On diagrams accompanying reductio ad absurdum proofs in Euclid's Elements book I. Reviewing Hartshorne and Manders

TL;DR

This paper examines Euclid's Book 1 proofs, revealing the use of non-constructible figures and challenging existing claims about geometric diagram construction and attribute distinctions.

Contribution

It provides a detailed analysis of Euclidean diagrams, questioning prior assumptions about their constructibility and the nature of geometric attributes.

Findings

Euclidean proofs include non-constructible figures.

Challenging Hartshorne's claim on constructibility.

Questioning Manders' attribute distinction in diagrams.

Abstract

Exploring selected reductio ad absurdum proofs in Book 1 of the Elements, we show they include figures that are not constructed. It is squarely at odds with Hartshorne's claim that "in Euclid's geometry, only those geometrical figures exist that can be constructed with ruler and compass". We also present diagrams questioning Manders' distinction between exact and co-exact attributes of a diagram, specifically, a model of semi-Euclidean geometry which satisfies straightness of lines and equality of angles and does not satisfy the parallel postulate.