Abu al-Wafa' al-Buzjani, Muhammad ibn Muhammad

Abu al-Wafa Muhammad ibn Muhammad ibn Yahya Isma'il al-Buzjani, one of the leading Muslim scholars of mathematics and astronomy. He was born in Buzjan in Khurasan in 328 AH (around 940 CE), and died in 388 AH (around 998 CE). ¹

The Encyclopaedia of Islam (2nd ed.), unsigned, calls Abu 'l-Wafa' Muhammad b. Muhammad b. Yahya b. Isma'il b. al-'Abbas al-Buzdjani one of the greatest Arab mathematicians, of Persian origin, born at Buzdjan in Kuhistan on 1 Ramadan 328 (10 June 940). His first teachers were his uncles Abu 'Amr al-Mughazili and Abu 'Abd Allah Muhammad b. 'Anbasa; Abu 'Amr had studied geometry under Yahya al-Marwazi and Abu 'l-'Ala' b. Karnib. In 348/959 he emigrated to Iraq and lived in Baghdad until his death there in Rajab 388 (July 998). Ibn al-Athir and Ibn Khallikan, followed by the article, give 387/997 as the year. He introduced Abu Hayyan al-Tawhidi to the vizier Ibn Sa'dan, for whom Abu Hayyan wrote al-Imta' wa'l-mu'anasa. ²

Three mathematical and astronomical works survive. The arithmetic book Fima yahtaji ilayh al-kuttab wa'l-'ummal min 'ilm al-hisab is identical with the al-Manazil fi'l-hisab that Ibn al-Qifti mentions; Woepke published the titles of its stations and chapters in 1855. The al-Kamil is probably the Almadjist that Ibn al-Qifti names, and parts of it were translated by Carra de Vaux in 1892. The al-Handasa, in Arabic and Persian, is probably the Persian book of geometrical constructions in Paris; Woepke thought it was written by a pupil summing up his lectures, and partly based on Indian models. His commentaries on Euclid, Diophantus and al-Khwarizmi, and his astronomical tables al-Wadih, are lost. The anonymous tables al-Zidj al-shamil in Florence, Paris and London are very likely an adaptation of his tables. ²

The article's chief claim is that he developed trigonometry further. For the right-angled spherical triangle he replaced the perfect quadrilateral with the proposition of Menelaus by the "rule of the four magnitudes" (sine a : sine c = sine A : 1) and a tangent theorem (tan a : tan A = sine b : 1), from which he derived cos c = cos a cos b. He probably also established the sine proposition for the oblique spherical triangle. He gave a method for calculating the sine of 30' that agrees to eight decimal places with the true value. The article denies two claims: that he introduced the tangent, cotangent, secant and cosecant, which Habash al-Hasib already used, and that he discovered the variation of the moon, which L. A. Sedillot asserted in 1836 and which Carra de Vaux settled against him in 1892. ²