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*.
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Błaszczyk, P., & Petiurenko, A. (2021). Commentary to Book I of the Elements. Hartshorne and beyond. Annales Universitatis Paedagogicae Cracoviensis | Studia Ad Didacticam Mathematicae Pertinentia, 13, 43–99. https://doi.org/10.24917/20809751.13.7
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