Book I of Euclid's Elements and application of areas
Jordan Bell
TL;DR
This paper surveys Euclid's Book I with an emphasis on the application of areas, especially $I.42$, $I.44$, and $I.45$, and compiles alternate medieval constructions that illuminate how this approach can be realized without superposition. It analyzes the construction of parallelograms equal to given triangles within a fixed angle, the role of complements and the diameter, and how these ideas underpin the broader theory of area equality. By aggregating editorial variants and commentaries (e.g., Proclus, Al-Nayrizi, Adelard, Campanus, Albertus) and linking them to the modern digitization of editions, the work clarifies the conceptual and historical development of area application and its connection to squaring the circle. The study thus enhances understanding of Euclid's geometry, the evolution of geometric reasoning, and the role of historical interpretation in the pedagogy of Book I.
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) Österreichische Nationalbibliothek, Cod. Phil. gr. 31, (b) Biblioteca Comunale dell'Archiginnasio, Collocazione A 19, (p) Bibliothèque nationale de France, Grec 2466.