An analysis of Euclid's geometrical foundations
Peter M Johnson
TL;DR
The paper addresses the problem of clarifying Euclid's geometric foundations in Book I without overreliance on the parallel postulate, by reinterpreting implicit notions of equality, betweenness, and intersection within a modern axiomatic framework. It develops an absolute-geometry approach grounded in right-angle principles, using segment and angle transport to define congruence and measure, and constructs a coordinatization that supports rigid motions. Key contributions include a transport-based treatment of congruence, rigorous handling of angles through triangle transport, and a coordinatization scheme that links geometric objects to an ordered abelian group, along with foundational rigidity results. The work aims to reconcile Euclid's constructive methods with modern formalism, enabling formal verification and connecting ancient geometry to contemporary axiom systems and coordinate geometry.
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.