 Riemann’s account, which was stated in \(n\) dimensions, agrees with the one that Poincaré was to use in many short publications in 1880 and 1881 but only describe explicitly in his major paper (Poincaré 1882). In this so-called conformal model, the plane is again a disk, geodesics are arcs of circles perpendicular to the boundary of the disk and angles are true (Euclidean). The Riemann and Beltrami disks swiftly persuaded mathematicians that the non-Euclidean geometry of Bolyai and Lobachevskii did, after all, make rigorous mathematical sense. Poincaré’s contribution a decade later was to make non-Euclidean geometry the natural geometry for certain topics elsewhere in mathematics, chiefly the developing and important subject of Riemann surfaces.