Teoria Fondamentale Degli Spazii Di Curvatura Costante, Pp. 232-255 In Annali Di Matematica Pura Et Applicata, Serie Ii, Tomo Ii. NON-EUCLIDEAN Geometry Beltrami, Eugenio

First edition, complete journal volume, of this crucially important memoir on non-euclidean geometry, the second of two published by Beltrami in the same year. In this paper, Beltrami "gave an abstract proof of equiconsistency of hyperbolic and Euclidean geometry for any dimension. He accomplished this by introducing several models of non-Euclidean geometry that are now known as the Beltrami Klein model, the Poincaré disk model, and the Poincaré half-plane model, together with transformations that relate them. For the half-plane model, Beltrami cited a note by Liouville in the treatise of Monge on differential geometry. Beltrami also showed that n-dimensional Euclidean geometry is realized on a horosphere of the (n + 1)-dimensional hyperbolic space, so the logical relation between consistency of the Euclidean and the non-Euclidean geometries is symmetric. Beltrami acknowledged the influence of Riemann's groundbreaking Habilitation lecture "On the hypotheses on which geometry is based" (1854; published posthumously in 1868)" (Wikipedia). "Thus Beltrami showed how possible contradictions in non-Euclidean geometry would reveal themselves in the Euclidean geometry of surfaces; and this removed for most, or probably all, mathematicians the feeling that non-Euclidean geometry might be wrong. Beltrami, by mapping one geometry upon another, made non-Euclidean geometry respectable. His method was soon followed by others, including Felix Klein, a development that opened entirely new fields of mathematical thinking" (DSB). 4to, pp. [iv], 348. Contemporary quarter-calf and marbled boards (spine and corners rubbed, library stamp on title recto and verso and library labels on front paste-down).