CLASSIFICATION OF GENERAL ABSOLUTE GEOMETRIES WITH LAMBERT-SACCHERI QUADRANGLES

Without claiming any kind of continuity we show that an absolute geometry has either a singular, a hyperbolic or an elliptic congruence, i.e. for any quadruple (a, b, c, d) of distinct points contained in a plane with d, a ⊥ a, b ⊥ b, c ⊥ c, d ((a, b, c, d) is then called Lambert-Saccheri quadrangle) if a �: = (a ⊥ c, d)∩c, d then either a � = d ( singular case), or a � ∈]c, d [ (hyperbolic case) or d ∈]c, a �[ (elliptic case). Introduction. In their attempt, to deduce the parallel postulate from the other ax-ioms of Euclid, G. Saccheri and H. J. Lambert considered quadrangles (a, b, c, d) with three orthogonal angles α, β, γ and for the forth angle δ they made the three hypotheses: δ = R, δ < R and δ> R (where