Metrical Geometry is, then, a necessary part of the science of space, and a part not included in descriptive Geometry. Its _à priori_ element, nevertheless, so far as this is spatial and not arithmetical, is the same as the postulate of projective Geometry, namely, the homogeneity of space, or its equivalent, the relativity of position. We can see, in fact, that the _à priori_ element in both is likely to be the same. For the _à priori_ in metrical Geometry will be whatever is presupposed in the possibility of spatial measurement, _i.e._ of quantitative spatial comparison. But such comparison presupposes simply a known identity of quality, the determination of which is precisely the problem of projective Geometry. Hence the conditions for the possibility of measurement, in so far as they are not arithmetical, will be precisely the same as those for projective Geometry. =142.= Metrical Geometry, therefore, though distinct from projective Geometry, is not independent of it, but presupposes it, and arises from its combination with the extraneous idea of _quantity_. Nevertheless the mathematical form of the axioms, in metrical Geometry, is slightly different from their form in projective Geometry. The homogeneity of space is replaced by its equivalent, the axiom of Free Mobility. The axiom of the straight line is replaced by the axiom of distance: Two points determine a unique quantity, distance, which is unaltered in any motion of the two points as a single figure. This axiom, indeed, will be found to involve the axiom of the straight line--such a quantity could not exist unless the two points determined a unique curve--but its mathematical form is changed. Another important change is the collapse of the principle of duality: quantity can be applied to the straight line, because it is divisible into similar parts, but cannot be applied to the indivisible point. We thus obtain a reason, which was wanting in descriptive Geometry, for preferring points, as spatial elements, to straight lines or planes[146]. Finally, an entirely new idea is introduced with quantity, namely, the idea of _Motion_. Not that we study motion, or that any of our results have reference to motion, but that they cannot, though in projective Geometry they could, be obtained without at least an ideal motion of our figures through space.