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Ibn al-Banna and his “Concise exposition of arithmetical operations”. (Russian. English summary) Zbl 1084.01003

Ibn al-Banna (Abu-l-Abbas Ahmad ibn Muhammad ibn Usman al-Azdi al-Marrakeshi) (1256–1321) was born in Granada according to some authors and in Fez or Marrakesh according to other sources. He was the author of fourteen works in mathematics and twelve in astronomy, the most well-known mathematical work being A short exposition of arithmetical operations (Talhis a’mal al-hisab); the manuscript is kept in the scientific library of the St. Petersburg’s State University, Faculty for Eastern Studies, ms. No.757/23; the authors also used manuscripts of West-Arabian scholars of the 14th to the 15th centuries kept in the Institute for the History of Arabic Science in Aleppo, Syria. This work was well known since its publication and commentaries were written about it. It consists of separate parts dealing with arithmetic (integers and operations on them, fractions and operations on them, taking square roots of whole numbers and fractions, operations with quadratic and higher irrationals) and algebra (ratios and proportions and operations on them, solving problems by weighing designs; algebra proper (al dzhabr and al-mukabala), i.e., description, classification and solutions of linear and quadratic equations by methods of geometric algebra; these chapters also cover rules on algebraic expressions and the rules of signs. Among numerous topics one can find an expanded concept of number and using mechanical methods (such as the balance method and weighing) to solve linear equations. Formulations of results are concise and the proofs are mostly not given, save for a few geometrical proofs. One can also find, for instance, the methods for summation of powers of successive natural numbers; one of the results mentioned is that the sum of cubes of (the first \(n\)) natural numbers is the square of the sum of their first powers. Perhaps best known is the exposition of the (previously known) method of false values (method of corrections). The authors discuss the influences of al-Chorezmi as well as earlier Greek and Indian mathematics building on their earlier work on the same subject. The authors may add to their bibliography R. Rashed [“The development of Arabic mathematics: Between arithmetic and algebra” (1994; Zbl 1073.01525) and “Entre arithmétique et algèbre: Recherches sur l’histoire des mathématiques arabes” (1984; Zbl 0944.01019).

MSC:

01A30 History of mathematics in the Golden Age of Islam

Biographic References:

Ibn al-Banna