 As soon as Poincaré began to write his popular philosophical essays about geometry, he made it clear that his chief concern was with how we could rely on any geometry at all. Mathematical advances were the basis for his reflections, but the key question was about the epistemological status of the hypotheses on which geometry is founded. And here he could not agree with Kant, nor Helmholtz, nor with Riemann. He was well aware of the great range of Riemannian geometries, and of the conclusion of Helmholtz’s speculations, by then made rigorous in the work of Sophus Lie, that a very limited number of geometries admitted rigid body motions. His concern in his “On the foundations of geometry” (1898) was with epistemology.