Abū Kāmil: Algèbre et Analyse Diophantienne

Abū Kāmil, who lived in Egypt from around 850 to around 930, is remembered mostly for his Algebra. In this book Roshdi Rashed presents a new edition (apparently the first critical edition) of the complete Arabic text with a French translation on facing pages, together with an extensive commentary. This is the only complete translation into a language I can read, so I am very glad to have it.

Abū Kāmil’s Algebra starts out, as do most Arabic algebras, with a discussion of the six kinds of quadratic problems. Then follow a large number of problems that can be solved using these methods. Many of them involve dividing 10 into two summands or two factors that have to satisfy another constraint, and so have something of a “recreational” flavor.

The next two sections are quite interesting. Apparently inspired by Book X of Euclid’s elements, Abū Kāmil considers problems in which various kinds of irrational numbers appear. The numbers he deals with are all either quadratic irrationals or generalizations: things like \(\sqrt{a+\sqrt{b}}\) or \(\sqrt{a}+\sqrt{b}\). This may well be the earliest text in Arabic where such quantities are treated as numbers. He then, still following Euclid, applies this to the study of regular polygons (specifically the pentagon and decagon).

Abū Kāmil follows that with a section on indeterminate problems. Rashed claims that this is the first example of the explicit distinction between determinate problems, with a unique solution, and indeterminate problems, which have many solutions and typically are to be solved under restrictions; here the solutions are supposed to be positive rational numbers. (It seems clear that Abū Kāmil did not know Diophantus’s Arithmetica, though a translation into Arabic did exist in his time.)

Rashed also includes another short work, The Book of Birds, which is dedicated to problems of the “hundred fowls” type: someone buys a hundred birds of various kinds for a hundred units of money; given the price of each kind of bird, determine how many of each were bought. Abū Kāmil seems particularly interested in the fact that such problems can have a large (but finite) number of solutions.

The edition and translation of these texts is very well done, and everyone interested in this historical period or in the history of algebra will be grateful for having it. Things are less clear when it comes to Rashed’s commentary and interpretation of this material, which is very much open to debate. Those interested in a detailed discussion of the issues might start by looking at the review in Aestimatio, written by Jeff Oaks.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME. He loves reading old books.