On Lobachevsky's trigonometric formulae

Abstract

We elaborate on some important ideas contained in Lobachevsky's Pangeometry and in some of his other memoirs. The ideas include the following: (1) The trigonometric formulae, which express the dependence between angles and edges of triangles, are not only tools, but they are used as the basic elements of any geometry. In fact, Lobachevsky developed a large set of analytical and geometrical theorems in non-Euclidean geometry using these formulae. (2) Differential and integral calculus are developed in hyperbolic space without the use of any Euclidean model of hyperbolic space. (3) There exist models of spherical and of Euclidean geometry within hyperbolic geometry, and these models are used to prove the hyperbolic trigonometry formulae. (4) If hyperbolic geometry were contradictory, then either Euclidean or spherical geometry would be contradictory. We shall also see that some of these ideas were rediscovered by later mathematicians.