Legendre-like Theorems in a General Absolute Geometry

Abstract.

In this paper the axiomatic basis will be a general absolute plane A = (${\mathcal{P}}, {\mathcal{L}}, \alpha, \equiv$) in the sense of [6], where ${\mathcal{P}}$ and ${\mathcal{L}}$ denote respectively the set of points and the set of lines, α the order structure and ≡ the congruence, and where furthermore the word “general” means that no claim is made on any kind of continuity assumptions. Starting from the classification of general absolute geometries introduced in [5] by means of the notion of congruence, singular or hyperbolic or elliptic, we get now a complete characterization of the different possibilities which can occur in a general absolute plane studying the value of the angle δ defined in any Lambert–Saccheri quadrangle or, equivalently, the sum of the angles of any triangle. This yelds, in particular, a Archimedes-free proof of a statement generalizing the classical “first Legendre theorem” for absolute planes.