no longer equal when separated: absolute magnitude is an absurdity, and the magnitude resulting from comparison does not differ from that which would result if the dimensions of bodies were unchanged in motion. Therefore, since magnitude is only intelligible as the result of comparison, the dimensions of bodies _are_ unchanged in motion, and the suggested hypothesis is unmeaning. On this subject I shall have more to say in Chapter III.[96] =82.= This hypothesis, however, is not introduced for its own sake, but only to usher in the Helmholtzian _deus ex machina_, Mechanics. For Mechanics proves--so Erdmann confidently continues--that rigidity must hold, not merely as to ratios, in the above restricted geometrical sense, but as to absolute magnitudes (p. 62). Hence we get at last true Congruence, empirical as Mechanics is empirical, and impossible to prove apart from Mechanics. I have already criticized Helmholtz's view of the dependence of Geometry on Mechanics, and need not here speak of it at length. It is a pity that Erdmann has in no way specified the procedure by which Mechanics decides the geometrical alternatives--indeed he seems to rely on the _ipse dixit_ of Helmholtz. How, if Geometry would be totally unable to discover a change in dimensions of the kind suggested, the Laws of Motion, which throughout depend on Geometry, should be able to discover it if it existed, I am wholly at a loss to understand. Uniform motion in a straight line, for example, presupposes geometrical measurement; if this measurement is mistaken, what Mechanics imagines to be uniform motion is not really such, but Mechanics can never discover the discrepancy. If the Laws of Motion had been regarded as _à priori_, Geometry might possibly have been reinforced by them; but so long as they are empirical, they presuppose geometrical measurement, and cannot therefore condition or affect it.