibility of distinguishing various points is an axiom 119 107. The qualitative relations between points, dealt with by projective Geometry, are presupposed by the quantitative treatment 119 108. The only qualitative relation between two points is the straight line, and all straight lines are qualitatively similar 120 109. Hence follows, by extension, the principle of projective transformation 121 110. By which figures qualitatively indistinguishable from a given figure are obtained 122 111. Anharmonic ratio may and must be descriptively defined 122 112. The quadrilateral construction is essential to the projective definition of points, 123 113. And can be projectively defined, 124 114.