David A. King

Khalīlī: Shams al‐Dīn Abū ʿAbdallāh Muḥammad ibn Muḥammad al‐Khalīlī

Flourished Damascus, (Syria), circa 1365

Khalīlī was an astronomer associated with the Umayyad Mosque in the center of Damascus. A colleague of the astronomer Ibn al‐Shāṭir, he was also a muwaqqit – i. e., an astronomer concerned with ʿilm al‐mīqāt, the science of timekeeping by the Sun and stars and regulating the astronomically defined times of Muslim prayer. Khalīlī's major work, which represents the culmination of the medieval Islamic achievement in the mathematical solution of the problems of spherical astronomy, was a set of tables for astronomical timekeeping. Some of these tables were used in Damascus until the 19th century, and they were also used in Cairo and Istanbul for several centuries. The main sets of tables survive in numerous manuscripts, but they were not investigated until the 1970s.

Khalīlī's tables can be categorized as follows:

(Paris, Bibliothèque Nationale MS ar. 2558, copied in 1408, contains all of the tables in Khalīlī's major set [1, 2, 5 and 6]. Dublin, Chester Beatty MS 4091 and Bursa, Haraççioğlu MS 1177,4 are unique copies of the minor auxiliary tables [3] and [4], respectively.)

The first two sets of tables correspond to those in the large corpus of spherical astronomical tables computed for Cairo that are generally attributed to the 10th‐century Egyptian astronomer Ibn Yūnus.

Khalīlī's fifth set of tables was designed to solve all the standard problems of spherical astronomy, and they are particularly useful for those problems that, in modern terms, involve the use of the cosine rule for spherical triangles. Khalīlī tabulated three functions and gave detailed instructions for their application. The functions are the following:

f_ϕ = sin θ/ cos ϕ and g_ϕ = sin θ tan ϕ,

K(x,y) = arc cos {x/cos y},

computed for appropriate domains. The entries in these tables, which number over 13,000, were computed to two sexagesimal digits and are invariably accurate. An example of the use of these functions is the rule outlined by Khalīlī for finding the hour angle t for given solar or stellar altitude h, declination δ, and terrestrial latitude φ. This may be represented as:

t(h,δ,ϕ) = K {[ f_ϕ (h) – g_ϕ (δ)],δ},

and it is not difficult to show the equivalence of Khalīlī's rule to the modern formula

t = arc cos {[sin h – sin δ sin ϕ ] / [ cos δ cos ϕ ]} .

These auxiliary tables were used for several centuries in Damascus, Cairo, and Istanbul, the three main centers of astronomical timekeeping in the Muslim world. They were first described in 1973. In 1991 Glen Van Brummelen, in his statistical investigation of the errors in the entries, determined that the tables of (7) had been computed first and the tables of (6) were computed from these. In 2000, the fourth set of Khalīlī's tables was discovered in a manuscript in Bursa. These were compiled before the fifth set and also contain a set of tables of (7); when compiling his main set (5), Khalīlī simply took over the tables of (7) from this earlier set (4). So Van Brummelen's hypothesis was confirmed.

Khalīlī's computational ability is best revealed by his qibla table. The determination of the qibla for a given locality is one of the most complicated problems of medieval Islamic trigonometry. If (L,φ) and (LM,φ_M) represent the longitude and latitude of a given locality and of Mecca, respectively, and ΔL = |L−LM|, then the modern formula for q(L,φ), the direction of Mecca for the locality, measured from the south, is

q = arc cot {[ sin ϕ cos ΔL – cos ϕ tan ϕ_M ] / sin ΔL}.

Khalīlī computed q(L,φ) to two sexagesimal digits for the domains φ = 10°, 11°,…, 56° and ΔL = 1°, 2°,…, 60°; the vast majority of the 2,880 entries are either accurately computed or in error by ±1’ or ± 2’. Several other qibla tables based on approximate formulas are known from the medieval period. Khalīlī's splendid qibla table does not appear to have been widely used by later Muslim astronomers.

Selected References

King, David A. (1973). “Al‐Khalīlī's Auxiliary Tables for Solving Problems of Spherical Astronomy.” Journal for the History of Astronomy 4: 99–110. (Reprinted in King, Islamic Mathematical Astronomy, XI. London: Variorum Reprints, 1986; 2nd rev. ed., Aldershot: Variorum, 1993.)

——— (1975). “Al‐Khalīlī's Qibla Table.” Journal of Near Eastern Studies 34: 81–122. (Reprinted in King, Islamic Mathematical Astronomy, XIII. London: Variorum Reprints, 1986; 2nd rev. ed., Aldershot: Variorum, 1993.)

——— (1978). “Al‐Khalīlī.” In Dictionary of Scientific Biography, edited by Charles Coulston Gillispie. Vol. 15, pp. 259–261. New York: Charles Scribner's Sons.

——— (1983). “The Astronomy of the Mamluks.” Isis 74: 531–555. (Reprinted in King, Islamic Mathematical Astronomy, III. London: Variorum Reprints, 1986; 2nd rev. ed., Aldershot: Variorum, 1993.)

——— (1993). “L'astronomie en Syrie à l'époque islamique.” In Syrie, mémoire et civilization, [exhibition catalogue] edited by Sophie Cluzan, Eric Delpont and Jeanne Mouliérac, pp. 392–394, 440. Paris: Institut du monde arabe and Flammarion.

——— (2004). In Synchrony with the Heavens: Studies in Astronomical Timekeeping and Instrumentation in Medieval Islamic Civilization. Vol. 1, The Call of the Muezzin (Studies I–IX). Leiden: Brill, II–10.

Van Brummelen, Glen (1991). “The Numerical Structure of al‐Khalīlī's Auxiliary Tables.” Physis, n.s., 28: 667–697.