 Riemann’s “Über die Hypothesen …” (given as a lecture in 1854, published posthumously in 1868) and Beltrami’s “Saggio di interpretazione” (1868) gave different but equivalent accounts of 2-dimensional non-Euclidean geometry. Beltrami employed differential geometry to justify the hyperbolic geometry of Lobachevskii-Bolyai. By studying the pseudosphere and how it can be mapped onto the plane (in such a way that its geodesics went to straight lines), Beltrami realized that the unit disk can be viewed as an ‘infinite plane’. It had been impossible to find a surface in \(\mathbb{R}^3\) that behaved like the hyperbolic plane (the closest, but partial, being the pseudosphere); now it was possible to realize non-Euclidean geometry as the geometry of the interior of a disk with a novel metric. The disk is the whole ‘plane’, its circumference a line at infinity, line segments in the disk are ‘lines’, and one can define a distance function \(d(P,Q)\) that makes sense and validates Lobachevskii’s results. From the logical point of view, this was an enormous step forward, as the consistency of non-Euclidean geometry was now proven.