The Tendency to Generalize: A Feature of Late Antique and Medieval Mathematics, or a Flaw in Modern Historiography?*
Alain G. Bernard
Jan 1, 2006
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Classical Philology
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Semantic Scholar
Key Takeaway Key Takeaway: The book concludes by examining the generalization of mathematics from Archimedes to al-Khayyam, focusing on the transformation of geometry from the third to the eleventh centuries.
Abstract
The title of Netz’s book promises a wide-ranging study on the transformation of mathematics from the third century b.c.e. (Archimedes’ time) to the eleventh century c.e. (al-Khayyam’s time). It is actually in no way an exhaustive study of the subject; it rather focuses on the solutions found during this long span of time to a particular geometrical problem. As we shall see, this particular study is given a paradigmatic value, hence the title. The paradigmatic case is presented in the first of the three parts of N.’s book, entitled “The Problem in the World of Archimedes.” In it, N. comments on various solutions found in early Hellenistic time to the famous fourth geometrical problem listed by Archimedes in SC 2.4:1 “to cut a given sphere, so that its segments have to each other a given ratio” (Pb1).2 Archimedes’ solution consists in a long and complex reduction,3 through the ancient technique of analysis, to a more general problem, which is the following: “to cut a given line DZ at X, so that we should have the following proportion: the one segment XZ should be to some given line as some given area is to the square on the other segment DX” (Pb2).4 N. very quickly discusses Archimedes’ solution (part 1.1, pp. 11–13), and then goes on to propose a translation of what the sixth-century c.e. commentator Eutocius says (with N.’s approval, as we shall see) was Archimedes’ solution to Pb2 (part 1.2).5 N. goes on