"Riemann... made the important distinction, which had escaped previous writers, between the infinite and the unlimited. All of our experience tends to show that the universe is unlimited; a given segment may be extended indefinitely in either direction, but we know nothing as to whether it is infinite or not. If space have constant positive curvature, a geodesic surface is applicable to a Euclidean sphere where a geodesic is a circle, unlimited but not infinite. This possibility destroys the validity of Euclid's proof that an exterior angle of a triangle is greater than either opposite interior angle. Of all methods devised for attacking the problem of the bases of geometry Riemann's has proved by far to be the most fruitful. That is probably because it is the most flexible, and applicable to the greatest number of problems. In the twentieth century reverence for Euclid has been replaced by reverence for the differential equation{{center|1=ds^2 = \sum_{ij}^{} a_{ij} dx_i dx_j.}}"