The asymptotic: Appolonius and his readers (Q2895498)

This is a study of Proposition II.14 of Appolonius's \textit{Conics}, which states that the asymptote and the hyperbola approach each other indefinitely, the distance between them becoming smaller than any given segment, without ever meeting. It is first shown why the Proposition (there are, in fact, two versions of it, which the author believes originate in two editions of the \textit{Conics}, one of which was available to its Arab translator) caused a stir in the minds of philosophers, such as Proclus, with its mention (in the version of Eutocius) of an ``extension towards infinity'' of the asymptote, then what solutions were offered to the perceived problems by various mathematicians, including al-Sijzī\, in the second half of the 10th century, and by his successor al-Qummī\, (whose work on Proposition II.14 can be found in Arabic and in French translation in an appendix to this paper).