It is shown in this article that the two sides of an equation in the medieval Arabic algebra are aggregations of the algebraic “numbers” (powers) with no operations present. Unlike an expression such as our 3x + 4, the Arabic polynomial “three things and four dirhams” is merely a collection of seven objects of two different types. Ideally, the two sides of an equation were polynomials so the Arabic algebraists preferred to work out all operations of the enunciation to a (...) problem before stating an equation. Some difficult problems which involve square roots and divisions cannot be handled nicely by this basic method, so we do find square roots of polynomials and expressions of the form “A divided by B” in some equations. But rather than initiate a reconsideration of the notion of equation, these developments were used only for particularly complex problems. Also, the algebraic notation practiced in the Maghreb in the later middle ages was developed with the “aggregations” interpretation in mind, so it had no noticeable impact on the concept of polynomial. Arabic algebraists continued to solve problems by working operations before setting up an equation to the end of the medieval period. (shrink)

In rich detail Jonathan Berkey interprets the social and cultural consequences of Islam's regard for knowledge, showing how education in the Middle Ages played a central part in the religious experience of nearly all Muslims. Focusing on Cairo, which under Mamluk rule was a vital intellectual center with a complex social system, the author describes the transmission of religious knowledge there as a highly personal process, one dependent on the relationships between individual scholars and students. The great variety of institutional (...) structures, he argues, supported educational efforts without ever becoming essential to them. By not being locked into formal channels, religious education was never exclusively for the elite but was open to all. Berkey explores the varying educational opportunities offered to the full run of the Muslim population--including Mamluks, women, and the "common people." Drawing on medieval chronicles, biographical dictionaries, and treatises on education, as well as the deeds of endowment that established many of Cairo's schools, he explains how education drew groups of outsiders into the cultural center and forged a common Muslim cultural identity. Originally published in 1992. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. (shrink)

An examination of the emergence of the phenomenon of deductive argument in classical Greek mathematics.

The transformation of mathematics from ancient Greece to the medieval Arab-speaking world is here approached by focusing on a single problem proposed by Archimedes and the many solutions offered. In this trajectory Reviel Netz follows the change in the task from solving a geometrical problem to its expression as an equation, still formulated geometrically, and then on to an algebraic problem, now handled by procedures that are more like rules of manipulation. From a practice of mathematics based on the localized (...) solution we see a transition to a practice of mathematics based on the systematic approach. With three chapters ranging chronologically from Hellenistic mathematics, through late Antiquity, to the medieval world, Reviel Netz offers an alternate interpretation of the historical journey of pre-modern mathematics. (shrink)