ABU’L-WAFĀʾ MOḤAMMAD B. MOḤAMMAD BŪZJĀNĪ, mathematician and astronomer, b. Wednesday, on the new moon of Ramażān, 328/10 June 940, at Būzǰān in the region of Nīšāpūr. He studied arithmetic under his paternal uncle, Abū ʿAmr Moḡāzelī, and his maternal uncle, Abū ʿAbdallāh Moḥammad b. ʿAnbasa, presumably at Būzǰān. He moved to Iraq at the age of nineteen (348/959-60) and soon became a leading mathematician and astronomer at the Buyid court in Baghdad. He is known to have made observations there in A.D. 974 (Bīrūnī, al-Qānūn al-Masʿūdī, Hyderabad, 1954-56, II, pp. 640, 654-55, 677; and Taḥdīd al-amāken, ed. P. Bulgakov, Cairo, 1964, p. 301) and in 976 (Qānūn II, p. 658). Bīrūnī asserts (Taḥdīd, p. 100) he knows that Abu’l-Wafāʾ made most of his observations in Baghdad at the Bāb al-Tebn, during the reign of ʿEzz-al-dawla (356-67/967-78), in A.H. 365-66/Yazdeǰerdī 345-46; these two pairs of years overlap between 22 March 976 and 18 August 977. Abu’l-Wafāʾ was also associated with the observatory established by Šaraf-al-dawla (see Abū Sahl Kūhī). In one of the few recorded medieval attempts to establish the difference in longitude between two places, he observed the lunar eclipse of 24 May 997 at Baghdad, while Bīrūnī observed it in Ḵᵛārazm. According to Ebn al-Qefṭī (Taʾrīḵ al-ḥokamāʾ, ed. J. Lippert, Leipzig, 1903, p. 288), he remained in Baghdad until his death on 3 Raǰab 388/1 July 998. This date seems more likely than Ebn al-Aṯīr’s report (IX, p. 137), repeated by Ebn Ḵallekān (tr. de Slane, III, pp. 320-21), that he died in 387/997-98. Among his pupils, the most illustrious was Bīrūnī’s teacher, Abū Naṣr Manṣūr.

Abu’l-Wafāʾ was certainly one of the greatest mathematicians ever born in Persia. Especially imaginative was his work in plane and solid geometry, in computing sine-tables to an interval of 0;30°, and in simplifying the solution of problems in spherical trigonometry. In astronomy the new parameters established by his observations continued in use for centuries after his death. It is greatly to be regretted that so little of his work on algebra has as yet been discovered. Of his great volume of writing, much has been lost or mutilated.

The following works are listed by Ebn al-Nadīm (Fehrest, p. 283) and repeated, in part, by Ebn al-Qefṭī: 1. Ketāb la-manāzel fī mā yaḥtāǰ elayh al-kottāb wa’l-ʿommāl men ʿelm al-ḥesāb (“Book of the stations on what scribes and secretaries need of the science of calculation”). Its seven sections (“stations”) are devoted respectively to proportion, multiplication and division, operations of measurement, operations of taxation, operations of apportionments, money exchange, and mercantile transactions. It was written some time between ʿAżod-al-dawla’s assumption of the title Tāǰ-al-mella (368/979) and that ruler’s death (8 Šawwāl 372/26 March 983); see A. S. Ehrenkreutz, “The Taṣrīf and Tasʾīr Calculations in Mediaeval Mesopotamian Fiscal Operations,” JESHO 7, 1964, p. 46, n. 1. Ed. in A. Saʿīdān, ʿElm al-ḥesāb al-ʿarabī, ʿAmmān, 1971, pp. 64-368. Specialized studies on parts of the treatise include: P. Luckey, “Beiträge zur Erforschung der islamischen Mathematik. II,” Orientalia N.S. 22, 1953, pp. 166-89 (especially 175-79); M. I. Medovoy, “Ob odnom sluchae primeneniya otritsatel’nykh chisel u Abu-l-Vafy,” Istoriko-matematicheskie issledavaniya 11, 1958, pp. 593-98; idem, “Ob arifmeticheskom traktate Abu-l-Vafy,” Ist.-mat. issled. 13, 1960, pp. 253-324; Ehrenkreutz, “The Kurr System in Medieval Iraq,” JESHO 5, 1962, pp. 309-14; idem, “Al-Būzaǰānī (A.D. 939-997) on the " Maʾsīr’,” JESHO 8, 1965, pp. 90-92; idem, “Taṣrīf.” (See also below, no. 13.) 2. Tafsīr ketāb al-Ḵᵛārazmī fi’l-ǰabr wa’l-moqābala (“Commentary on Ḵᵛārazmī’s Algebra”), lost. 3. Tafsīr ketāb Ḏīūfanṭes fi’l-ǰabr (“Commentary on Diophantus’ Algebra”), lost. 4. Tafsīr ketāb Ebarḵos fi’l-ǰabr (“Commentary on Hipparchus’ Algebra”), lost. Although Hipparchus did not write an Algebra, Ebn al-Nadīm also refers to Abu’l-Wafāʾ’s translation and commentary on the work elsewhere (p. 269); cf., however, Ebn al-Qefṭī (p. 70) and Ḥāǰǰī Ḵalīfa (Kašf al-ẓonūn [Leipzig] V, p. 73), where the original work is ascribed to Aristippus. 5. Al-Modḵal elā al-areṯmāṭīqī (“Introduction to arithmetic”); a manuscript is extant at Rampur (Sezgin, GAS V, p. 403). 6. Ketāb fī mā yanbaḡī yoḥfaẓ qabl ketāb areṯmāṭīqī (“On what one must memorize before studying [Diophantus’] Arithmētikē”), lost. 7. Ketāb al-barāhīn ʿalā al-qażāyā allatī estaʿmala Ḏīūfanṭes fī ketābehe wa ʿalā mā estaʿmalahu howa fi’l-tafsīr (“Proofs of the theorems which Diophantus used in his book and what he [Abu’l-Wafāʾ] used in the commentary”), lost. 8. Ketāb esteḵrāǰ żeḷʿ al-mokaʿʿab be māl māl wa mā yatarakkab menhomā (“Derivation of the side of a cube, of the square of a square, and of what is composed of these two”), lost. 9. Ketāb maʿrefat al-dawāʾer men al-falak (“Knowledge of the circles of the sphere”); this may be no. 20, below. 10. Ketāb al-kāmel (“The perfect book”); this is probably the Ketāb al-maǰesṭī (Almagest), which is not noticed by Ebn al-Nadīm but is mentioned after Ketāb al-kāmel by Ebn al-Qefṭī; see also Abu’l-Faraǰ, Taʾrīḵ moḵtaṣar al-dowal, Beirut, 1958, p. 181. According to Ebn al-Nadīm the Ketāb al-kāmel (like the Almagest) consists of three books dealing, respectively, with preliminaries to the study of the motions of the celestial bodies (Almagest: trigonometry), the motions themselves (Almagest: applied spherical trigonometry), and the things which concern these motions (Almagest: planetary theory). An imperfect copy of the Almagest survives in Paris. A misinterpretation of one section of it led L. A. Sédillot to assert that Abu’l-Wafāʾ had discovered the variation of the moon’s motion (“Découverte de la variation, par Aboul-Wafā, astronome du Xe siècle,” JA sér. 2, 16, 1835, pp. 420-38). This claim led to a long controversy, which was summed up and concluded against Sédillot’s thesis by Carra de Vaux (“L’Almageste d’Abū ʾlWéfa alBūzdjāni,” JA sér. 8, 19, 1892, pp. 408-71). On the spherical trigonometry in the Almagest, a field to which Abu’l-Wafāʾ made important contributions, see M. Delambre, Histoire de l’astronomie du moyen âge, Paris, 1819, pp. 156-66; and P. Luckey, “Zur Entstehung der Kugeldreiecksrechnung,” Deutsche Mathematik 5, 1940, pp. 405-46. On his instruments see L. A. Sédillot, Mémoire sur les instruments astronomiques des Arabes, Paris, 1842, pp. 195-97. Bīrūnī often cites the Almagest; he distinguishes it from the Zīǰ in his On Shadows (the second treatise in his Rasāʾel, Hyderabad, 1948, p. 43). 11. Zīǰ al-wāżeḥ (“Lucid astronomical tables” = E. S. Kennedy, A Survey of Islamic Astronomical Tables, Philadelphia, 1956, no. 73). Now lost, it apparently was extremely influential. Its elements were used in a Zīǰ al-ʿalāʾī (Kennedy, no. 42), whence they entered the Zīǰ al-šāmel (Kennedy, no. 29). The latter may be identical with Aṯīr-al-dīn’s Zīǰ al-molaḵḵaṣ ʿalā al-raṣad al-ʿalāʾī (ca. 638/1240; Kennedy, nos. 40, 56; Sezgin, GAS V, pp. 324-25) and Cyriacus’ Zīǰ al-ǰadīd (ca. 885/1480; Kennedy, no. 81).

In a further passage (p. 266), Ebn al-Nadīm attributes the following to Abu’l-Wafāʾ: 12. A Commentary on Euclid’s Elements (Ebn al-Qefṭī, p. 64; Kašf al-ẓonūn [Leipzig] I, p. 382), lost. Ebn al-Qefṭī adds: 13. Ketāb al-ʿamal bi’l-ǰadwal al-settīnī (“Operating with a sexigesimal table”); this work is probably part of the Ketāb al-manāzel. Ebn Ḵallekān (tr. de Slane, III, p. 320) adds: 14. Ketāb fī esteḵrāǰ al-awtār (“On the derivation of chords”). See F. Woepcke, “Sur une mésure de la circonférence du cercle, due aux astronomes arabes, et fondée sur un calcul d’Aboûl Wafâ,” JA ser. 5, 15, 1860, pp. 281-320. 15. Ketāb fī mā yaḥtāǰ elayh al-ṣāneʿ men aʿmāl al-handasa (“On what artisans need of geometrical constructions”) in thirteen chapters (Kašf al-ẓonūn [Leipzig] V, p. 172); the first seven chapters are preserved in a ms. in Milan and were translated in H. Suter, Beiträge zur Geschichte der Mathematik bei den Griechen und Arabern, Erlangen, 1922, pp. 94-109. The entire work survives in a ms. which once belonged to Uluḡ Beg and is now in Istanbul. It is translated in S. A. Krasnova, “Abu-l-Vafa al-Buzdzhani, Kniga o tom, chto neobkhodimo remeslenniku iz geometricheskikh postroeniĭ,” Fiziko-matematicheskie nauki v stranakh 1, 1966, pp. 42-140. A Persian translation was analyzed in F. Woepcke, “Analyse et extrait d’un recueil de constructions géométriques par Aboûl Wafâ,” JA sér. 5, 5, 1855, pp. 218-56, 310-59. (See also Storey, II/1, pp. 2-3.) 16. Resāla fī tarkīb ʿadad al-wafq fi’l-morabbaʿāt (“Epistle on constructing magic squares”). 17. Fī mesāḥat al-moṯallaṯāt (“On the area of triangles”), a response to a question put by Abū ʿAlī Ḥasan b. al-Ḥāreṯ. This is referred to by Abū Naṣr Manṣūr in his Resāla fī maʿrefat al-qosī al-falakīya (eighth treatise in his Rasāʾel, Hyderabad, 1948, p. 2). 18. Resāla al-areṯmāṭīqī (“Epistle on arithmetic”). 19. Resāla fi’l-nesab wa’l-taʿrīfāt (“Epistle on proportions and determinations”). 20. Resāla fī eqāmat al-borhān ʿalā al-dawāʾer men al-falak (“Epistle on establishing proof on the basis of the circles of a sphere”), addressed to Abū ʿAlī Aḥmad b. ʿAlī b. al-Sakr; published as the fifth treatise in al-Rasāʾel al-motafarreqa fi’l-hayʾa (Hyderabad, 1948) and expounded by N. Nadir, “Abū al-Wafāʾ on the Solar Altitude,” The Mathematics Teacher 53, 1960, pp. 460-63. 21. Barāhīn al-aʿmāl al-handasīya (“Proofs of geometrical constructions”); extant in a Persian translation with a commentary by Moḥammad Bāqer Zayn-al-ʿābedīn. 22. Resāla fī ǰamʿ ażlāʿ al-morabbaʿāt wa’l-mokaʿʿabāt (“Epistle on adding together the sides of squares and cubes”). I owe to my colleague, Professor G. Toomer, the information that Abu’l-Wafāʾ in this epistle answers a question put by Abū Bešr al-Ḥasan b. Sahl, and that he mentions as his patron al-Moʾayyed al-Manṣūr, who is probably ʿAżod-al-dawla’s brother and regent in Isfahan, Moʾayyed al-dawla (366-73/977-83). For some further works in manuscript, see GAS V, pp. 324-25.

Bibliography: See also A. P. Youschkevitch in Dictionary of Scientific Biography I, New York, 1970, pp. 39-43. EI2 I, p. 159.

(D. Pingree)

Originally Published: December 15, 1983

Last Updated: July 21, 2011

This article is available in print.
Vol. I, Fasc. 4, pp. 392-394

Cite this entry:

David Pingree, “ABU’L-WAFĀ BŪZJĀNI,” Encyclopædia Iranica, I/4, pp. 392-394; an updated version is available online at http://www.iranicaonline.org/articles/abul-wafa-buzjani (accessed on 2 February 2014).