  In a lecture “On the hypotheses that lie at the foundation of geometry”, delivered to the Faculty of Philosophy at Göttingen in 1854 and posthumously published in 1867, Bernhard Riemann (b. 1826, d. 1866) presented some radically innovative views on this matter. He noted that the measurable properties of a discrete manifold can be readily determined by counting. (Think of the population of a country, and the proportion of born-again Christians, or of couples who divorced within the first year of their marriage.) But continuous manifolds do not admit this approach. In particular, the measurable properties of physical space, which are the subject of geometry, depend on the binding forces that act on it. The distance between two points in space can be ascertained with a rod, or a tape, or by optical means, and the result depends essentially on the physical behavior of the instruments used. Up to now, the measurable properties of space have been successfully described in accordance with Euclidean geometry. However, “the empirical concepts on which the metric determinations of space are based—the concepts of a rigid body and a light ray—lose their validity in the infinitely small; it is therefore quite likely that the metric relations of space in the infinitely small do not agree with the assumptions of geometry, and in fact one would have to accept this as soon as the phenomena can thereby be explained in a simpler way” (Riemann 1854, p. 149). To prepare physicists for this eventuality, Riemann proposed a more general conception of geometry. Riemann's basic scheme makes allowance for much greater generality than he actually reaches for; but, in his judgment, it should be enough for the time being to characterize the geometry of continuous manifolds in such a way that it agrees optimally with Euclidean geometry on a small neighborhood of each point. 