J. Geom. 98 (2010), 151–170 c  2010 Springer Basel AG 0047-2468/10/010151-20 published online November 25, 2010 DOI 10.1007/s00022-010-0053-z Journal of Geometry Non-euclidean geometries: the Cayley-Klein approach Horst Struve and Rolf Struve Abstract. A. Cayley and F. Klein discovered in the nineteenth century that euclidean and non-euclidean geometries can be considered as mathe- matical structures living inside projective-metric spaces. They outlined this idea with respect to the real projective plane and established (“begr¨ undeten”) in this way the hyperbolic and elliptic geometry. The generalization of this approach to projective spaces over arbitrary fields and of arbitrary dimensions requires two steps, the introduction of a met- ric in a pappian projective space and the definition of substructures as Cayley-Klein geometries. While the first step is taken in H. Struve and R. Struve (J Geom 81:155–167, 2004), the second step is made in this article. We show that the concept of a Cayley-Klein geometry leads to a unified description and classification of a wide range of non-euclidean geometries including the main geometries studied in the foundations of geometry by D. Hilbert, J. Hjelmslev, F. Bachmann, R. Lingenberg, H. Karzel et al. Mathematics Subject Classification (2010). 06B25, 51F10, 20G15. Keywords. Cayley-Klein geometry, non-euclidean geometry, projective-metric spaces, Klein’s model. 1. Introduction In 1859 A. Cayley discovered that euclidean geometry can be considered as a special case of projective geometry 1 which led him to the famous statement that “descriptive geometry (his term for projective geometry) is all geometry” [9]. Ten years later F. Klein [22] took up the ideas of A. Cayley and showed that projective geometry as well provides a framework for the development of hyperbolic and elliptic geometry. 2 Thus from a projective point of view 1 He introduced in the real projective plane an euclidean metric by specializing two imaginary points known as the circular points at infinity. 2 F. Klein replaced the two imaginary circular points by a real and by an imaginary non- degenerated conic to get a model of the hyperbolic and elliptic plane, respectively.