Note 12 — Introductory material

The originator of the whole system, Gauss, does not appear, as regards strictly non-Euclidean Geometry, in any of his hitherto published papers, to have given more than results; his proofs remain unknown to us. Nevertheless he was the first to investigate the consequences of denying the axiom of parallels[8], and in his letters he communicated these consequences to some of his friends, among whom was Wolfgang Bolyai. The first mention of the subject in his letters occurs when he was only 18; four years later, in 1799, writing to W. Bolyai, he enunciates the important theorem that, in hyperbolic Geometry, there is a maximum to the area of a triangle. From later writings it appears that he had worked out a system nearly, if not quite, as complete as those of Lobatchewsky and Bolyai[9].