« Al-Māhānī's Commentary on the Concept of Ratio ». Arabic Sciences and Philosophy, 12 (2002), pp. 9-52.
Abū ‘Abdallāh Muḥammad ibn ‘Īsā ibn Aḥmad al-Māhānī (fl. 865) was the author of one of the earliest Arabic commentaries on the treatment of ratio and proportionality in Book V of Euclid's Elements. In this commentary, extant in nine manuscripts, he provides an alternative, equivalent definition of ratio by means of anthyphairesis, a process similar to the Euclidean algorithm for finding the greatest common divisor of two numbers. In this article al-Māhānī's definition is explained and some possible reasons why Muslim mathematicians considered it “to express the essence of the sameness of ratios” and why they preferred it to Euclid's definition are given. Three appendixes contain an edition of the Arabic text with an English translation and an extensive mathematical commentary. Al-Māhānī's approach to anthyphairetic ratios has been compared to those of al-Nayrīzī and ‘Umar al-Khayyām by Jan Hogendijk (cf. Abs. Ir. 25 (2002), c.r. n°271).