On Klein's So-called Non-Euclidean geometry

Abstract

In two papers titled "On the so-called non-Euclidean geometry", I and II, Felix Klein proposed a construction of the spaces of constant curvature -1, 0 and and 1 (that is, hyperbolic, Euclidean and spherical geometry) within the realm of projective geometry. Klein's work was inspired by ideas of Cayley who derived the distance between two points and the angle between two planes in terms of an arbitrary fixed conic in projective space. We comment on these two papers of Klein and we make relations with other works.

To appear in : Sophus Lie and Felix Klein: The Erlangen program and its impact in mathematics and physics (ed. L. Ji and A. Papadopoulos), European Mathematical Society Publishing House, 2014