This conclusion, I believe, is valid of all actual measurement. But the possibility of such empirical and approximate rigidity, I must insist, depends on the _à priori_ law that _mere_ motion, apart from the action of other matter, cannot effect a change of shape. For without this law, the effect of other matter would not be discoverable; the laws of motion would be absurd, and Physics would be impossible. Consider the second law, for example: How could we measure the change of motion, if motion itself produced a change in our measures? Or consider the law of gravitation: How could we establish the inverse square, unless we were able, independently of Dynamics, to measure distances? The whole science of Dynamics, in short, is fundamentally dependent on Geometry, and but for the independent possibility of measuring spatial magnitudes, none of the magnitudes of Dynamics could be measured. Time, force, and mass are alike measured by spatial correlates: these correlates are given, for time, by the first law, for force and mass, by the second and third. It is true, then, that an empirical element appears unavoidably in all actual measurement, inasmuch as we can only know empirically that a given piece of matter preserves its shape throughout the necessary change of dynamical relations to other matter involved in motion; but it is further true that, for Geometry--which regards matter simply as supplying the necessary breach in the homogeneity of space, and the necessary term for spatial relations, not as the bearer of forces which change the configuration of other material systems--for Geometry, which deals with this abstract and merely kinematical matter, rigidity is _à priori_, in so far as the only changes with which it is cognizant--changes of mere position, namely--are incapable of affecting the shapes of the imaginary and abstract bodies with which it deals. To use a scholastic distinction, we may say that matter is the _causa essendi_ of space, but Geometry is the _causa cognoscendi_ of Physics. Without a Geometry independent of Physics, Physics itself, which necessarily assumes the results of Geometry, could never arise; but when Geometry is used in Physics, it loses some of its _à priori_ certainty, and acquires the empirical and approximate character which belongs to all accounts of actual phenomena.