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MATHEMATICAL METHODS IN ABŪ AL-WAFĀʾ'S ALMAGEST AND THE QIBLA DETERMINATIONS

Published online by Cambridge University Press:  18 February 2011

Ali Moussa
Affiliation:
King Fahd University of Petroleum and MineralsP.O. Box 749, Dhahran 31261, Saudi Arabia Email: amoussa@dcc.kfupm.edu.sa

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 (to construct the trigonometric tables) and spherical trigonometry (as the problem belongs to spherical astronomy). Observation and its instruments were needed to find the geographical coordinates of Mecca and the given location; these coordinates (latitude, longitude) will be the input data in the formulas of the Qibla. In his Almagest, Abū al-Wafāʾ 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 (which was based on the new definitions and this rule allowed him to work out the easiest solution, as will be shown). In observation, he described three instruments which he used over several years in Baghdad. This paper is a detailed technical and analytical description of Abū al-Wafāʾ's mathematical methods and the Qibla determinations, supplemented with many important original Arabic texts with translation and commentary.

Résumé

Le problème de la détermination de la Qibla est l'une des questions cruciales qui se posent à la culture scientifique de l'Islam médiéval; le résoudre correctement nécessite tant des théories mathématiques que des observations. Les mathématiques relèvent de deux chapitres: la trigonométrie plane (nécessaire à la construction de tables trigonométriques) et la trigonométrie sphérique (puisque le problème relève de l'astronomie sphérique). L'observation et les instruments d'observation sont indispensables à la détermination des coordonnées géographiques de La Mecque et du lieu donné; ces coordonnées (latitude et longitude) sont en effet les données que l'on entre dans les formules donnant la Qibla. Dans son Almageste, Abū al-Wafāʾ résout brillamment ce problème. Son travail porte tant sur les mathématiques que sur l'observation; il obtient des solutions modernes, adéquates et faciles. En trigonométrie plane, il donne de nouvelles définitions des fonctions trigonométriques, démontre les formules des sinus, donne une approximation de sin 1° grâce à laquelle il construit des tables de sinus et de tangentes d'une grande précision. En trigonométrie sphérique, il démontre quatre nouveaux théorèmes dont la règle des tangentes (basée sur les nouvelles définitions des fonctions trigonométriques); cette règle lui permet de trouver, comme nous le montrerons, des solutions simples au problème de la détermination de la Qibla. En ce qui concerne les observations, il décrit trois instruments utilisés par lui plusieurs années de suite à Bagdad. Cet article, nourri de nombreux textes arabes originaux traduits et commentés, donne une description détaillée, technique et analytique, des méthodes mathématiques d'Abū al-Wafāʾ pour la détermination de la Qibla.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2011

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