We are thus brought back to the point from which we started, namely, the falsity of Riemann's initial disjunction, and the consequent fallacy in his proof of the empirical nature of the axioms. His philosophy is chiefly vitiated, to my mind, by this fallacy, and by the uncritical assumption that a metrical coordinate system can be set up independently of any axioms as to space-measurement[82]. Riemann has failed to observe, what I have endeavoured to prove in the next chapter, that, unless space had a strictly constant measure of curvature, Geometry would become impossible; also that the absence of constant measure of curvature involves absolute position, which is an absurdity. Hence he is led to the conclusion that all geometrical axioms are empirical, and may not hold in the infinitesimal, where observation is impossible. Thus he says (p. 267): "Now the empirical conceptions, on which spatial measurements are based, the conceptions of the rigid body and the light-ray, appear to lose their validity in the infinitesimal: it is therefore quite conceivable that the relations of spatial magnitudes in the infinitesimal do not correspond to the presuppositions of Geometry, and this would, in fact, have to be assumed, as soon as it would enable us to explain the phenomena more simply." From this conclusion I must entirely dissent. In very large spaces, there might be a departure from Euclid; for they depend upon the axiom of parallels, which is not contained in the axiom of Free Mobility; but in the infinitesimal, departures from Euclid could only be due to the absence of Free Mobility, which, as I hope my third chapter will show, is once for all impossible. Helmholtz. =66.= Helmholtz, like Riemann, was important both in the mathematics and in the philosophy of Geometry. From the mathematical point of view, his work has been already considered in Chapter I.; the consideration of his philosophy, which must occupy us here, will be a more serious task. Like Riemann, he endeavoured to prove that all the axioms are empirical, and like Riemann, he based his proof chiefly on Metageometry. He had an additional resource, however, in the physiology of the senses, which first led him to reject the Transcendental Aesthetic, and enabled him to attack Kant from the psychological as well as the mathematical side[83]. The principal topics, for a criticism of Helmholtz, are three: First, his criterion of the _à priori_; second, his discussion with Land as to the "imaginability" of non-Euclidean spaces; third--and this is by far the most important of the three--his theory of the dependence of Geometry on Mechanics. Let us discuss these three points successively.