Some time in the tenth century AD, a group of educated men in Basra and Baghdad known as the Brethren of Purity composed a large reference work — an encyclopedia of sorts ― encompassing most of the intellectual pursuits of its time. Of the 52 epistles in this compendium, 14 are devoted to the mathematical sciences; the others deal with natural philosophy, the intellect, and theology. The volume under review produces editions and English translations of the first two epistles, on arithmetic and geometry, part of a massive project encompassing all 52 epistles. The treatise on arithmetic had already been treated, on its own, in 1964 by Bernard Goldstein.
The mathematics in these first two epistles, suited to its foundational role, is quite elementary. On arithmetic, included are definitions of types of numbers, the basic calculations, perfect and amicable numbers, series, and some meta-mathematical ruminations. On geometry, the Brethren classify the basic geometric entities (points, lines, planes, various polygons), give a description of geometry in surveying, introduce magic squares, and make connections with philosophy. Of some interest is the fact that the triangle is considered to be the primitive object in the study of polygons just as the unit is the primitive object in arithmetic, because every polygon may be broken into triangles. Underlying the definitions in these epistles are the works of Greek mathematicians such as Euclid and Nicomachus.
The translation is accompanied by about 40 pages of helpful commentary, which nevertheless could be improved. Some of the equations in this section are typeset incorrectly. More awkwardly, the commentary reorders the text’s mathematical content without providing linkages, making it difficult to go back and forth. Some topics receive much greater treatment in the commentary than in the text, while a few topics (such as magic squares) are not commented upon at all. Even so, reading through the commentary (or, even better, the text itself) gives a nice quick snapshot of how the medieval Islamic scholar began his study of the mathematical sciences.
Glen van Brummelen is a founding faculty member at Quest University Canada. He is the author of Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry and The Mathematics of the Heavens and the Earth: The Early History of Trigonometry.