Book I of Euclid's Elements and application of areas

TL;DR

This paper analyzes Book I of Euclid's Elements focusing on the application of areas, comparing medieval and modern constructions, and examining the use of superposition in Euclid's proofs.

Contribution

It provides a detailed comparison of Euclid's constructions with medieval editions and highlights alternative methods avoiding superposition.

Findings

Medieval editions often avoid superposition in Euclid's proofs.

Euclid's original proof of I.44 involves superposition, but alternatives exist.

The paper collates figures from various digitized editions of Euclid.

Abstract

We work through Book I of Euclid's Elements with our focus on application of areas (I.42, I.44, I.45). We summarize alternate constructions from medieval editions of Euclid's elements and ancient and medieval commentaries. We remark that Euclid's proof of I.44 involves a seldom commented on use of superposition, but that several medieval editions of Euclid give constructions that avoid the use of superposition. This use of superposition is also avoided in Ralph Abraham's ``VCE: The Visual Constructions of Euclid'' C#12, C#12B at http://www.visual-euclid.org/vce/contents.html We collate the figures with the digitized editions of Euclid at (P) Biblioteca Apostolica Vaticana (BAV), Vat. gr. 190, (F) Biblioteca Medicea Laurenziana (BML), Plut. 28.03, (B) Bodleian, MS. D'Orville 301, (V) \"Osterreichische Nationalbibliothek, Cod. Phil. gr. 31, (b) Biblioteca Comunale dell'Archiginnasio, Collocazione A 19, (p) Biblioth\`eque nationale de France, Grec 2466.