Al-KhalīLĪ's Auxiliary Tables for Solving Problems of Spherical Astronomy

Irani's R. thesis (Irani [1]) remains to be published. On Habash, see Kennedy [1], 126–7 (nos. 15 and 16).

See Jensen [1].

See King [1], 29–33.

On al-Dīn Abū Shams 'Abd Allāh Muhammad b. Muhammad al-Khalīlī, see Suter [1], no. 418.

Suter (cf. ref. 4) states that al-Khalīlī was associated with the Yalbaghā Mosque outside Damascus. The title folio of MS B.N. ar. 2558 states that he was a timekeeper (muwaqqit) at the Umayyad Mosque.

Sayili A. has already pointed to the need for further research in this area. Cf. Sayili [1], 245. On Ibn al-Shātir and his work, see Kennedy [1], 125 (no. 11); Wiedemann [1], ii, 729–38; and the recent studies listed by Swerdlow N. in Journal for the history of astronomy, iii (1972), 48, ref. 3.

Jensen C. has published a few entries from Abū Nasr's table. The following table shows the corresponding recomputed values, to be compared with those in Jensen [1], 4. I II III IV V VI VII 1° 2°;24,16,10,41;0,27,47,16;1,54,54,58;2, 5,38,17;54,52,50,12 2 4;24,16,50,37;0,55,35,33;3,49,45,39;4,11, 7,24;54,54, 0,38 3 6;24,17,57,15;1,23,25,52;5,44,27,44;6,16,18, 9;54,55,57,36 4 8;24,19,30,41;1,51,19,15;7,38,56,50;8,21, 1,23;54,58,40,27 44 88;33,44, 1, 2;25,37,20,18;59,57,22,43;59,57,48,25;59,59,38,29 45 90;34,19, 1,59;26,31,57,38;60, 0, 0, 0;60, 0, 0, 0;60, 0, 0, 0 It is clear that in numerous entries the third of the four digits is in error, and the fourth is meaningless. Furthermore, had Abū Nasr used the value 23;35° for the obliquity, which was generally accepted in his time, rather than Ptolemy's value 23;51,20°, the entries in all but column VI (which are for the sine function) would generally differ even in the second digit from those given in his table. It is perhaps worth remarking that al-Nayrīlzī (ca 900) also tabulated some auxiliary functions. Two tables attributed to him, based on the Khanda Khādyaka value R = 2,30, are located in MS Berlin 5750, fols 83r–85r, of Habash's Zīj. Cf. King [1], 29.

Cf. King [2] for a detailed analysis.

Cf. King [1], 43 and [2].

An analysis is in preparation.

This table is analyzed in detail in King [3]. It may be that al-Khalīlī computed the entries in the qibla table using his auxiliary tables.

I have not inspected MSS Berlin 5754–56, British Museum 977, 31, and Oxford I. 961, 1039,2, listed by Suter. Cf. ref. 4 above.

Cf. Slane de [1], 460–1.

I am indebted to Prof. Mach R. of Princeton University for kindly allowing me to consult his unpublished catalogue of the Yahuda Collection.

Cf. Renaud [1], 42–3. In this MS the tables are attributed to Sharaf al-Dīn Abū 'Abd Allāh al-Khalīlī, who lived about the year 1400. Cf. Suter [1], no. 427. Probably Sharaf al-Dīn should be restored to Shams al-Dīn.

Cf. ref. 10 above.

An analysis is in preparation. On al-Wafā'ī see Suter [1], no. 437, and King [1], 43 and 49.

The capital letter notation, now standard, is used to denote medieval trigonometric functions. Cf. Kennedy [1], 139b–140a.

In the Escorial MS, the maximum value of ϕ is 50°. As stated by Abū Nasr, beyond latitude 45° there were few people who knew about spherical astronomy or even thought about it. Cf. Jensen [1], 5.

On which, see Irani [2].

Cf. refs 26, 29, and 31 below.

For some of the modern formulae, consult Smart [1], 25–52. For the derivation of these formulae by medieval methods, see, for example, King [1], 90–316.

On which, see Kennedy [1], 132–3 (no. 55), and the edition of Nallino C. (Milan-Rome, 1899–1907). al-Battānī does not, for example, discuss the calculation of the duration of twilight.

With the reservation that Ibn Yūnus discusses the calculation of time since sunrise rather than the hour-angle. Cf. King [1], 147–52.

Generally , I use δ for solar declination (mayl) and Δ for the declination of a star (bu'd). For most of the problems under discussion, al-Khalīlī notes the validity of his procedures for both the Sun and the fixed stars.

al-Khalīlī's instructions (MS B.N. ar. 2558, fol. 62r, 11. 1–7) read: Section on the hour-angle. Find the value of the first function for the altitude (as argument) on the page corresponding to the latitude, and find the value of the second function (for the declination as argument) as we described above for the half diurnal arc. Then, if the Sun is in the southern signs, add the two functions; otherwise, take the difference between them. The result will be the auxiliary Sine. Enter it in the table with the auxiliary Sine (as horizontal argument) and enter the declination as (vertical) argument: You will find the hour-angle. However, if the Sun is in the northern signs and the value of the first function is less than the value of the second, subtract the result from 180°: The remainder will be the hour-angle. If the two functions are equal, the hour-angle will be 90°.

On Islamic values for these parameters, consult Wiedemann [1], ii, 769 and 777, and King [2].

See, in particular, Smart [1], 51–52.

al-Khalīlī's instructions for these operations (MS B.N. ar. 2558, fol. 62v, 11. 15–23) read as follows:

If either one of the meridian or the prime vertical is known, then the azimuth of an observed solar altitude can be found. Enter the altitude and azimuth in the auxiliary Sine tables in the following way. Look for a value in the table equal to the azimuth, which has the altitude as (vertical) argument. The auxiliary Sine is then found (as the horizontal argument) at the head of the appropriate table. Next find the value of the second function for the altitude on the appropriate page for the latitude, and subtract the value from the auxiliary Sine if the Sun is in the southern signs. If it is in the northern signs and the azimuth is northern, add the two values. Otherwise take the difference between them. The result is a value of the first function. Enter the value on the appropriate latitude page and find the corresponding vertical argument. This will be the declination.

Cf. King [1], 196, for Ibn Yūnus's treatment of this problem.

al-Khalīlī describes this method as follows (MS B.N. ar. 2558, fols. 63r, 1. 27–63v, 1. 4): … Enter the latitude of the star as argument for the second function for latitude 49°. Then double the result and add it to the auxiliary Sine…. Enter the result in the table of the first function for latitude 37° and find the corresponding vertical argument. It will be the declination of the star…. He does not explain the reason for using the tables for latitudes 49° and 37°.

The normed right ascension is defined by: α'(λ) = α(λ) + 90°. This function is tabulated in the Handy tables, and similar tables are common in Islamic zījes. It is useful for finding the ascendant, H, if the longitude of upper midheaven, M, is known, since:, where αϕ denotes oblique ascension for latitude ϕ.

Aumer [1]: Aumer J. , Die arabischen Handschriften der kaiserlichen Hof- und Staatsbibliothek in München (Munich, 1866).

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Irani [2]: Irani R. A. K. , “Arabic Numeral Forms”, Centaurus, iv (1955), 1–12.

Irani [3]: Irani R. A. K. , “A Sexagesimal Multiplication Table in the Arabic Alphabetical System”, Scripta mathematica, xviii (1952), 92–3.

Jensen [1]: Jensen C. , “Abū Nasr Mansūr's Approach to Spherical Astronomy as Developed in His Treatise The Table of Minutes”, Centaurus, xvi (1971), 1–19.

Kennedy [1]: Kennedy E. S. , “A Survey of Islamic Astronomical Tables”, Transactions of the American Philosophical Society, N.S., xlvi, 2 (1956), 123–77.

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King [2]: King D. A. , “Ibn Yūnus' Very Useful Tables for Reckoning Time by the Sun”, to appear in Archive for the history of exact sciences.

King [3]: King D. A. , “al-Khalīlī's Qibla Table”, to appear in Journal of Near Eastern studies.

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Renaud [1]: Renaud H. P. J. , Les manuscrits arabes de l'Escorial. Tome ii, Fasc. 3: Sciences exactes et sciences occultes (Paris, 1941).

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de Slane [1]: de Slane MacG. , Catalogue des manuscrits arabes (Paris, 1883–95).

Smart [1]: Smart W. M. , Textbook on spherical astronomy (London, 1972 edn).

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Wiedemann [1]: Wiedemann E. , Aufsätze zur arabischen Wissenschaftsgeschichte (2 vols, Hildesheim, 1970).