Abstract
In this paper the axiomatic basis will be a general absolute plane
A = (P,L, α,≡) in the sense of [6], 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 [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.
Lingua originale | English |
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pagine (da-a) | 61-71 |
Numero di pagine | 11 |
Rivista | Results in Mathematics |
Stato di pubblicazione | Pubblicato - 2007 |
Keywords
- Legendre theorem
- absolute geometry
- classification of absolute geometry