Mathematical methods in abū al-wafāʾ's almagest and the qibla determinations

Arabic Sciences and Philosophy 21 (1):1-56 (2011)
Abstract
The problem of the Qibla was one of the central issues in the scientific culture of Medieval Islam, and to solve it properly, one needed mathematics and observation. The mathematics consisted of two parts: plane trigonometry and spherical trigonometry . Observation and its instruments were needed to find the geographical coordinates of Mecca and the given location; these coordinates will be the input data in the formulas of the Qibla . In his Almagest , Ab?? al-Waf???? produced a brilliant work to solve the problem. He worked on both mathematics and observation, and reached accurate and easy ???modern??? solutions. In plane trigonometry, he introduced the trigonometric functions with new definitions, proved the formulas for sines, approximated the sine of degree one, and thus constructed the tables of sines and tangents with high accuracy. In spherical trigonometry, he proved four new spherical theorems, including the tangent rule . In observation, he described three instruments which he used over several years in Baghdad. This paper is a detailed technical and analytical description of Ab?? al-Waf????'s mathematical methods and the Qibla determinations, supplemented with many important original Arabic texts with translation and commentary. R??sum?? Le probl??me de la d??termination de la Qibla est l'une des questions cruciales qui se posent ?? la culture scientifique de l'Islam m??di??val; le r??soudre correctement n??cessite tant des th??ories math??matiques que des observations. Les math??matiques rel??vent de deux chapitres: la trigonom??trie plane et la trigonom??trie sph??rique . L'observation et les instruments d'observation sont indispensables ?? la d??termination des coordonn??es g??ographiques de La Mecque et du lieu donn??; ces coordonn??es sont en effet les donn??es que l'on entre dans les formules donnant la Qibla . Dans son Almageste , Ab?? al-Waf???? r??sout brillamment ce probl??me. Son travail porte tant sur les math??matiques que sur l'observation; il obtient des solutions modernes, ad??quates et faciles. En trigonom??trie plane, il donne de nouvelles d??finitions des fonctions trigonom??triques, d??montre les formules des sinus, donne une approximation de sin??1?? gr??ce ?? laquelle il construit des tables de sinus et de tangentes d'une grande pr??cision. En trigonom??trie sph??rique, il d??montre quatre nouveaux th??or??mes dont la r??gle des tangentes ; cette r??gle lui permet de trouver, comme nous le montrerons, des solutions simples au probl??me de la d??termination de la Qibla . En ce qui concerne les observations, il d??crit trois instruments utilis??s par lui plusieurs ann??es de suite ?? Bagdad. Cet article, nourri de nombreux textes arabes originaux traduits et comment??s, donne une description d??taill??e, technique et analytique, des m??thodes math??matiques d'Ab?? al-Waf???? pour la d??termination de la Qibla
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David Pingree (1985). Ptolemy's Almagest. Ancient Philosophy 5 (2):348-349.
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