One broad result of non-Euclidean geometry, even in its earliest form, was that the geometry of actual space is, at least in part, an empirical study, not a branch of pure mathematics. It may be said that empiricists, such as J. S. Mill, always based geometry upon empirical observation. But they did the same with arithmetic, in which they were certainly mistaken. No one before the non-Euclideans perceived that arithmetic and geometry stand on a quite different footing, the former being continuous with pure logic and independent of experience, the latter being continuous with physics and dependent upon physical data. Geometry can, it is true, be still studied as a branch of pure mathematics, but it is then hypothetical, and cannot claim that its initial hypotheses (which replace the axioms) are true in fact, since this is a question outside the scope of pure mathematics. The geometry which is required by the engineer or the astronomer is not a branch of pure mathematics, but a branch of physics. Indeed, in the hands of Einstein geometry has become identical with the whole of the general part of theoretical physics: the two are united in the general theory of relativity. Riemann, who was logically the immediate predecessor of Einstein, brought in a new idea of which the importance was not perceived for half a century. He considered that geometry ought to start from the infinitesimal, and depend upon integration for statements about finite lengths, areas, or volumes. This requires, inter alia, the replacement of the straight line by the geodesic: the latter has a definition depending upon infinitesimal distances, while the former has not. The traditional[Pg 22] view was that, while the length of a curve could, in general, only be defined by integration, the length of the straight line between two points could be defined as a whole, not as the limit of a sum of little bits. Riemann's view was that a straight line does not differ from a curve in this respect. Moreover, measurement, being performed by means of bodies, is a physical operation, and its results depend for their interpretation upon the laws of physics. This point of view has turned out to be of very great importance. Its scope has been extended by the theory of relativity, but in essence it is to be found in Riemann's dissertation.