The birth of non-euclidean geometry

Christian Houzel

pp. 1-21

in: Luciano Boi, Dominique Flament, Jean-Michel Salanskis (eds), 1830–1930: a century of geometry, Berlin, Springer, 1992

Abstract

When reading Bolyai and Lobachevsky, one is struck by the similarities of their ways. The importance of the trigonometrical developments must be emphasised; they gave an analytical basis to the new Geometry, guaranteeing, to some extent, the consistence of the construction. This construction is not axiomatic; on the contrary, modern axiomatics came from a reflexion made necessary by the existence of the non-Euclidean Geometry.The Differential Geometry is independant of, but very close to, the non-Euclidean Geometry: they communicate through the spherical Geometry and the interpretation of the sum of the angles of a triangle in terms of an area. It is difficult to think that Gauss had not seen the relation between his researches on the theory of parallels and his intrinsic Geometry of surfaces. And we saw that Bolyai and Lobachevsky made explicit the ds² of their plane Geometry. Nevertheless, the time was not yet ripe, in the first half of the XIX^th century, to interpret the non-Euclidean Geometry (of a synthetic origin) in terms of the Differential Geometry.