The Role of the Fifth Postulate in the Euclidean Construction of Parallels

TL;DR

This paper explores the constructive role of Euclid's Fifth Postulate in parallel line constructions, revealing its essential epistemological function and implications for Euclidean and non-Euclidean geometries.

Contribution

It provides a reconstruction of Euclidean principles underlying parallel constructions, emphasizing the postulate's constructive and epistemological significance.

Findings

Euclidean constructions depend critically on the Fifth Postulate

Reconstruction clarifies the epistemological role of Euclidean geometry

Implications for understanding non-Euclidean geometries

Abstract

We ascribe to the Euclidean Fifth Postulate a genuine constructive role, which makes it absolutely necessary in the parallel construction. For that, we present a reconstruction of the general principles underlying the Euclidean construction of a geometric property. As a consequence, the epistemological role of Euclidean constructions is revealed. We also examine some first implications of our interpretation to the relation between Euclidean and non-Euclidean geometries. The Bolyai construction of limiting parallels is also discussed from the Euclidean point of view, as this is reconstructed here.