From DSB 1:599-600 (Beltrami, Eugenio):

...In a paper of 1865 Beltrami had shown that on surfaces of constant curvature, and only on them, the line elements ds² ...can be written in such a form that the geodesics, and only these, are represented by linear expressions in u and v. For positive curvature R^-2 this form is . . . The geodesics in this case behave, locally speaking, like the great circles on a sphere. It now occurred to Beltrami that, by changing R to iR and a to ia, the line element thus obtained . . ., which defines surfaces of constant curvature -R^-2, offers a new type of geometry for its geodesics inside the region u² + v² < a². This geometry is exactly that of the so-called non-Euclidean geometry of Lobachevski, if geodesics on such a surface are identified with the "straight lines" of non-Euclidean geometry.