In this paper the axiomatic basis will be a general absolute plane A = (P,L, α,≡) in the sense of , where P and 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  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.
|Numero di pagine||11|
|Rivista||Results in Mathematics|
|Stato di pubblicazione||Pubblicato - 2007|
- Legendre theorem
- absolute geometry
- classification of absolute geometry