A set of postulates for general projective geometry

Abstract

Since Klein promulgated his famous Erlangen Programmet it has been known that the various types of geometry are such that each is characterized by a group of transformations. In view of the importance of the concept of transformation in nearly all mathematics and perhaps especially in geometry, geometers may properly seek to develop the various types of geometry in terms of point and transformation. For euclidean geometry this has been done by Pieri.:t This paper is devoted to a similar treatment of general projective geometry.§ One would naturally lay such postulates on the system of transformations so as to make the system form the group associated with the geometry. This was the scheme that Pieri used. His postulates make his transformations form the group of motions. In general projective geometry, however, this method is not necessary. If we are given the group of all projective traDsformations we can deduce the geometry from it but it will be shown in the sequel that we can also do that from a properly chosen semi-group belonging to that group. Our basis, to repeat, is a class of undefined elements called points and a class of undefined functions on point to pointll or transformations called collineations. For notation we will use small Roman letters to designate