Saccheri's Rectilinear Quadrilaterals

TL;DR

This paper investigates Saccheri's hypotheses on rectilinear quadrilaterals within Euclidean and hyperbolic geometries, concluding that only rectangles satisfy these hypotheses and discussing the philosophical implications of Euclidean parallel postulate independence.

Contribution

It demonstrates that Saccheri's rectilinear quadrilaterals can only be rectangles and explores the logical independence of the Euclidean parallel postulate.

Findings

Saccheri's quadrilaterals are only rectangles.

Euclidean parallelism is a theorem in Hilbert's geometry.

Hyperbolic parallelism maps to Euclidean parallelism under transformations.

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.