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.