Saccheri's Rectilinear Quadrilaterals

Abstract

We study Saccheri`s three hypotheses on a two right-angled isosceles quadrilateral, with a rectilinear summit side. We claim that in the Hilbert`s foundation of geometry the euclidean parallelism is a theorem, and in the h-plane the hyperbolic parallelism under a hyperbolic transformation has image the euclidean parallelism. We prove that Saccheri`s rectilinear quadrilaterals can be only rectangle. Finally we believe that the independence of the euclidean parallel postulate is just a matter of philosophy of logic.