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The Computation of Planetary Longitudes in the Zīj of Ibn al-Bannā'

  • Julio Samsó (a1) and Eduardo Millás (a1)

Ibn al-Bannā' of Marrakesh (1256–1321) is the author of one of the four extant “editions” of the unfinished zīj of Ibn Isḥāq (fl. Tunis and Marrakesh ca. 1193–1222): it contains a selection of his tables accompanied by a collection of canons, easy to understand, which makes the zīj accessible for the computation of planetary longitudes. The present paper studies some modifications of the structure of the tables the purpose of which is to make calculations easier. The tables of the planetary and lunar equations of the centre are “displaced." The tables of the equation of the anomaly of Mars, Venus and Mercury, are standard, while, in the cases of Jupiter and Saturn, the equation of the anomaly is calculated in the same way as that for the Moon. Ibn al-Bannā' appears as a clever adapter, who displays a clear ingenuity allowing him to introduce formal modifications which give his work an appearance of novelty which does not correspond to reality.

Ibn al-Bannā' de Marrakech (1256–1321) est l'auteur de l'une des quatre “éditions” existantes du Zīj inachevé d'lbn Isḥāq (Tunis et Marrakech autour de 1193–1222). Cette édition contient une sélection des tables du Zīj d'lbn Isḥāq, accompagnée d'une collection de canons faciles à comprendre, ce qui en fait un ouvrage accessible, permettant d'effectuer les calculs des longitudes planétaires. Le présent article étudie quelques-unes des modifications subies par la structure des tables dont le but est de faciliter les calculs. Les tables des équations planétaires et lunaires du centre sont “déplacées.” Les tables de l'équation concernant l'anomalie de Mars, Vénus et Mercure sont classiques alors que, dans le cas de Jupiter et de Saturne, l'équation de l'anomalie est calculée de la même manière que pour la lune. Ibn al-Bannā' apparaît comme un adaptateur intelligent faisant montre d'une indéniable ingéniosité qui lui permet d'introduire des modifications formelles qui donnent à son ouvrage un semblant de nouveauté, ce qui ne correspond pas à la réalité.

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1 The canons were edited by Vernet, J., Contribución al estudio de la labor astronómica de Ibn al-Bannā' (Tetu´n, 1952). On Ibn al-Bannā' see also Renaud, H.P.J., “Ibn al-Bannā' de Marrakech sûfî et mathématicien (XIIIe - XIVe s. J.C.)Hespéris, 25 (1938): 1342;Renaud, , “Sur les dates de la vie du mathématicien arabe macrocian Ibn al-Bannā’, Isis, 27 (1937): 216–18;al-Bannā', Ibn, Talkhīṣ a'māl al-ḥisāb, Texte établi, annoté et traduit par Souissi, M. (Tunis, 1969);Al-maqālāt fi ‘ilm al-nisāb li-Ibn al-Bannā’ al-Marrākushl, ed. by Sa'īdān, A.S. (Ammān, 1984;)Puig, R., “El Taqbīl ‘alā risālat al-safiḥa al-zarqāliyya de Ibn al-Bannā' de Marrākush,” Al-Qantara, 8 (1987): 4564;Calvo, E., “La Risālat al-safiḥa al-muštaraka ‘alà al-šakkāziyya de Ibn al-Bannā’ de Marrākuš,”, Al-Qantara, 10 (1989): 2150. This latter text has also been edited by al-Khaṭṭābī, Muḥammad al-'Arabī, ‘Ilm al-mawāqīt. Usūluhu wa manāhijuhu (Munammadiyya, 1407/1986), pp. 136–74.

2 See, for example, King, David A., “An overview of the sources for the history of astronomy in the Medieval Maghrib,” Deuxième Colloque Maghrébin sur l'Histoire des mathématiques arabes (Tunis, 1988), pp. 125–57. See now a detailes survey of the contents of the manuscript in Metres, A., “Maghribī astronomy in the 13th century: a description of manuscript Hyderabad Andra Pradesh State Library 298,” in Casulleras, J. and Samsó, J. (eds.), From Baghdad to Barcelona.Studies in the Islamic Exact Sciences in Honour of Prof. Juan Vernet (Barcelona, 1996), vol. I, pp.383443.

3 Cf. Samsó, J., Las Ciencias de los Antiguos en al-Andalus (Madrid, 1992), pp. 324–6.

4 A partial edition of this zīj, including a commentary and recomputation of the tables, was presented by al-Ranmān, Munammad 'Abd (Institute for the History of Arabic Science, Aleppo) as a doctoral dissertation (Hisāb atwāl al-kawākib fi l-Zīj al-Shāmil fi Tahdhīb al-Kāmil li-Ibn al-Raqqām) in the University of Barcelona (September 1996). See also Kennedy, E.S., “The astronomical tables of Ibn al-Raqqām, a scientist of Granada,’ Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften, 11 (1997): 3572.

5 Vernet, J., “La supervivencia de la astronomía de Ibn al-Bannā,” Al-Qantara, 1 (1980): 445–51.

6 Comes, Mercè, “Deux échos andalous à Ibn al-Bannā' de Marrākush,” in Le patrimoine andalou dans la culture arabe et espagnole (Tunis, 1991), pp. 8194. This paper has been discussed recently by Goldstein, Bernard R. and Chab´s, José, “Ibn al-Kammād's star list,” Centaurus, 38 (1996): 317–34. See, however, Samsó, J., “Andalusian astronomy in 14th century Fez: the al-Zīl al-Muwāfiq of Ibn 'Azzūz al- Qusantīnī,” Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften,”, 11, (1997): 73110.

7 Samsó, J. & Millás, E., “Ibn Isnāq and Ibn al-Zarqālluh's solar theory,” in Samsó, J., Islamic Astronomy and Medievel Spain, Variorum (Aldershot, 1994), no. X (35 p.).

8 These parameters were calculated using a sexagesimal calculator and a computer programme which reconstructs the mean motion tables from a given parameter (TAPJ). Both were prepared by Honorino Mielgo.

9 Toomer, G.J., “A survey of the Toledan tables,” Osiris, 15 (1968): 5174, p. 44.

10 Chabás, José & Goldstein, Bernard R., “Andalusian astronomy: al-Zîj al-Muqtabis of Ibn al-Kammâd,” Archive for History of Exact Sciences, 48 (1994): 141.

11 Kennedy, E.S. & King, D.A., “Indian astronomy in fourteenth century Fez: the Versified Zīj of al-Qusanṭīnī,” Journal for the History of Arabic Science, 6 (1982): 345 (see p. 10). Reprinted in King, D.A., Islamic Mathematical Astronomy, Variorum Reprints (London, 1986: 2nd revised edn, Aldershot, 1993).

12 See Mestres, “Maghribī astronomy in the 13th century,” pp. 412–18. These parameters have been calculated using Benno van Dalen's programme for the analysis of mean motion tables.

13 We are using the unpublished doctral thesis by al-Ranmān., Mu' 'Abd

14 Samsó & Millás, “Ibn Ishāq, Ibn al-Bannā' and Ibn al-Zarqālluh's solar theory”, pp. 7–9.

15 Ibn al-Raqqām's al-Zīj al-Shāmil gives two radix positions for the meridians of Arīn and Bijāya which are mutually coherent. Only the positions for Bijāya have been considered here.

16 MS M (Madrid, Museo Naval, without number), table 30; MS A (Alger 1454), fol. 43r; MS E (Escorial 909), fol. 26v.

17 Cf. Vernet, Contribución p. 91, who follows MS E fol. 27r. MS A fol. 45v and MS M, table 32, give 49.

18 4s 2; 12,49° in MS M, table 34. See Vernet, Contribución, p. 92, who follows MS E fol. 28r. The apogee of Mars seems to be missing in MS A fols. 46r-47v.

19 Ibn al-Bannā' states, in chapter 9 of his canons, that the apogee of Venus is the same as that of the Sun: see Vernet, Contribución, p. 34 (Ar. text) and 92 (Sp. translation). As for the solar apogee see Samsó-Millás, “Ibn al-Bannā',”, pp. 7–9.

20 See Vernet, Contribución, p. 34 (Ar.Text) and p. 92 (Sp. tr.): the apoges of Mercury is placed at a fixed distance of 4s 1;40° from the solar apogee.

21 See Mestres, “Maghribī astronomy in the 13th century,” p. 412. The only exception is the apogee of Mercury, which seems corrupt in the Hyderabad MS (68 14;43).

22 In his al-Zīj al-qawīm fi funūn al-ta'dīl wa-al-taqwīm (MS Rabat, General Library 260; fragments of this same zīj can also be found in our MS M). The same values appear also in his al-Zīj al-shāmil fi al-Kāmil (MS Istanbul, Kandilli 249), studied by Muḥammad 'Abd al-Raḥmān. Mercury's apogee in the same as that of Ibn al-Bann.ā'.

23 With the exception of Venus and the Sun, we find the same set of planetary apogees in the zīj of Abū al-Hasan 'Alī al-Qusunṭīnī: see Kennedy & King, “Indian astronomy 10.

24 See Toomer, G.J., Ptolemy' Almagest (New York, Berlin, Heidelberg, Tokyo, 1984).

25 See Stahlman, W.D., The Astronomical Tables of Codex Vaticanus Graecus Astronomical Tables of Codex Vaticanus Graecus 1291, available through University Microfilms, Ann Arbor, Michigan, microfilm no. 62–5761.

26 Ed. and Latin translation by Nallino, C.A., Al-Battānī sive Albatenii Opus Astronomicum, 3 vols. (Mediolani Insubrum, 1899, 1903 et 1907).

27 Neugebauer, O., The Astronomical Tables of al-Khwārizmī.Translation with Commentaries of the Latin Version edited by Suter, H.supplemented by Corpus Christi College MS 283 (Copenhagen, 1962), pp. 41, 99.

28 Toomer, “Toledan tables”, p.45.

29 Chabás & Goldstein, “Muqtabis”, p. 33.

30 Cf. Poulle, E., Les Tables alphonsines avec les canons de Jean de Saxe (Paris, 1984).

31 Goldstein, B.R., “Remarks on Ptolemy's model in Islamic astronomy,” Prismata. Festschrift für Willy Hartner (Wiesbaden, 1977), pp. 165–81.

32 Samsó & Millás, “Ibn Isḥāq, Ibn al-Bannā'…,” pp. 7–9.

33 This hypothesis contradicts the one formulated in Samsó, J. and Mielgo, Honorino, “Ibn al-Zarqālluh on Mercury,” Journal for the History of Astronomy, 25 (1994): 289–96.

34 See Samsó, J., “Trepidation in al-Andalus in the 11th century,” in Samsó, J., Islamic Astronomy and Medievel Spain, Variorum (Aldershot, 1994), no. VIII. p. 25.

35 E fol. 15r; A fol. 31v; M fol. 12v.

36 Comes, Mercè, “Deux échos andalous à Ibn al-Bannā'.”

37 See MS Madrid, Biblioteca Nacional 10023, fol. 14v. We have used a provisional edition of this text prepared by Angel Mestres and other undergraduate students as a part of a course on Latin Palegraphy by Dr. Mercè Viladrich. The italicization is ours.

38 Al-Zīj al-Kāmil fi al-Ta'ālīm, MS Bodleian Marsh 618, fol. 34v. This text does not state clearly which are the planets affected by the motion of the apogee, but it defines the meaning of awj mu'addal (“corrected apogee”): “Obtain the mean motion of the apogee for any moment, using any era (ta'rīkh) you wish. Add the result to the radix position for the begining of the era. You have obtained, then, the position of the corrected apogee (awj mu'addal) on the ecliptic, that is to say its distance from the point of the Head of Aries for that moment.” The question becomes absolutely clear if we read the chapters in which Ibn al-Hā'im describes the procedure to calculate the true longitude of the planets: there, both for the superior (fol. 37r-v) and the inferior planets (fol. 38r), he says that the awj mu'addal has to be substracted from the mean longitude of the planet.

39 Mestres, “Maghribī astronomy in the 13th century,” pp. 394–5 and 412.

40 We are using here al-Raḥmān's, Muḥammad 'Abd unpublished doctoral dissertation. The manuscripts used are Istanbul, Kandilli 249 (Shāmil, fols, 13r and 14v) and Rabat General Library 260 (Qawīm, pp. 1516).

41 See Samsó, J., “Andalusian astronomy in 14th century Fez,” pp. 82–3.

42 Vallicrosa, J.M. Millás, Las tablas astronómicas del rey Don Pedro el Ceremonioso. Edición crítica de los textos hebraico, catalán y latino con estudio y notas (Barcelona, 1962), pp. 128, 147 and 190;on these tables see now Chabás, J., “Astronomía andalusí en Cateluña: les Tables de Barcelona”, in From Baghdad to Barcelona, pp. 477–525.

43 Samsó & Millás, “Ibn Isḥāq, Ibn al-Bannā'…,” pp. 18–26.

44 Cf. Kennedy, E.S. and Salam, H., “Solar and lunar tables in early Islamic astronomy”, in Kennedy, E.S., Colleagues and Former Students, Studies in the Islamic Exact Sciences (Beirut, 1983). pp. 108–13;Kennedy, E.S., “The astronomical tables of Ibn al-A'lam,” Journal for the History of Arabic Science, 1 (1977): 1323.Kennedy, E.S., “AL-Bīrūnī's Masudic Canon,” in Studies, pp. 573–95;King, D.A., “A double argument table for the lunar equation attributed to Ibn Yūnus,” Centaurus, 18 (1974): 126–46 (reprint in King, , Isalmic Mathematical Astromomy, Variorum (London, 1986; 2nd edn, Aldershot, 1993);Jensen, C., “The lunar theories of al-Baghdādī,” Archive for the History of the Exact Sciences, 8 (19711972): 321–8;Saliba, G.A., “The double-argument lunar tables of Cyriacus,” Journal for the History of Astronomy, 7 (1976): 41–6;Saliba, , “The planetary tables of Cyriacus,” Journal for the History of Arabic Science, 2 (1978): 5365;Saliba, , “Computational techniques in a set of late Medieval astronomical tables,” Journal for the History of Arabic Science, 1 (1977): 2432.

45 Cf. Samsó-Millás, “Ibn Isḥāq, Ibn al-Bannā'…,” p. 20.

46 This table of the solar equation (no. 30–32 of Ibn Isḥāq's zīj) bears the following title: Jadwal ākhar bi-ta'dīl al-shams al-murakkab li-Ibn Isḥāq bi-nass al-Zarqālluh lā bi-jadwalihi (“Second table of the solar equation, calculated by Ibn Isḥāq following [Ibn] al-Zarqālluh's instructions not his tables”). Table 27 of this same zīj attains a maximum value of 1;49, 7° and it has been calculated “according to the opinion of Ibn Isḥāq” ('alā ra'y Ibn Isḥāq). See Mestres, “Maghribī astronomy,” p. 414.

47 Chabás & Goldstein, “Muqtabis,” pp. 4–10.

48 Pedersen, Olaf, A Survey of the Almagest (Odense, 1974), pp. 291–2, 309, 325–8;Neugebauer, Otto, A History of Ancient Mathematical Astronomy (Berlin, Heidelberg, New York, 1975), vol. I, pp. 183–90, vol. II, pp. 1002–4.

49 For all this terminology see Vernet, Contribución, p. 34, for example (Mars).

50 Pedersen, Survey, pp. 195–9; Neugebauer, H.A.M.A., I, 93–8 and II, 988–9. The symbols c4, c5 and c6 correspond to the columns in the Almagest.

51 Cf. Vernet, Contribución, pp. 31–2 (Arabic text) and 87–9 (Spanish translation). On the Zarqāllian correction cf. Samsó, Ciencias de los Antiguos, pp. 218–19.

52 Cf. Haddad, F. I., Kennedy, E.S. and Pingree, D., The Book of the Reasons behind Astronomical Tables (Kitāb fi 'ilal al-zījāt) by 'Alī ibn Sulaymān al-Hāshimī (New York, 1981), pp. 167, 210, 220, 300, 308, 313.

53 Pingree, D., The Thousands of Abū Ma’shar (London, 1968), pp. 47–8; Kennedy et al., Studies in the Islamic Exact Sciences, pp. 329 and 349.

54 Samsó, J., Ciencias de los Antiguos, p. 157.

55 Cf. also Goldstein, B.R., “Lunar velocity in the Middle Ages: A comparative study,” From Baghdad to Barcelona, I, 184–5.

56 See, for example, for Saturn, Vernet, Contribución, pp. 33–5 (Arabic text) and 91–3 (Spanish translation).

57 It is the standard value in the tradition of the Handy Tables. If we use Ibn al-Bannā's tables, we would have: (6;37° + 5;53°)/ 2 = 6;15° We do not use this value because the results obtained are clearly worse.

58 As in the case of Saturn, it is the standard value of the tradition of the Handy Tables. With the zīj of Ibn al-Bannā', we would have: (10;34° + 11;35°) / 2 = 11;4,30°. Here also the results obtained would be worse with this value.

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