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 _à 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

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 _à 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
_à 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 _à priori_, actual measurement is
empirical 154

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 _à 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

175. Hence the axiom of distance, also, is _à 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