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² Ibid., p. 45.

³ See al-Haytham, Ibn, al-Shukūk 'alā Baṭlamyūs, ed. Sabra, A.I. and Shehabi, N. (Cairo, 1971).Google Scholar

⁴ See, for example, on Bitṭūjī's work, Kitāb fi al-hay'a, trans. Goldstein, Bernard R., in Al-Biṭrūjī: On the Principles of Astronomy, 2 vols. (New Haven, 1971); also, for a more general survey of these activities see,Google ScholarSaliba, George, “Islamic planetary theories after the eleventh century A.D.,” to appear, pp. 42–5, where an overview of the contributions of such astronomers as Jäbir ibn Aflah (c. 1150 A.D.), al-Biṭrūjī (c. 1200 A.D.), and Ibn Rushd (d. 1198 A.D.) is given.Google Scholar

⁵ On the work of Urḍī see The Astronomical Work of Mu'ayyad al-Dīn al-'Urḍī: A Thirteenth-Century Reform of Ptolemaic Astronomy. Kitāb al-Hay'ah, Edition and Introduction by Saliba, George (Beirut, 1990), pp. 30–43.Google Scholar

⁶ On the work of Ṭūsī see de Vaux, Carra, “Lea sphères célestes selon Nasīr-EddīnAttūsī,” in Tannery, Paul, Recherches sur l'histoire de l'astronomie ancienne (Paris, 1893), Appendix VI, pp. 337–61;Google ScholarHartner, Willy, “Na⋅ir al-Dīn al-Ṭūsī's lunar theory,” Physis, 11 (1969): 287–304;Google Scholar and Ragep, Faiz Jamil, Cosmography in the Tadhkira of Nasir al-Din al-Tusi, unpublished dissertation (Ann Arbor, 1982).Google Scholar

⁷ On the work of Shīrāzī see Kennedy, Studies in the Islamic Exact Sciences, pp. 84–97.Google Scholar

⁸ Hulagu of the Ilkhanid dynasty commissioned Naṣīr al-Dīn al-Ṭūsī to establish and direct an observatory at Maragha, to which he invited many Muslim and non- Muslim astronomers. Among these were Mu'ayyad al-Dīn al- 'Urḍī who came from Syria, and Quṭb al-Dīn al-Shīrāzī. These three scholars produced a number of works which dealt with the problems of Ptolemaic astronomy, and proposed a number of significant solutions to these problems. Moreover, a major part of the discussion at this stage was devoted to questioning the adequacy of the newly proposed models. On the history of this observatory seeGoogle ScholarSayili, Aydin, The Observatory in Islam (Ankara, 1960), pp. 370–3.Google Scholar

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¹⁰ On the life and work of Ṣadr see Baṭūṭa, Ibn, Voyages d'Ibn Batoutah, trans. of Defrémery, C. and Sanguinetti, B.R., 3 vols. (Paris, 1977), vol. III, p. 28;Google Scholar‘al-Luknawī, Abd al-Hayy, al-Fawā'id al-bahiy-ya fi tarājim al-ḥanafiyya (India, 1967), PP. 91–5;Google ScholarTāshkoprūzāde, Ahmad b. Muṣṭfā, Miftāh al-sa 'ādah wa mi⋅bā al-siyādah, 3 vols. (Cairo, 1968), vol. I, pp. 60–1, vol. II, pp. 182, 191–2;Google ScholarKutlūbughā, Zein-ad-dīn K¯sim ibn, Tāj al-tarājim fi ⃛abāqāt al-hanafiyya, ed. Flūgel, Gustav (Leipzig, 1862), pp. 29, 30, 115;Google Scholar‘Abd al-ahdib 'Imrān al-Dujaylī,Google ScholarA'lām al-'Arab fī al-'lūm wa al-funūn, 2 vols. (Bagdad, 1966), vol. II, pp. 162–3;Google Scholaral-Zarkalī, Khayr al-Dīn, al-A'lām (Beirut, 1969), p. 354;Google Scholar'Kaḥḥāla, Umar Riḍā, Mu‘Jam al-mu’allifin, 2 vols. (Damascus, 1958), p. 246;Google ScholarSarkīs, Yūsuf Ilyās, Mu'jam al-maṭbū 'āt al-'arabiyya wa al-mu'arraba (Cairo, 1928), pp. 1199–200;Google ScholarZaydān, Jurjī, Tārīkh ādāb al-lugha al-'arabiyya, 3 vols. (Cairo, 1913), vol. III, p. 239;Google ScholarKhalīfa, Kātib Jelebī Ḥājī, Kashf al-ẓunūn 'an asāmī al-kutub wa al-funūn, ed. Flūgel, Gustav, 16 vols. (London, 1852), vol. II, pp. 315, 417, 601, vol. III, p. 37, vol. IV, pp. 439–40, vol. VI, pp. 373–6, 443, 458–66; andGoogle ScholarAs'ad Ṭiās, Muḥammad, al-Kashshāf ‘an khazā’in kutub al-awqāf (Bagdad, 1953), pp. 61, 69–70, 99, 100.Google Scholar

¹¹ See, for example, Ḥājī Khalīfa, Kashf al-żunūn, I, 417.Google Scholar

¹² For references to this work see Brockelmann, C., Geschichte der arabischen Litteratur (Leiden, 1937), vol. II, pp. 277–9, and Suppl. II, pp. 300–1; al-Dujaylī, A'lām al- 'Arab, II, 162;Google ScholarHājī Khalīfa, Kashf al-ẓnūn, II, 315; Kaḥḥā1a, Mu ‘jam almu’allifin, VI, 246;, al-Luknawī al-Fawā'id al-bahiyya, p. 94;Google ScholarSuter, Heinrich, Die Mathematiker und Astronomen der Araber und ihre Werke (Leipzig, 1900), # 404; Ṭāshkoprūzāde, Miftāḥ al-sa ādah, II, 182; al-Zarkalī, al-A lām, IV, 354; and Zaydān, Tārīkh ādāb al-lugha, II, 239.Google Scholar

¹³ The present author wishes to express his gratitude to those librarians who cooperated with him by supplying microfilm copies of the MSS in their possession.Google Scholar

¹⁴ For a catelogue reference to this MS see Catalogus codicum manuscriptorum orientlium qui in Museo Britannico asservantur, Part 2. Cod. Arab. Amplect. (London, 1846–1871), # 400.Google Scholar

¹⁵ For a catalogue reference to this MS see Ahlwardt, W., Die HandschriftenVerzeichnisse der königlichen Bibliothek zu Berlin: Verzeichniss der arabischen Handschriften (Berlin, 1982), p. 432.Google Scholar

¹⁶ For a catalogue reference to this MS see ibid., p. 165.

¹⁷ For a catalogue reference to this MS see Flügel, Gustav, Die Arabischen, Persischen und Türkischen Handschriften der Kaiserlich-Königlichen Hofbibliothek zu Wien (Wien, 1865), vol. XI, p. 13.Google Scholar

¹⁸ For a catalogue reference to this MS see Loth, Otto, A Catalogue of the Arabic Manuscripts in the Library of the India Office (London, 1877), p. 145.Google Scholar

¹⁹ For a catalogue reference to this MS see Karatay, , Topkapi Sarayi Müzesi Kütüphanesi Arapca Yazmalar Katalogu (Istanbul, 1966) vol. III, # 6760 E.H. 1669.Google Scholar

²⁰ For the corresponding sections in the Almagest see Ptolemy's Almagest, English trans. Toomer, G.J. (New York, 1984), Books IV–V, pp. 173–216.Google Scholar

²¹ See Pedersen, Olaf, A Survey of the Almagest (Denmark, 1974), P. 161.Google Scholar

²² See Ragep, Cosmography in the Tadhkira, VI, pp. 91, 95; and Quṭb al-Dīn al-Shīrāzī, al-Tuhfa al-shāhiyya, Mawṣil, MS Jāmi' al-Bāshā 287, fols. 78^r, 81^r.Google Scholar

²³ For the corresponding sections see Ragep, Cosmography in the Tadhkira, VI, pp. 93–4, and Shīrāzī, al-Tuḥfa, fols. –80^r, 81^r.Google Scholar

²⁴ See Ragep, Cosmography in the Tadhkira, VI, pp. 93, 96, and Shīrāzī, al-Tuhfa, fols. 80^r, 81^v-82^r.Google Scholar

²⁵ See Ragep, Cosmography in the Tadhkira, VI, pp. 91–3, and Shīrāzīrāzīrāzī, al-Tuḥfa, fols. 80^v, 84^v.Google Scholar

²⁶ See Ragep, Cosmography in the Tadhkira, VI, p. 91, and Shīrāz¯i, al-Tuhfa, fol. 78^V.Google Scholar

²⁷ For these numbers see Ragep, Cosmography in the Tadhkira, VI, p. 99, and Shīrāz¯i, al-Tuhfa, fol. 84^V. For the equivalent sections see Ragep, Cosmography in the Tadhkira, VI, pp. 91–2, and Shīrāz¯i, al-Tuhfa, fol. 84^V.Google Scholar

²⁸ See Ragep, Cosmography in the Tadhkira, VI, p. 102.Google Scholar

²⁹ For the corresponding sections see Ragep, Cosmography in the Tadhkira, VI, pp. 101–2, and Shīrāz¯i, al-Tuhfa, fol. 87^r.Google Scholar

³⁰ See below, paragraph [13].Google Scholar

³¹ For the corresponding sections see Ragep, Cosmography in the Tadhkira, VI, p. 102, and Shīrāzī, al-Tuḥfa, fol, 87^r–87^v.Google Scholar

³² For the corresponding sections see Ragep, Cosmography in the Tadhkira, VI, p. 102, and Shīrāzī, al-Tuḥfa, fols. 87^r-87^v.Google Scholar

³³ See Neugebauer, Otto, The Exact Sciences in Antiquity (New York, 1969), pp. 193, 198.Google Scholar

³⁴ Incidentally, the same discrepancy is noted between the interpolation function used by Copernicus and that used by Ptyolemy, resultiong in similar graphs, See graph for c₄ in Noel Swerdlow and Negebauer, Otto, Mathematival Astronomy in Copernicus' De Revolutionaibus, 2 vols. (New York, 1984), Part 2, p. 600, figure 15, and compare with graph of Ṣadr.Google Scholar

³⁵ For the corresponding sections see Ragep, Cosmography in the Tadhkira, VI, p. 104. and Shīrāzī, al-Tuḥfa, fol. 88^r.Google Scholar

³⁶ For the sections discussing the first variation see Shīrāzī, al-Tuḥfa, fols. 88^r-88^v, and Ragep, Cosmography in the Tadhkira VI, pp. 103–5.Google Scholar

³⁷ For an illustration of the method given by Ptolemy to prove this point see Neugebauer, Otto, A History of Ancient Mathematical Astronomy (New York, 1975), Part 1, p. 57. also see Ptolemy's Almagest, Book III, p. 156, and Pedersen, A Survey of the Almagest, pp. 141–3.CrossRefGoogle Scholar

³⁸ See Ragep, Cosmography in the Tadhkira, V, p. 61; VI, p. 103, and Shīrāzī, al-Tuḥfa, fol. 88^v.Google Scholar

³⁹ SeeShīrāzī, al-Tuḥfa, fol. 90^v.Google Scholar

⁴⁰ See Ptolemy's Almagest, Book V, p. 238.Google Scholar

⁴¹ For the corresponding sections see Ragep, Cosmography in the Tadhkira, VI, pp. 104–5, and Shīrāzī, al-Tuḥfa, fol. 88^v.Google Scholar

⁴² See MS A, fol. 21^v.Google Scholar

⁴³ For the corresponding sections see Shīrāzī, al-Tuḥfa, fols. 90^v−91^v, and Ragep, Cosmography in the Tadhkira, VI, pp. 176–7.Google Scholar

⁴⁴ On objections to Ptoemy's models, and on the attempts to correct them see Saliba, George, “Arabic astronomy and Copernicus,” Zeitschrift für Geschichte der Arabisch-Isamischen Wissenschaften 1 (1981): 73–87, pp. 75–84, and Saliba, “Islamic planetary theories.”Google Scholar

⁴⁵ For the corresponding sections see Ragep, Cosmography in the Tadhkira, VI, p. 107, and Shīrāzī, al-Tuḥfa, fol; 95^r.Google Scholar

⁴⁶ See Ragep, Cosmography in the Tadhkira, VI, p. 170.Google Scholar

⁴⁷ See Ibid., VI, p.172.

⁴⁸ See MS A, fol. 23^r.Google Scholar

⁴⁹ For an explanation and proof of Ṭūs's point see the comentary in Ragep, Cosmography in the Tadhkira, IV, pp. 274–8.Google Scholar

⁵⁰ See, for example, Pedersen, A Survey of the Almageast, V, p. 223.Google Scholar

⁵¹ See Saliba, Geoge, “A medieval Arabic reform of the Ptolemaic lunar model”, Journal for the Histry of Astronomy 20 (1989): 157–64.CrossRefGoogle Scholar

⁵² See Ragep, Cosmography in the Tadhkira, VI, pp. 156–61.Google Scholar

⁵³ For futher discussion of the Ṭūsī couple see, for example, Neugebauer, Ancient Mathematical Astronomy, pp. 1035, 1456.Google Scholar

⁵⁴ For comparison see Ragep, Cosmography in the Tadhkira, VI, pp. 164–8.Google Scholar

⁵⁵ For the corresponding section see Ragep, Cosmography in the Tadhkira, VI, pp. 185–90; also for a discussion of this method see same source, VI, pp. 283–98.Google Scholar

⁵⁶ For the earliest use of this method of homocentric spheres by Eudoxus see Neugebauer, Exact Sciences in Antiquity, pp. 153–6.Google Scholar

⁵⁷ See Ragep, Cosmography in the Tadhkira, VI, pp. 175–83.Google Scholar

⁵⁸ For a full discussion of the spherical couple see Saliba, and Kenndy, E.S. “The spherical case of the Ṭūsī couple”, Arabic Sciences and Philosophy, 1 (1991): 285–91.CrossRefGoogle Scholar

⁵⁹ On the contributions of 'Urḍī and his influence on Shīrāzī see Saliba, “Astronomical work of al-'Urḍ”, and Saliba, , “Falakī min Dimashq yaruddu ‘alā hay’at Baṭlamyūs”, Journal for the Histry of Arabic Science, 4 (1980): 3–17;on 'Urḍī's lemma seeGoogle ScholarSaliba, “Arabic astronomy and Copernicus”, For a discussion of the lunar models of both Shīrāzī and 'Urḍī see Saliba, “Islamic planetary theories,” pp. 49–60. also for 'Urḍī's lunar model see Salina, “A medieval Arabic reform”. For Shīrāzī's discussion of 'Urḍī, and the presentation of his own model see Tuḥfa, fol. 97^v–98^v, and fol. 98^v–100^v.Google Scholar

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⁶¹ On the work of Ibn al-Shāṭir see Abbud, Fund, “The planetary theory of Ibn al-Shāṭir: reduction of the geometric models to numerical tables”, Isis, 53 (1962): 492–9; on Copernicus see Swerdlow-Neugebauer, Mathematical Astronomy, pp. 196–7.CrossRefGoogle Scholar

⁶² For the corresponding sections see Ragep, Cosmography in the Tadhira, VI, p. 105. and Shīrāzī, al-Tuḥfa fols. 89^v–90^r. This same problem is raised by Ibn al-Shāṭir in his al-Zīj al-jadīd; in this zīj the above variation, for which a table is computed, is called naql al-qamar.Google Scholar