(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