On Klein's So-called Non-Euclidean geometry

TL;DR

Felix Klein's papers explore the construction of hyperbolic, Euclidean, and spherical geometries within projective geometry, building on Cayley's ideas to relate distances and angles via conics.

Contribution

The paper analyzes Klein's method of constructing constant curvature geometries within projective geometry, connecting it with Cayley's approach and other related works.

Findings

Klein's construction unifies different geometries within projective space

Relations established between Klein's work and Cayley's ideas on metric definitions

Commentary on the influence and connections of Klein's approach

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.