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