Al-Buzjani: A Great Astronomer

A crater near lunar equator on the far side of moon was named after him, and on June 10, 2015, Google changed its logo in memory of Abu al-Wafa’ Buzjani
Abul Wafa Muhammad Al-Buzjani
(940–997/98 AD)
Abul Wafa Muhammad Al-Buzjani (940–997/98 AD)


Abul Wafa Muhammad Ibn Muhammad ibn Yahya Ibn Ismail al-Buzjani, a great Persian Muslim scientist and scholar, was a mathematician, astronomer and geometrician. He was born in Buzjan or Buzhgan, Nisahpur, Iran on 10 June, 940 AD. He was influenced by Al-Batani, and Abu Nasr Mansur. He flourished and worked as a great mathematician, astronomer and geometrician in Baghdad and died on 15 July, 998 AD.

He learnt mathematics in Baghdad. In 959 AD, he migrated to Iraq and lived there till his death. His main contribution lies in several branches of mathematics, especially geometry and trigonometry. He worked in a private observatory in Baghdad, where he made observations to determine, among other astronomical parameters, the obliquity of the ecliptic, the length of the seasons, and the latitude of the city. He soon rose to prominence as a leading astronomer and mathematician at the Būyid court, conducting observations and research in the Bāb al‐Tibn observatory. Al-Būzjānī became actively involved in the construction of a new observatory in Baghdad. His collaborator was Al-Kūhī, another celebrated astronomer, who was excellent in mathematics, physics and in making astronomical instruments. The astronomical work of Al-Būzjānī and his colleagues in Baghdad mark the revival of the “Baghdad school,” a tradition with much vitality in the preceding century.

In honor of his astronomical work, a crater Abul Wafa on the Moon was named for him”. Abul Wafa is an impact crater located near the lunar equator on the far side of the Moon, named after the Persian Astronomer, Abu al-Wafa Buzjani. On June 10, 2015, Google changed its logo in memory of Abu al-Wafa’ Buzjani.

Apart from being a mathematician, al-Buzjani also contributed to astronomy. In this field, he discussed different movements of the moon, and discovered ‘variation’. He was also one of the last Arabic translators and commentators of the 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.

He made important innovations in spherical trigonometry, and his work on arithmetic for business contains the first instance of using negative numbers in a Medieval Islamic Text. His notable work is Almagest of Abu al-Wafa. His notable ideas include Tangent function, Law of Sines, several trigonometric identities. He is also credited with compiling the tables of sines and tangent at 15’ intervals. He also introduced the secant (sec) and cosecant (cosec) functions, as well studied the inter-relations between the six trigonometric lines associated with an arc. His Almagest was widely read by medieval Arabic astronomers in the centuries after his death. He is known to have written several other books that have not survived.

He was the first to build a wall quadrant to observe the sky. It has been suggested that he was influenced by the work of al-Batani as the latter described a quadrant instrument in his Book, Kitab az-Zij. His use of the concept of the tangent helped solve problems involving right-angled spherical triangles. He developed a new technique to calculate sine tables, allowing him to construct more accurate tables than his predecessors.

In 997, he participated in an experiment to determine the difference in local time between his location, Baghdad, and that of al-Biruni (who was living in Kath, now a part of Uzbekistan). The result was very close to present-day calculations, showing a difference of approximately 1 hour between the two longitudes. Abu al-Wafa is also known to have worked with Abu Sahl al-Quhi, who was a famous maker of astronomical instruments. While what is extant from his works lacks theoretical innovation, his observational data were used by many later astronomers, including al-Biruni.

Among his works on astronomy, only the first seven treatises of his Almagest (Kitab al-Majisti) are now extant. The work covers numerous topics in the fields plane and spherical trigonometry, planetary theory, and solutions to determine the direction Qibla.

 In geometry, his contribution comprises solution of geometrical problems of the compass; construction of a square equivalent to other squares; regular polyhedral; construction of regular hectogon 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:

x4 = a and x4 + ax3 = b

Al-Buzjani 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 relationship for sine (a+b) and the formula:

2 sin 2 (a / 2) = 1 – cos a,             


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

In addition, he made a special study of the tangent and calculated a table of tangent. 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.

He defined the tangent function, and he established several trigonometric identities in their modern from, where the ancient Greek mathematicians had expressed the equivalent identities in terms of chords. The trigonometric identities he introduced were:

sin (a_+b) = sin (a) cos (b) _+ cos (a) sin (b)

cos (2a) = 1 – 2 sin2 (a)

sin (2a) = 2 sin (a) cos (a)

He may have developed the law of sines for spherical triangles, though others like Abu-Mahmud Khojandi have been credited with the same achievements:

sin A = sin B = sin C

-------     ------      ------

sin a       sin b       sin c


where A, B, C are the sides of the triangle (measured in radians on the unit sphere) and a, b, c are opposing angles.

His main works include Almagest (Kitab al-Majisti), a Book of Zij, called, Zij al-waadih, (no longer extant); a Book on Those Geometric Constructions Which Are necessary for a Craftsman; Applied Geometry (Kitab fi maa yahtaaj ilayh al-saani’min al-a’maal al-handasiyya) containing over one hundred Geometric constructions, including for a regular heptagon, which have been reviewed and compared with other mathematical treatises. The legacy of this text in Latin Europe is still debated; a Book on What is Necessary for the Science of Arithmetic for Scribes and Businessmen; a Practical Book of Arithmetic (Kitab fi maa yahtaaj ilayh al-kuttaab wa’l-ummaal min’ilm al-hisaab). The first book of this kind where negative numbers have been used in the medieval Islamic texts; Al-Kitab al-Kamil (the complete Book); Book on Translation and Commentaries on the Algebraic works of Diophantus, al-Khawarizmi, and Euclid’s Elements.

“His astronomical knowledge on the movements of the moon has been criticized in that, in the case of ‘variation’, the third inequality 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”

Al-Būzjānī was one of the leading astronomers and mathematicians of the Islamic scientific tradition (Islamic Golden Age), with significant contributions in observational astronomy. His achievements in trigonometry paved the way for more precise astronomical calculations.

In view of his outstanding contribution in the field of mathematics, geometry, trigonometry, and astronomy, it needs to be explored and experimented with available latest scientific tools and experimental protocols to prove their relevance in the modern scientific world. It is also wonderful to know that how far scientific was he in the so-called unscientific period that he would do mathematics, geometry, trigonometry, and astronomy in his time during 940 – 997/998 AD.

Prof. Dr. Naquibul Islam, Senior Unani Medical Consultant, Solina Bazar, Srinagar, Kashmir. He can be reached at:

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