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Ab%C5%AB Al-Waf%C4%81' B%C5%ABzj%C4%81n%C4%AB
Abu al-Wafa' al-Buzjani
Buzjani, the Persian.jpg
Born(940-06-10)June 10, 940
Buzhgan, Iran
DiedJuly 15, 998(998-07-15) (aged 58)
Academic background
Academic work
EraIslamic Golden Age
Main interestsMathematics and Astronomy
Notable worksAlmagest of Ab? al-Waf?'
Notable ideas
InfluencedAl-Biruni, Abu Nasr Mansur

Ab? al-Waf, Mu?ammad ibn Mu?ammad ibn Ya?y? ibn Isml ibn al-?Abb?s al-B?zj?n? or Ab? al-Waf? B?zhj?n? (Persian: ? or ?‎)[1] (10 June 940 - 15 July 998)[2] was a Persian[3][4]mathematician and astronomer who worked in Baghdad. He made important innovations in spherical trigonometry, and his work on arithmetics for businessmen contains the first instance of using negative numbers in a medieval Islamic text.

He is also credited with compiling the tables of sines and tangents at 15 ' intervals. He also introduced the secant and cosecant functions, as well studied the interrelations between the six trigonometric lines associated with an arc.[2] 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 born in Buzhgan, (now Torbat-e Jam) in Khorasan (in today's Iran). At age 19, in 959 AD, he moved to Baghdad and remained there for the next forty years, and died there in 998.[2] He was a contemporary of the distinguished scientists Ab? Sahl al-Q?h? and Al-Sijzi who were in Baghdad at the time and others like Abu Nasr ibn Iraq, Abu-Mahmud Khojandi, Kushyar ibn Labban and Al-Biruni.[5] In Baghdad, he received patronage by members of the Buyid court.[6]


Abu Al-Wafa' was the first to build a wall quadrant to observe the sky.[5] It has been suggested that he was influenced by the works of Al-Battani as the latter describes a quadrant instrument in his Kit?b az-Z?j.[5] His use of tangent helped to solve problems involving right-angled spherical triangles, and developed a new technique to calculate sine tables, allowing him to construct more accurate tables than his predecessors.[6]

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 Ab? Sahl al-Q?h?, who was a famous maker of astronomical instruments.[6] While what is extant from his works lacks theoretical innovation, his observational data were used by many later astronomers, including al-Biruni.[6]


Among his works on astronomy, only the first seven treatises of his Almagest (Kit?b al-Majis) are now extant.[7] The work covers numerous topics in the fields of plane and spherical trigonometry, planetary theory, and solutions to determine the direction of Qibla.[5][6]


He established several trigonometric identities such as sin(a ± b) in their modern form, where the Ancient Greek mathematicians had expressed the equivalent identities in terms of chords.[8]

He also discovered the law of sines for spherical triangles:

where A, B, C are the sides (measured in radians on the unit sphere) and a, b, c are the opposing angles.[8]

Some sources suggest that he introduced the tangent function, although other sources give the credit for this innovation to al-Marwazi.[8]


  • Almagest (? ? Kit?b al-Majis).
  • A book of zij called Z?j al-wi? ( ), no longer extant.[6]
  • "A Book on Those Geometric Constructions Which Are Necessary for a Craftsman", (? ? ? Kit?b f? m? ya?t?j ilayh al-ni? min al-a?m?l al-handasiyya).[9] This text contains 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.[10][11]
  • "A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen", (? ? ? Kit?b f? m? ya?t?j ilayh al-kutt?b wa'l-?umm?l min ?ilm al-?is?b).[9] This is the first book where negative numbers have been used in the medieval Islamic texts.[6]

He also wrote translations and commentaries on the algebraic works of Diophantus, al-Khw?rizm?, and Euclid's Elements.[6]


  • The crater Abul Wáfa on the Moon is named after him.[12]
  • On 10 June 2015, Google changed its logo in memory of Abu al-Wafa' Buzjani.[13]


  1. ^ "?". Encyclopaediaislamica.com. Archived from the original on 2008-10-25. Retrieved .
  2. ^ a b c O'Connor, John J.; Robertson, Edmund F., "Mohammad Abu'l-Wafa Al-Buzjani", MacTutor History of Mathematics archive, University of St Andrews.
  3. ^ Ben-Menahem, A. (2009). Historical encyclopedia of natural and mathematical sciences (1st ed.). Berlin: Springer. p. 559. ISBN 978-3-540-68831-0. 970 CE Abu al-Wafa al-Buzjani (940-998, Baghdad). Persian astronomer and mathematician.
  4. ^ Sigfried J. de Laet (1994). History of Humanity: From the seventh to the sixteenth century. UNESCO. p. 931. ISBN 978-92-3-102813-7. The science of trigonometry as known today was established by Islamic mathematicians. One of the most important of these was the Persian Abu' l-Wafa' Buzjani (d. 997 or 998), who wrote a work called the Almagest dealing mostly with trigonometry
  5. ^ a b c d Moussa, Ali (2011). "Mathematical Methods in Ab? al-Waf's Almagest and the Qibla Determinations". Arabic Sciences and Philosophy. Cambridge University Press. 21 (1): 1-56. doi:10.1017/S095742391000007X.
  6. ^ a b c d e f g h Hashemipour 2007.
  7. ^ Kennedy, E. S. (1956). Survey of Islamic Astronomical Tables. American Philosophical Society. p. 12.
  8. ^ a b c Jacques Sesiano, "Islamic mathematics", p. 157, in Selin, Helaine; D'Ambrosio, Ubiratan, eds. (2000), Mathematics Across Cultures: The History of Non-western Mathematics, Springer, ISBN 1-4020-0260-2
  9. ^ a b Youschkevitch 1970.
  10. ^ Raynaud 2012.
  11. ^ Gamwell, Lynn (2 December 2015). "Why the history of maths is also the history of art". The Guardian. Retrieved 2015.
  12. ^ "Abul Wáfa". Gazetteer of Planetary Nomenclature. USGS Astrogeology Research Program.
  13. ^ "Abu al-Wafa' al-Buzjani's 1075th Birthday". Google. 10 June 2015.


External links

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