AL-KHAL?L?, SHAMS AL-D?N ABU? ‘ABDALL?H MUH?AMMAD IBN MUH?AMMAD

(fl. Damacus, Syria,ca.1365)

astronomy, mathematics.

Al-Khal?l? (suter,no,418)was an astronomer associated with the Umayyad Mosque in the center of Damascus. A colleague of the astronomer Ibn al-Sh?tir, he was also a muwaqqit–that is,an astronomer concerned with ?ilm al-m?q?t, the science of timekeeping by the sun and stars and regulating the astronomically defined times of Muslim prayer. Al-Khal?l?’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 these tables for astronomical timekeeping. Some of these tables were used in Damascus until the nineteenth 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 in modern times until the 1970’s.

Al-Khal?l?’s tables can be categorized as follows; tables for reckoning time by the sun, for the latitude of Damascus; tables for regulating the times of Muslim prayer,for the latitude of Damascus; tables of auxiliary mathematical functions for timekeeping by the sun for all latitudes; table displaying the qibla, that is, the direction of Mecca,as a function of terrestrial latitude and longitude; and tables for converting lunar ecliptic coordinates to eqatiorial coordinates.

The first two sets of tables correspond to those in the large corpus of spherical astronomical tables computed for Cairo that are generally attributeed to the tenth-century Egyptian astronomer Ibn Y?nus. They are recomputed for Ibn al-Sh?tir’s parameters; 33;30° for the latitude of Damascus and 23:31° for the obliquity of the ecliptic. Al-Khal?l? does not mention any of his Egyptian predecessors. We know, however, that an elder colleague of his, the instrument maker al-Mizz? (d.ca.1350;Suter no.406), who spent the first part of his life in Egypt and then moved to Damascus, had already compiled a set of hour-angle tables and prayer tables similar to those used in Egypt and based on 33;27° for the latitude of Damascus and 23:33° for the obliquity, a pair of parameters used by earlier Syrian astronomers. Al-Khal?l?’s first and second tables were thus intended to replace al-Mizz?’s set. These tables were used in Damascus until the nineteeth century. The Damascus muwaqqit Muhammad ibn Mustaf? al- Tanat?wi, who died in 1889, was one of the last to use them; he also converted the entries from equatorial degrees and minutes to equinatial hours and minutes.

The third set of tables compiled by al-Khal?l? consisted of auxiliary tables for timekeeping by the sun and a table of the solar azimuth as a function of the solar meridian altitude and instantaneous altitude. The auxiliary tables, which contain over 9,000 entries, are intended specifically for facilitating the computation of the hour angle for given solar altitude,and solar longitude, and any terrestrial latitude. They were plagiarized by later astronomers in both Egypt and Tunis.

Al-Khal?l?’s fourth 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. Al-Khal?l? tabulated three functions and gave detailed instructions for their application. The functions are the following(the capital notation indicates that the medieval trigonometric functions are computed to base R=60, thus Sin ? = R Sin ?, and so on):

and

computed for the domains

?= 1°,2°,…, 90°

?= 1°,2°,…, 55°, as well as 21;30°(Mecca) and 33;30°(Damascus)

x=1, 2, … ,59

y=0°, 1°,… ,n(x)

where n(x) is the largest integer such that

R x ? Cos n(x).

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 al-Khalili for finding the hour angle t for given solar or stellar altitude h, declination $, and terrestrial latitude This may be represented as and it is not difficult to show the equivalence of al-Khal?l?’s for finding the hour angle t for given solar or stellar altitude h, declination ?, and terrestrial latitudeø.This may be represented as

t(h,?,?) =G{[f_?(h)-g_?(?)],?},

and it is not difficult to show the equivalence of al-Khal?l?’s rule to the modern formula

These auxiliary tables were used for several centuries in Damascus, Cairo, and Istanbul, the three main centers of astronomical timekeeping in the Muslim world.

Al-Khal?l?’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 medieal Islamic trigonometry. If (L,?)and (L_M,?_M) represent the longitude and latitude of a given locality and of Mecca,respectively, and ?L= ?L–L_M?,then the modern formula for q(L, ?), the direction of Mecca for the locality, measured from the south, is

Al-Khal?l? computed q(?,L)to two sexagesimal digits for the domains ?=10°,11°,…,56°(also 33;30°)and ?L=1°,2°,…,60°; and the vast majority of the 2,880 entries are either accurately computed or in error by±0;1° or ±0;2°. He states that he used the method for finding the qibla expounded by the late thirteenth-century Cairo astronomer Ab? ?Ali al-Marr?kushi (Suter. no. 363); and it seems that he used his universal auxiliary tables to compute the qibla values, although they are generally more accurate than can be derived from the auxiliary tables in their present form. Several other qibla tables based on approximate formulas are known from the medieval period. Al-Khal?l?’s table does not appear to have been widely used by later Muslim astronomers.

The last set of tables known to have been compiled by al-Khal?l? is for converting lunar ecliptic coordinates to equatorial coordinates, in order to facilitate computations relating to the visibility of the lunar crescent.

Al-Khal?l? wrote at least one treatise on the use of the quadrant with a trigonometric grid (al-rub ?al-mujayyab), but his writings on this instrument have not yet been studied.

BIBLIOGRAPHY

I. Original Works. MS Paris B.N. ar. 2558, copied in 1408, contains all of the tables in al-Khal?l?’s major set (nos. 1, 2, 4, and 5). MS Berlin Ahlwardt 5753-6(Wetzstein 1138) contains all but the hour-angle tables, MS Oxford Bodleian Seld. sup. 100 contains the prayer tables and the hour-angle tables; MSS Oxford Marsh 39 contains the hour-angle tables. MSS Oxford Marsh 95 and Escorial ar. 931 contain only the universal auxiliary tables; and MS Gotha Forschungsbibliothek A1406, only the prayer tables. MSS Damascus Z?hiriyya 3116 and 10378 also contain tables from the corpus. MS Cairo Talaat m?q?t 228 contains al-Khal?l?’s universal auxiliary tables and hour-angle tables for Damascus, as well as an anonymous set of prayer tables for the latitude of Tripoli (Lebanon). MS Cairo D?r al-Kutub m?q?t 71M contains al-Khal?l?’s tables of the hour angle and time since sunrise, plus tables of the solar azimuth computed by al-Halab??i (d. 1455; Suter, no. 434). Egyptian copies of the auxiliary tables exist in MSS Princeton Yahuda 861, 2 and Cairo D?r al-Kutub m?q?t 43M. MS Istanbul Aya Sofya 2590 consists of a recension of the auxiliary tables by the Ottoman astronomer Muhammad ibn K?tib Sin?n (ca. 1500; Suter, no, no 455). Al-Tant?w?’s prayer tables and hour-angle tables are extant in MSS Damascus Z?hiriyya 9233, and Cairo Taym?r riu?diyy?t129 and D?r al-Kutub m?q?t 1007.

MS Dublin Chester Beatty 4091 is an apparently unique copy of al-Khal?l?’s auxiliary tables for timekeeping by the sun and azimuth tables. Later Egyptian and Tunisian copies of the auxiliary tables are MSS Cairo D?r al-Kutub m?q?t 644 and Istanbul S. Esad Ef. Madresesi 119,2 and MS Cairo D?r al-Kutub m?q?t 689, respectively.

Al-Khal?l?’s tables for crescent visibility are in the treatise entitled ? lqd al-durar, by the later Egyptian astronomer Ibn al-Majdi (Suter, no. 432), and his criteria for crescent visibility are outlined in the writings of his nephew Sharaf al-D?n al-Khal?l? (Suter, no. 427).

Al-Khal?l?’s treatise on the quadrant with trigonometric grid is in MSS Cairo D?r al-Kutub m?q?t 138,9 and 201M. Another treatise of considerable interest on the same subject, preserved in MS Cairo D?r al-Kutub m?q?t 167M,8, may be by al-Khal?l?. A treatise describing a horizontal sundial is attributed in MS Princeton Yahuda 373, fols. 131v-135r, to Ab? ?Abd All?h al-Khal?l?, and in MS Manchester 361G to the later Egyptian astronomer ’?Abd al-?Az?z ibn Muhammad al-Waf??? (Suter, no. 437).

II . Secondary Literature. See D. A. King, “AlKhal?l?’s Auxiliary Tables for Solving Problems of Spherical Astronomy,” in Journal for the History of Astronomy, 4 (1973), 99-110; and “Al-Khal?l?’s QiblaTable,” in Journal of Near Eastern Studies, 34 (1975), 81-122; and H.Suter, Die Mathematiker and Astronomen der Araber und ihre Werke(Leipzig, 1900), 169.

David A. King