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Ibn al-Haytham's Universal Solution for Finding the Direction of the Qibla by Calculation

Published online by Cambridge University Press:  24 October 2008

Ahmad S. Dallal
Ahmad S. Dallal
Affiliation:
Department of Near Eastern Languages and Civilizations, Yale University, New Haven, CT 06520, U.S.A.
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Abstract

This paper presents an edition of al-Hasan ibn al-Ḥasan ibn al-Haytham's (d. 1040) treatise, Qawl fi samt al-qibla bi-al-ḥisāb (on finding the azimuth of the Qibla by calculation) with translation and commentary. In it Ibn al-Haytham provides a universal method for finding the direction of the qibla at any location on the surface of the earth by using spherical trigonometry and accurate calculation. Ibn al-Haytham's computational solution has not been studied before, and it has often been confused with another work of his in which he uses an analemma construction to solve the problem of the qibla graphically. As a result of this confusion, contemporary scholars have mistakenly attributed the first universal solution of the qibla problem to Jamshīd al-Kāshī (15th century), some four centuries after the introduction of this method by Ibn al-Haytham. The present treatise represents an important juncture in the history of the development of mathematics of the qibla, and sheds more light on the contributions of one of the most important scientists of medieval Islam.

Dans cet article, on présente une édition, une traduction et un commentaire d'un traité d'al-Ḥasan ibn al-Ḥasan ibn al-Haytham (m. 1040), dont l'objet est de déterminer, par le calcul, l'azimuth de la qibla. Ibn al-Haytham y propose une méthode universelle pour trouver la direction de la qibla, en n'importe quel emplacement sur la surface de la terre, en utilisant la trigonométrie sphérique et le calcul exact. La solution par calcul d'lbn al-Haytham n'a pas été étudiée auparavant et le présent traité a souvent été confondu avec une autre oeuvre d'Ibn al-Haytham, dans laquelle ce dernier utilise une construction par analemme pour résoudre le même probléme. Cette confusion a eu pour effet de conduire les chercheurs modernes á attribuer, á tort, la première solution universelle du probléme de la qibla à Jamshīd al-Kāshī (XVe siècle), solution conçue quelque quatre siècles après la méthode introduite par Ibn al-Haytham. Le présent traité d'Ibn al-Haytham est ainsi remarquable pour deux raisons: il représente un jalon important dans l'histoire du développement des méthodes mathématiques pour résoudre le problème de la qibla, et il jette des lumières nouvelles sur la contribution de l'un des plus grands savants de I'Islam médiéval.

Type
Research Article
Information
Arabic Sciences and Philosophy , Volume 5 , Issue 2 , September 1995 , pp. 145 - 193
Copyright
Copyright © Cambridge University Press 1995

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References

1 On the life and work of Ibn al-Haytham see the latest analysis in Rashed, R., Les mathématiques inflnitésimales du IXe au XIe siècle. Vol II: Ibn al-Haytham (London, 1993).Google Scholar In the introduction to this work, Rashed resolves a long standing ambiguity in the biographical studies on Ibn al-Haytham, and proves that there are in fact two historical figures by this name; these are al-Ḥasan ibn al-Hasan, and Muhammad ibn alHasan. The former is the most famous of the two, and is the author of the work studied here, while the later is a philosopher and physician. For an earlier account of Ibn al-Haytham's life and work see Sabra, A. I., “Ibn al-Haytham,” Dictionaiy of Scientific Biography, 18 vols. (New York, 1970–1990), vol. 6 (1972), pp. 189–210.Google Scholar For lists of his works and available manuscripts, see Sezgin, F., Geschichte des arabischen Schrifttums, 9 vols. (Leiden, 1967 Onwards), especiallyGoogle Scholar Band V, Mathematik (1974), pp. 358–73Google Scholar, and Band VI (1978),Google Scholar Astronomie, pp. 251–61.Google Scholar

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4 For exact solutions of the problem of the qibla using rectilinear configurations inside the sphere see Ali, J., The Determination of the Coordinates of Cities: al-Bīrūnī's Taḥdīd al-Amākin (Beirut, 1967), pp. 243–52;Google Scholar Kennedy, E. S., A Commentary upon al-Bīrūnī's Kitāb Taḥdīd al-Amākin (Beirut, 1973), pp. 202–9;Google Scholar and King, “Earliest,” pp. 112–15.

5 On different analemmas used for solving the qibla problem see Kennedy, E. S. and Id, Y., “A letter of al-Bīrūnī: Ḥabash al-Ḥāsib's analemma for the qibla,” in Kennedy, E. S., Colleagues and Former Students, Studies in the Islamic Exact Sciences (Beirut, 1983), pp. 621–9;Google Scholar Ali, Bīrūnī's Taḥdīd, pp. 255–6;Google Scholar Kennedy, Commentary upon al-Bīrūnī's Tahdīd, pp. 209–11;Google Scholar For a comparison of four analemmas including, in addition to the above two by Habash and Bīrūnī, one analemma from Bīrūnī's Al-Qānūn al-Mas'ūdī and one by Ibn al-Haytham see Berggren, J. L., “A comparison of four analemmas for determining the azimuth of the qibla”, Journal of the History of Arabic Science, 4 (1980): 69–80.Google Scholar Also, on the analemmas of Habash and Ibn al-Haytham, see King, “Kibla,” p. 85, and King, “Earliest,” pp. 115–18.Google Scholar For tables based on analemmas see King, “Kibla,” p. 87, and King, “Al-Khalīhī's qibla table” pp. 101–8, 110–11.Google Scholar

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7 See, in addition to references listed in the above footnotes, King, “Earliest,” pp. 130–41. It should be noted that the working inside the sphere, analemmas, and work on the surface of the sphere all started in Hellenistic times, but these methods were greatly augmented in the Islamic Middle Ages.Google Scholar

8 For a list of available manuscripts see Sezgin, Geschichte, Band V, # 15, p. 368Google Scholar, and Band VI, # 18, p. 259. Also for a reference to this treatise see Sabra, “Ibn al-Haytham,” p. 205.Google Scholar

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12 See King, “The sacred direction,” p. 317, and King, “Earliest,” p. 116.Google Scholar

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25 For example, Bīrūnī Proposes in his al-Qānūn al-Mas 'ūdī a refind computational method in which he uses the rule of foue quantities, and the law of since for spherical triangles instead of the cumbersome application of the theorem of Menelaos.Google Scholar See King, “Kibla”, p. 86, and Berggren, “Origins”, pp. 14–15. On the rule of four and other developments in Islamic trigonometry see Kennedy, “The history of trigonometry”Google Scholar, in Kennedy, colleagues and students, Studies in the Islamic Exact sciences. pp. 3–29.Google Scholar

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27 See Berggren, “Birūnī's ‘Method’,” p. 245Google Scholar, and Berggren, “Origins”, pp. 5–6, 8–9.Google Scholar

28 See Berggren, “Origins”, p. 16.Google Scholar

29 For the earlist extant method where the arcs are called “the first quantity”, etc., see King, “Earlist”, pp. 112–15.Google Scholar

30 See Berggren, “Origins”, pp. 11–14.Google Scholar

31 See King, “Earlist”, p. 117.Google Scholar

32 On Bīrūnī's methods see footnotes 4–6 above.Google Scholar

33 On the use of base 60 see Kennedy, “The history of trigonometry”, pp. 3–29;Google Scholar also on the use of R see Neugebaure, Otto, A History of Ancient Mathematical Astronomy. 3 vols. (New York, 1975), part 3, pp. 1115–6.CrossRefGoogle Scholar

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* The author would like to express his gratitude to professor F. Sezgin for supplying related material which was of use during the study, and to professor E. S. Kennedy and G. Saliba, who read this paper and made valuable comments on it. Special thanks go to Professor D. A. King who familiarized me with the topic in the first place, supplied copies of the Cairo, and Berlin manuscripts, and guided me through the study. The author remains, however, solely responsible for any errors that may appear in this paper.Google Scholar