Commentary to Book I of the Elements. Hartshorne and beyond

Abstract

(Hartshorne, 2000) interprets Euclid’s Elements provides an interpretation of Euclid’s Elements in the Hilbert system of axioms, specifically propositions I.1-I.27, covering the so-called absolute geometry. We develop an alternative interpretation that explores Euclid’s practice concerning the relation greater-than. Discussing the Postulate 5, we present a model of nonEuclidean plane in which angles in a triangle sum up to π. It is a subspace of the Cartesian plane over the ordered field of hyperreal numbers R*.