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*.