An analysis of Euclid's geometrical foundations

TL;DR

This paper reexamines Euclid's foundational techniques, clarifies key concepts like equality and superposition, and sketches a rigorous approach to absolute geometry based on simple axioms, connecting classical and modern geometric ideas.

Contribution

It provides a detailed analysis of Euclid's original methods and proposes a rigorous framework for absolute geometry inspired by Euclidean principles.

Findings

Clarification of Euclid's techniques on equality and superposition

A sketch of a rigorous absolute geometry framework

Connection between Euclidean methods and modern geometric axioms

Abstract

The initial techniques developed in Euclid's Elements, well before the use of the parallel postulate, are reexamined in order to clarify even the most obscure details, particularly those related to equality, superposition and angle comparison. Some commentary on modern developments is included. The known but often misunderstood implicit handling of betweenness and points of intersection is briefly treated. We also sketch a rigorous treatment of absolute geometry in a spirit similar to Euclid's, one that allows properties of angles and triangles to be derived from two simple axioms on right angles, which then leads to rigid motions of certain planar geometries.