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. By the general principle of projective transformation 126 115. The principle of duality is the mathematical form of a philosophical circle, 127 116. Which is an inevitable consequence of the relativity of space, and makes any definition of the point contradictory 128 117. We define the point as that which is spatial, but contains no space, whence other definitions follow 128 118. What is meant by qualitative equivalence in Geometry? 129 119. Two pairs of points on one straight line, or two pairs of straight lines through one point, are qualitatively equivalent 129 120. This explains why four collinear points are needed, to give an intrinsic relation by which the fourth can be descriptively defined when the first three are given 130 121. Any two projectively related figures are qualitatively equivalent, i.e. differ in no non-quantitative conceptual property 131 122. Three axioms are used by projective Geometry, 132 [xiii] 123. And are required for qualitative spatial comparison, 132 124. Which involves the homogeneity, relativity and passivity of space 133 125. The conception of a form of externality, 134 126. Being a creature of the intellect, can be dealt with by pure mathematics 134 127. The resulting doctrine of extension will be, for the moment, hypothetical 135 128. But is rendered assertorical by the necessity, for experience, of some form of externality 136 129. Any such form must be relational 136 130. And homogeneous 137 131. And the relations constituting it must appear infinitely divisible 137 132. It must have a finite integral number of dimensions, 139 133. Owing to its passivity and homogeneity 140 134. And to the systematic unity of the world 140 135. A one-dimensional form alone would not suffice for experience 141 136. Since its elements would be immovably fixed in a series 142 137. Two positions have a relation independent of other positions, 143 138. Since positions are wholly defined by mutually independent relations 143 139. Hence projective Geometry is wholly � priori, 146 140. Though metrical Geometry contains an empirical element 146 Section B. the axioms of metrical geometry. 141. Metrical Geometry is distinct from projective, but has the same fundamental postulate 147 142. It introduces the new idea of motion, and has three � priori axioms 148 I. The Axiom of Free Mobility. 143. Measurement requires a criterion of spatial equality 149 144. Which is given by superposition, and involves the axiom of Free Mobility 150 145. The denial of this axiom involves an action of empty space on things 151 146. There is a mathematically possible alternative to the axiom, 152 147. Which, however, is logically and philosophically untenable 153 148. Though Free Mobility is � priori, actual measurement is empirical 154 [xiv] 149. Some objections remain to be answered, concerning� 154 150. (1) The comparison of volumes and of Kant's symmetrical objects 154 151. (2) The measurement of time, where congruence is impossible 156 152. (3) The immediate perception of spatial magnitude; and 157 153. (4) The Geometry of non-congruent surfaces 158 154. Free Mobility includes Helmholtz's Monodromy 159 155. Free Mobility involves the relativity of space 159 156. From which, reciprocally, it can be deduced 160 157. Our axiom is therefore � priori in a double sense 160 II. The Axiom of Dimensions. 158. Space must have a finite integral number of dimensions 161 159. But the restriction to three is empirical 162 160. The general axiom follows from the relativity of position 162 161. The limitation to three dimensions, unlike most empirical knowledge, is accurate and certain 163 III. The Axiom of Distance. 162. The axiom of distance corresponds, here, to that of the straight line in projective Geometry 164 163. The possibility of spatial measurement involves a magnitude uniquely determined by two points, 164 164. Since two points must have some relation, and the passivity of space proves this to be independent of external reference 165 165. There can be only one such relation 166 166. This must be measured by a curve joining the two points, 166 167. And the curve must be uniquely determined by the two points 167 168. Spherical Geometry contains an exception to this axiom, 168 169. Which, however, is not quite equivalent to Euclid's 168 170. The exception is due to the fact that two points, in spherical space, may have an external relation unaltered by motion, 169 171. Which, however, being a relation of linear magnitude, presupposes the possibility of linear magnitude 170 172. A relation between two points must be a line joining them 170 173. Conversely, the existence of a unique line between two points can be deduced from the nature of a form of externality, 171 174. And necessarily leads to distance, when quantity is applied to it 172 [xv] 175. Hence the axiom of distance, also, is � priori in a double sense 172 176. No metrical coordinate system can be set up without the straight line 174 177. No axioms besides the above three are necessary to metrical Geometry 175 178. But these three are necessary to the direct measurement of any continuum 176 179. Two philosophical questions remain for a final chapter 177