ḤĀMED BAL-ḴEŻR AL-ḴOJANDI, ABU MAḤMUD, mathematician and astronomer of the late 4th/10th century who, according to Heinrich Suter (p. 74, no. 173), died in around 1000 C.E. His nesba suggests that he originated from Ḵojand (Qojand) in Ferḡāna, on the Jaxartes (Syr Darya) river in present-day Tajikistan. He is the author of the Ketāb fi ʿamal al-āla al-ʿāmma, also called Ketāb al-āla al-šāmela, in which he presents a description of a “universal” or “comprehensive” observational instrument; however, it could only be used at one latitude. This instrument was improved upon by Hebat-Allāh b. al-Ḥo-sayn al-Baḡdādi (al-Qefṭi, p. 339), who died in 534 /1139-40; indeed, one of the two extant manuscripts of the Ketāb al-āla al-šāmela was copied by Hebat-Allāh himself, who added an appendix (Sezgin, VI, p. 221).

Another of Ḵojandi’s works on instruments was his Ketāb samt al-qebla, which is referred to by Abu Rayḥān Biruni in his Maqāla fi tasṭiḥ al-ṣowar wa-tabṭiḵ al-kowar (p. 91 Arabic; p. 51 English), and is probably the source upon which his teacher, Abu Naṣr Manṣur b. ʿAli b. ʿErāq, drew in his Resālat dawāʾer al-somut fi al-asṭorlāb (pp. 3-9; Samsó, 1969, pp. 89-93, Spanish tr.). It is also probably the work referred to by Biruni in his Ketāb estiʿāb al-wojuh al-momkena fi ṣanʿat al-asṭorlāb (Wiedemann, 1970, II, p. 503).

In the last decade of the tenth century Ḵojandi was in Ray enjoying the patronage of the Buyid ruler, Faḵr-al-Dowla (r. 977-97). When Biruni visited him there, after his flight from Ḵᵛārazm, Ḵojandi showed him his book “on nocturnal labors concerning the fixed stars” in which he set out his proof of sine theorem for solving spherical triangles, which he called the qānun al-hayʾa, claiming to have discovered it himself (Biruni, 1985, pp. 100-103). This book was probably his Ketāb fi’l-sāʿāt al-māḍia men al-layl, which is cited in the anonymous Jāmeʿ qawānin ʿelm al-hayʾa (Sezgin, VI, p. 222). Ḵojandi’s proof is provided in extenso by Biruni (1985, pp. 138-41), whose account of the several proofs of this theorem is in turn the source of a passage in Naṣir-al-Din Ṭusi’s Ketāb šakl al-qaṭṭāʿ (pp. 108-19, Arabic; pp. 139-54, French; al-Ḵojandi’s proof is on p. 117, Arabic; pp. 151-52, French). There is a passage in the anonymous Masā-ʾel motafarreqa handasiya, attributing a slightly different proof to Ḵojandi (see Schoy, pp. 260-63). However, Debarnot (Biruni, 1985, p. 138, n. 1) has pointed out that this proof is derived from that of Gušyār ebn Labbān, which is also described by Biruni in his Ketāb maqālid (pp. 142-45). Ḵojandi is also cited by Moḥammad b. al-Ḥasan al-Ḥobubi for his work in spherical trigonometry (Sezgin, VII, pp. 414-15).

At the command of Faḵr-al-Dawla, Ḵojandi built a massive sextant with a diameter of 80 cubits on the hill called Ṭabrūk north of Ray (Biruni, 1964, pp. 101-2; English tr., 1967, pp. 70-71); every degree of the sextant was divided into 360 equal parts, so that each interval was 10 seconds of arc. He described this instrument, which he called al-suds al-faḵri, and some of the observations he made with it, in a work entitled Resāla fi taṣḥiḥ al-mayl wa-ʿarḍ al-balad (ed. Cheikho; tr. Schirmer (into German); see also Wiedemann, 1984,I, pp. 406-8; II, 1180-81; Sayılı, pp. 118-21). Birūni quotes extensively from Ḵojandi’s Resāla in his Ketāb taḥdidal-amāken (pp. 102-7; English tr., 1967, pp. 71-75 ), describing how he determined the noon solar altitudes before and after a summer solstice, on 5 and 6 Jumādā I in 384 A.H./16 and 17 June 994 (see also Biruni, 1954-56, II, p. 643), and before and after the following winter solstice, on 9 and 12 Ḏū’l-qaʿda 384 /14 and 17 December 994. He determined the obliquity of the ecliptic to be 23;32,21° (see Biruni, 1954-56, I, p. 364). In comparing this result with other determinations of it, Ḵojandi concluded that the obliquity was decreasing. Biruni, who did not believe in this diminution, explained Ḵojandi’s result by recounting that he had been directly informed by him that the aperture above the arch of this huge instrument had slipped down by about a span, causing an error in his observation (Biruni, 1964, pp. 107-9; English tr., 1967, 75-77; 1954-56, I, 364). Ḵojandi also calculated from the sun’s altitude at the summer solstice that the latitude of Rayy is 35;34,39° (Biruni, 1964, pp. 86-87; English tr., 1967, p. 56; he rounded this off to 34;35° in his Qānun, II, p. 612).

Toward the end of his Resāla fi taṣḥiḥ Ḵojandi says that he is making observations of the planets in preparation for composing an astronomical table (zij). Evidently he never completed this project, but there is in Tehran a manuscript of an anonymous Persian zij which may be based on parameters originally determined by Ḵojandi, though this matter has yet to be investigated; the epoch of this zij is 600/1231, at least 200 years after Ḵojandi’s death (Kennedy, “Survey,” p. 133, no. 60).

Finally, it should be mentioned that in a letter to Ḵojandi’s contemporary, Abu Moḥammad ʿAbd-Allāh b. ʿAli (Woepcke, pp. 301-24 and 345-56), Abu Jaʿfar Moḥ-ammad b. al-Ḥosayn states that he has shown the falsehood of Ḵojandi’s attempt to demonstrate that the sum of two cubes cannot be a cube (Woepcke, p. 301). This was the first of several attempts by Muslim mathematicians to demonstrate its impossibility (Rashed, pp. 82-83).


Abu Rayḥān al-Biruni, Ketāb maqālid ʿelm al-hayʾa, ed. and tr. M.-T. Debarnot, Damascus, 1985.

Idem, al-Qānun al-masʿudi, 3 vols., Hyderabad, 1954-56.

Idem, Ketāb taḥdid nehāyāt al-amāken, ed. P. Buljākuf, Cairo, 1964; tr. J. Ali, as The Determination of the Coordinates of Positions, Beirut, 1967.

Idem, Maqāla fi tasṭiḥ al-ṣowar wa tabṭiḵ al-kowar, ed. and tr. J. Lennart Berggren as “Al-Bīrūnī On Plane Maps of the Sphere,” Journal for the History of Arabic Science 6, 1982, pp. 47-169.

Abu Naṣr Manṣur b. ʿAli Ebn ʿErāq, Resāla dawāʾer al-somut fi’l-asṭorlāb, text no. 14 in Rasāʾel Abi Naṣr elā al-Biruni, Hyderabad, 1948.

Edward S. Kennedy, A Commentary upon Bīrūnī’s Kitāb taḥdīd al-amākin, Beirut, 1973.

Idem, “A Survey of Islamic Astronomical Tables,” Transactions of the American Philosophical Society, NS 46, 1956, pp. 121-77.

Abu Maḥmud Ḵojandi, Resāla fi taṣḥiḥ al-mayl wa-ʿarḍ al-balad, ed. L. Cheikho, Al Mashriq 11, 1908, pp. 60-69; tr. O. Schirmer as “Studien zur Astronomie der Araber,” Sb. der physikalischmedizinischen Sozietät zu Erlangen 58, 1926, pp. 33-88, esp. pp. 63-79.

F. Luckey, “Zur Entstehung der Kugeldreiecksrechnung,” Deutsche Mathematik 5, 1940, pp. 405-46.

Ebn al-Qefṭi, Taʾriḵ al-ḥokamāʾ, ed. J. Lippert, Leipzig, 1903.

Roshdi Rashed, “Analyse combinatoire, analyse numérique, analyse diophantienne et théorie des nombres,” Histoire des sciences arabes, II, Paris, 1997, pp. 55-91.

Julio Samsó, Estudios sobre Abu Naṣr Manṣur b. ʿAli b. Iṟāq, Barcelona, 1969.

Idem, “al-Ḵudjandī,” in EI2 V, pp. 46-47.

Aydın Sayılı, The Observatory in Islam, Ankara, 1960.

Carl Schoy, “Behandlung einiger geometrischen Fragepunkte durch muslimische Mathematiker,” Isis 8, 1926, pp. 256-63.

Sezgin, GAS V, pp. 307-8; VI, pp. 220-22; VII, pp. 414-15.

Heinrich Suter, Die Mathematiker und Astronomen der Araber und ihre Werke, Leipzig, 1900.

S. Tekeli, “Al-Ḵujandi,” Dictionary of Scientific Biography VII, New York, 1973, pp. 352-54.

Naṣir-al-Din Ṭusi, Ketāb šakl al-qaṭṭāʿ, ed. and tr. A. P. Caratheodory, Constantinople, 1891.

Eilhard Wiedemann, Aufsätze zur arabischen Wissenschaftsgeschichte, 2 vols., Hildesheim-New York, 1970.

Idem, Gesammelte Schriften zur arabische-islamischen Wissenschaftsgeschichte, 3 vols., Frankfurt am Main, 1984.

Franz Woepcke, “Recherches sur plusieurs ouvrages de Léonard de Pise découverts et publiés par M. le prince Balthasar Boncampagni et sur les rapports qui existent entre ces ouvrages et les travaux mathématiques des Arabes,” Atti dell’ Accademia Pontificia de’ nuovi Lincei 14, 1861, pp. 211-27, 241-69, 301-24, 345-56.

(David Pingree)

Originally Published: December 15, 2003

Last Updated: March 6, 2012

This article is available in print.
Vol. XI, Fasc. 6, pp. 636-637

Cite this entry:

David Pingree, “ḤĀMED BAL-ḴEŻR AL-ḴOJANDI,” Encyclopaedia Iranica, XI/6, pp. 636-637, available online at http://www.iranicaonline.org/articles/hamed-bal-kezralkojandi (accessed on 30 December 2012).