Apollonius of Perga, Cutting lines according to ratios. Historical and mathematical commentary, edition and translation of the Arabian text by Roshdi Rashed and Hélène Bellosta. With a preface by Rashed (Q1001935)

This is the only other treatise, beside the \textit{Conics}, written by Apollonius, that has reached us, and it is extant only in its Arabic translation (assumed to have happened in the 9th century, probably in the school of the Banū Mūsa and of Thābit ibn Qurra). The first translation from Arabic into Latin goes back to E. Halley (1706). The current work represents the first Arabic edition in print, as well as a French translation, preceded by an introduction, two meticulous commentaries of the text, one in geometric language, the other in algebraic language. and a history of the text, the manuscripts, its influence in the history of mathematics. The text is one vast study of a single geometrical problem: \(k\) being a given ratio, \(AB\) and \(CD\) being two given lines in the plane, \(H\) a point of that same plane, which lies neither on \(AB\) nor on \(CD\), \(E\) a point on \(AB\), \(G\) a point on \(CD\), determine a line through \(H\) that intersects \(AB\) in \(L\) and \(CD\) in \(K\), such that \(\frac{EL}{GK}=k\). Since the investigation is carried out in a synthetic geometric manner, Apollonius has to distinguish a great number of cases, 21 to be precise (7 in the first book, 14 in the second book). These are in turn subdivided according to certain incidences, resulting in a total of 87 cases, Apollonius presenting both their \textit{analysis} and their \textit{synthesis}. The main method used to solve these is the \textit{application of areas} (based on Propositions VI.28 and VI. 29 of Euclid's \textit{Elements}). This will remain the standard edition for mathematically inclined readers. The historical purist will lodge the obligatory protest against the explanation of its content in algebraic terms, and in general, against any attempt at a direct communication between the present and the distant past.