The sole surviving fragment of al-Khāzin’s commentary provides a mathematical proof for a statement that Ptolemy uses in Almagest I.3 to argue that the heavens are spherical. This statement is quoted literally from the Isḥāq/Thābit version; cf. Tunis, Dār al-kutub al-waṭaniyya, 7116, f. 3r, lines 5–4 from the bottom of the page. Al-Khāzin proves by means of ten numbered propositions that a circle has a greater area than a regular polygon of the same perimeter, and then by means of nine unnumbered propositions that the sphere has the largest volume of all solid bodies with the same surface-area.

The only other traces of al-Khāzin’s commentary are references in the works of later authors. In his Jadwal al-taqwīm, Abū Naṣr ibn ʿIrāq presents a proof for a calculation in spherical astronomy. In al-Maqālīd, al-Bīrūnī discusses al-Khāzin’s treatment of the sector figure (i.e., the Theorem of Menelaos) as explained in Almagest I.12, as well as its application to the calculation of declinations; in the Taḥdīd, al-Bīrūnī mentions al-Khāzin’s report on the determination of the obliquity of the ecliptic carried out by the Banū Mūsā in Baghdad in the years 868 and 869; and in al-Qānūn al-Masʿūdī, he refers to al-Khāzin’s report of the observations of the lengths of the seasons made by Khālid al-Marwarūdhī, ʿAlī b. ʿIsā and Sanad b. ʿAlī in Baghdad in the year 844.

Both Abū Naṣr and al-Bīrūnī generally speak of al-Khāzin’s commentary as a tafsīr (only in al-Maqālīd, al-Bīrūnī once calls it a sharḥ). Al-Bīrūnī in most cases quotes simultaneously from al-Nayrīzī’s and from al-Khāzin’s commentary on the Almagest; this makes it probable that the two works were quite close in content. In the Taḥdīd, al-Bīrūnī explicitly speaks of the ‘commentary on the first book of the Almagest’ of both authors. Since the subject of the only surviving section of al-Khāzin’s commentary, as well as all topics quoted by Abū Naṣr and al-Birūnī, belong to Book I of the Almagest, it seems very well possible that al-Khāzin only commented on the first book.

Text: [Paris, BnF, arabe 4821]

(47v) 〈…〉 قال بطلميوس وإنّ الأشكال المختلفة التي إحاطتها متساوية ما هو منها أكثر زوايا فهو أعظم قدرًا ولذلك وجب أنّ الدائرة أعظم السطوح والكرة أعظم المجسّمات. يعني أنّ الأشكال المختلفة من ذوات الأضلاع المستقيمة كالمثلّث والمربّع والمخمّس وسائر ذلك إلى ما لا ينتهي ... — (54v) ومن بعد ذلك فإنّا نبيّن أنّ الكرة أعظم الأشكال المجسّمة المتساوية الإحاطات كانت إحاطاتها سطوحًا مستوية كالمكعّب والمنشور والمخروط الذي قاعدته مستقيمة الأضلاع أو كانت سطوحاً مقوّسة كالكرة والأسطوانة ومخروط الأسطوانة. — (67r) 〈…〉 فنصف قطرها وهو مح أقصر من بع ولكن ضرب مح في ثلث سطح الكرة جسم الكرة التي عليها دائرة تث أعظم من المجسّم فإذن الكرة أعظم المجسّمات المتساوية الإحاطة.

Bibl.: al-Bīrūnī, Taḥdīd nihāyāt al-amākin (ed. Bulgakov & AḥmadPavel G. Bulgakov and Imām Ibrāhīm Aḥmad, Kitāb nihāyāt al-amākin li-taṣḥīḥ masāfāt al-masākin li-Abī Rayḥān Muḥammad ibn Aḥmad al-Bīrūnī al-Khwārizmī, Cairo: Maṭbaʿat Lajnat al-Taʾlīf wa-l-Tarjama wa-l-Nashr, 1964, p. 95; English tr. AliJamil Ali, The Determination of the Coordinates of Positions for the Correction of Distances between Cities. A Translation from the Arabic of al-Bīrūnī’s Kitāb Taḥdīd nihāyāt al-amākin litaṣḥīḥ masāfāt al-masākin, Beirut: American University of Beirut, 1967, p. 64); al-Bīrūnī, al-Qānūn al-Masʿūdī (Hyd. ed.Abū Rayḥān Muḥammad al-Bīrūnī, al-Qānunu’l-Mas‘ūdī (Canon Masudicus). An Encyclopaedia of Astronomical Sciences, 3 vols, Hyderabad: The Dāiratu’l-Ma‘ārifi’l-Oṣmānia (Osmania Oriental Publications Bureau), 1954–1956 vol. II, p. 653); al-Bīrūnī, al-Maqālīd ʿilm al-hayʾa (ed./tr. DebarnotMarie-Thérèse Debarnot, Al-Bīrūnī. Kitāb Maqālīd ʿilm al-hayʾa. La trigonométrie sphérique chez les arabes de l’Est à la fin du Xe siècle, Damascus: Institut Français de Damas, 1985, pp. 92/93 and 148–151); Abū Naṣr ibn ʿIrāq, Jadwal al-taqwīm (ed. RasāʾilAbū Naṣr Ibn ʿIrāq, Rasáil Abí Nasr ila’l-Bírúní by Abú Naṣr Mansúr b. Ali b. ‘Iráq (d. circa. 427 A.H.=1036 A.D.) for al-Bírúní, Hyderabad-Deccan: The Dáiratu’l-Ma‘árifi’l-Osmania (Osmania Oriental Publications Bureau), 1948, p. 67). — SuterHeinrich Suter, Die Mathematiker und Astronomen der Araber und ihre Werke, Leipzig: Teubner, 1900, p. 58 (no. 124) and p. 80 (no. 183); NachträgeHeinrich Suter, ‘Nachträge und Berichtigungen zu „Die Mathematiker und Astronomen der Araber und ihre Werke“’, Abhandlungen zur Geschichte der mathematischen Wissenschaften mit Einschluß ihrer Anwendungen 14 (1902), pp. 157–185, pp. 165 and 168; DSBCharles C. Gillispie (ed.), Dictionary of Scientific Biography, 14 vols plus 2 supplementary vols, New York: Scribner’s Sons, 1970–1990 article ‘al-Khāzin’ by Yvonne Dold-Samplonius; EI²P. J. Bearman et al. (eds), The Encyclopaedia of Islam. New Edition, 11 vols plus supplement and index, Leiden: Brill, 1960–2004 article ‘al-Khāzin’ by Julio Samsó; GhorbaniAbolghasem Ghorbani, Zindagīnāma-yi rīyāḍīdānān-i dawra-yi Islāmī. Az sada-yi siwum tā sada-yi yāzdahum-i Hijrī, Tehran: Markaz-i Nashr-i Dānishgāhī, 1986–1987, pp. 88–94 (2^nd ed.Abolghasem Ghorbani, Zindagīnāma-yi rīyāḍīdānān-i dawra-yi Islāmī. Az sada-yi siwum tā sada-yi yāzdahum-i Hijrī, 2nd ed., Tehran: Markaz-i Nashr-i Dānishgāhī, 1996 (1375 H.S.), pp. 63–68, no. 23); GAS, vol. VFuat Sezgin, Geschichte des arabischen Schrifttums. Vol. V: Mathematik bis ca. 430 H., Leiden: Brill, 1974, pp. 298–299 (with addendum in vol. VIIFuat Sezgin, Geschichte des arabischen Schrifttums. Vol. VII: Astrologie – Meteorologie und Verwandtes bis ca. 430 H., Leiden: Brill, 1979, p. 406), vol. VIFuat Sezgin, Geschichte des arabischen Schrifttums. Vol. VI: Astronomie bis ca. 430 H., Leiden: Brill, 1978, pp. 189–190; MAOSICBoris A. Rosenfeld and Ekmeleddin İhsanoğlu, Mathematicians, Astronomers, and other Scholars of Islamic Civilization and their Works (7th–19th c.), Istanbul: Research Centre for Islamic History, Art and Culture (IRCICA), 2003, pp. 81–82 (no. 194, A3); BEAThomas Hockey (ed.), The Biographical Encyclopedia of Astronomers, 2 vols, Dordrecht: Springer, 2007 article ‘Khāzin’ by Emilia Calvo. — Richard P. Lorch, ‘Abū Jaʿfar al-Khāzin on Isoperimetry and the Archimedean Tradition’, Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften 3 (1986), pp. 150–229; Roshdi Rashed, Les mathématiques infinitésimales du IX^e aux XI^e siècle. Vol. I: Fondateurs et commentateurs. Banū Mūsā, Ibn Qurra, Ibn Sinān, al-Khāzin, al-Qūhī, Ibn al-Samḥ, Ibn Hūd, London: Al-Furqān Islamic Heritage Foundation, 1996, ch. 4, pp. 737–833 (English tr. Roshdi Rashed and Nader El-Bizri, Founding Figures and Commentators in Arabic Mathematics. A History of Arabic Sciences and Mathematics, vol. I, New York: Routledge, 2012, ch. 4, pp. 503–578; Roshdi Rashed, Classical Mathematics from al-Khwārizmī to Descartes, Oxon / New York: Routledge, 2015, pp. 527–531.

Ed.: Edition of the Paris fragment with English translation and commentary in Lorch. Edition of the Paris fragment with French translation and mathematical commentary in Rashed.