Abul Wafa Muhammad al-Buzjani

(940-997 C.E.)

Mathematics, Astronomy, Geometry, Trigonometry.

Buzjani Maths Series - Problem and Solution - ABCD is a parallelogram (part a)

Abul Wafa Muhammad Ibn Muhammad Ibn Yahya Ibn Ismail al-Buzjani was born in Buzjan, Nishapur in 940 C.E. He flourished as a great mathematician and astronomer at Baghdad and died in 997/998 C.E. He learnt mathematics in Baghdad. In 959 C.E. he migrated to Iraq and lived there till his death.

Abul Wafa's main contribution lies in several branches of mathematics, especially geometry and trigonometry. In geometry his contribution comprises solution of geometrical problems with opening of the compass; construction of a square equivalent to other squares; regular polyhedra; construction of regular hectagon taking for its side half the side of the equilateral triangle inscribed in the same circle; constructions of parabola by points and geometrical solution of the equations:

x^4 = a and x^4 + ax^3 = b

Abul Wafa's contribution to the development of trigonometry was extensive. He was the first to show the generality of the sine theorem relative to spherical triangles. He developed a new method of constructing sine tables, the value of sin 30' being correct to the eighth decimal place. He also developed relations for sine (a+b) and the formula:


 2 sin^2 (a/2) = 1 - cos a ,


and sin a = 2 sin (a/2) cos (a/2)

In addition, he made a special study of the tangent and calculated a table of tangents. He introduced the secant and cosecant for the first time, knew the relations between the trigonometric lines, which are now used to define them, and undertook extensive studies on conics.

Apart from being a mathematician, Abul Wafa also contributed to astronomy. In this field he discussed different movernents of the moon, and discovered 'variation'. He was also one of the last Arabic translators and commentators of Greek works.

He wrote a large number of books on mathematics and other subjects, most of which have been lost or exist in modified forms. His contribution includes Kitab 'Ilm al-Hisab, a practical book of arithmetic, al-Kitab al-Kamil (the Complete Book), Kitab al-Handsa (Applied Geometry). Apart from this, he wrote rich commentaries on Euclid, Diophantos and al-Khawarizmi, but all of these have been lost. His books now extant include Kitab 'Ilm al-Hisab, Kitab al- Handsa and Kitab al-Kamil.

His astronomical knowledge on the movements of the moon has been criticized in that, in the case of 'variation' the third inequality of the moon as he discussed was the second part of the 'evection'. But, according to Sedat, what he discovered was the same that was discovered by Tycho Brache six centuries later. Nonetheless, his contribution to trigonometry was extremely significant in that he developed the knowledge on the tangent and introduced the secant and cosecant for the first time; in fact a sizeable part of today's trigonometry can be traced back to him.


abu al wafa al buzjani

Front cover of Majisti Abi al-Wafa’ al-Buzgani by Ali Musa, Beirut: Markaz dirasat al-wahda al-‘arabiyya, 2010.

Abu al-Wafa al-Buzjanî

- Al‐Qifṭī, Jamāl al‐Dīn (1903). Taʾrīkh al‐ḥukamāʾ, edited by J. Lippert. Leipzig: Theodor Weicher.

- Carra de Vaux (May–June 1892). “L'Almageste d'Abū ʾl Wéfā Alboūzdjānī.” Journal asiatique, 8th ser., vol. 19: pp. 408–471.

- Debarnot, Marie‐Thérèse (1996). “Trigonometry.” In Encyclopedia of the History of Arabic Science, edited by Roshdi Rashed. Vol. 2, pp. 495–538. London: Routledge.

- Ghorbani, A. and M. A. Sheykhan (1992). Buzdjānī Nāmeh: The Biography and a Survey of Buzdjānī's Mathematical Works. Tehran: Enqelab‐e Eslami Publishing and Educational Organization.

- Hashemipour, Behnaz, “Al-Būzjānī,” in Thomas Hockey et al. (eds.). The Biographical Encyclopedia of Astronomers, Springer Reference. New York: Springer, 2007, pp. 188-189.

- Ibn al‐Nadīm (1970). The Fihrist of al‐Nadīm: A Tenth‐Century Survey of Muslim Culture, edited and translated by Bayard Dodge. 2 Vols., Vol. 1, p. 83. New York: Columbia University Press.

- Kennedy, E. S. (1984). “Applied Mathematics in the Tenth Century: Abu ʾl‐Wafāʾ Calculates the Distance Baghdad–Mecca.” Historia Mathematica 11: 193–206.

- Kennedy, E. S. and Mustafa Mawaldi (1979). “Abū al‐Wafāʾ and the Heron Theorems.” Journal of the History of Arabic Science 3: 19–30.

- Medovoi, M. I. (1960). “On the Arithmetic Treatise of Abū ʾl‐Wafāʾ.” Studies in the History of Mathematics, vol. 13: pp. 253–324.

- Saidan, Ahmad S. (1974). “The Arithmetic of Abū ʾl‐Wafāʾ.” Isis, vol. 65: pp. 367–375.

- Sesiano, Jacques (1998). “Le traité d'Abū ʾl‐Wafāʾ sur les carrés magiques.” Geschichte der Arabisch‐Islamischen Wissenschaften, vol. 12: pp. 121–244.

- Suter, H. (1960). “Abu ʾl‐Wafāʾ al‐Būzadjānī.” In Encyclopaedia of Islam. 2nd ed. Vol. 1, p. 159. Leiden: E. J. Brill.

- Youschkevitch, A. P. (1970). “Abū ʾl‐Wafāʾ al‐Būzjānī.” In Dictionary of Scientific Biography, edited by Charles Coulston Gillispie. Vol. 1, pp. 39–43. New York: Charles Scribner's Sons.