A Foundation of Euclidean Geometry by Means of Congruence Mappings

Traditional teaching of geometry has been strongly influenced by the Euclid-Hilbert foundation of geometry. In his refinement of Euclid’s axiomatic system Hilbert1 used as undefined terms: to be a point, to be a line, to be a plane, the incidence relation, the betweenness relation, and the congruence relation (for explicitly defined objects such as line segments and angles). In this approach congruence is a basic (undefined) concept and the well known axioms and theorems on congruence play an important role in the whole development. The techniques of congruence proofs consist in a piece by piece comparison of the two figures which are to be proved congruent. In contrast to this piece by piece comparison Euclid also seemed to have in mind the idea of a motion of one figure onto another. This idea has been made precise during the last hundred years. One can reduce the cinematic concept of physical motions to that of mappings of the set of all points onto itself: the so called congruence mappings. These transformations of the space played an important role in F. Klein’s “Erlanger Programm” (1872). In this program the congruence mappings or euclidean transformations were specialized among a more general set of transformations, the set of all projective transformations. However Klein’s attitude was not directed towards a synthetic foundation of geometry. His underlying structure was that of analytic geometry: the number space.