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(940-997 A.D.)

Abul Wafa Muhammad Ibn Muhammad Ibn Yahya Ibn Ismail al-Buzjani was born in Buzjan, Nishapur in 940 A.D. He flourished as a great mathematician and astronomer at Baghdad and died in 997/998 A.D. He learnt mathematics in Baghdad. In 959 A.D. 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 geometri- cal solution of the equations:

x4 = a and x4 + ax3 = 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 sin2 (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 calcula- ted 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. Nonethe- less, 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.