Commentary to Book I of the Elements. Hartshorne and beyond

Abstract

(Hartshorne, 2000) interprets Euclid’s Elements in the Hilbert system of axioms, specifically propositions I.1–34 covering the foundations of Euclidean geometry. We develop an alternative interpretation that explores Euclid’s practice concerning the relation greater-than. Discussing the Fifth Postulate, we present a model of non-Euclidean plane in which angles in a triangle sum up to π. It is a subspace of the Cartesian plane over the field of hyperreal numbers R*.