Apollonius of Perga, Serenus of Antinoeia, Thabit ibn Qurra and the research on conical and cylindrical sections (Q2866538)

Probably in the 4th century CE, Serenus of Antinoë wrote (1) a commentary to Apollonius's \textit{Conics}, (2) an independent treatise on the topic of conic sections, and (3) a treatise \textit{On the section of a cylinder}. In the later 9th century, Thābit ibn Qurra wrote another treatise \textit{On the section of a cylinder and its lateral surface}.NEWLINENEWLINEIn the present article, the author presents a survey, with texts in parallel columns, of similarities and differences between the initial part of \textit{Conics} I (definitions, Propositions 1--9, 15 and 21), the definitions and Propositions 1--19 of Serenus's treatise, and the definitions and Propositions 1--11 of that of Thābit.NEWLINENEWLINEIt turns out that Serenus follows Apollonius's presentation very closely, at most replacing occasionally a synthetic ``Euclidean'' proof by a reductio ad absurdum. In the beginning, Thābit does as much, but then he introduces cylindrical projections as an alternative tool. The author explains this by Thābit's position at the intersection of the Apollonian and the Archimedean traditions and (following Roshdi Rashed) by his access to that theory of conic sections which Ahmad ibn Mūsā had developed independently at a moment when the Apollonian treatise was not yet at hand. He further sees it as a preparation of Thābit's further work on infinitesimal geometry.