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Abu al-Wafa' Buzjani

Abu al-Wafa' Buzjani

Abū al-Wafāʾ, Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī or Abū al-Wafā Būzhjānī (Persian: ابوالوفا بوزجانی or بوژگانی‎)[1] (10 June 940 – 15 July 998) was a Persian[2][3] 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.[4] 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.

Abu al-Wafa' al-Buzjani
BornJune 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 Abū al-Wafā'
Notable ideas
  • Tangent function
  • Law of sines
  • Several trigonometric identities
InfluencedAl-Biruni, Abu Nasr Mansur


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.[4] He was a contemporary of the distinguished scientists Abū Sahl al-Qūhī 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 Kitāb az-Zīj.[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 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 Abū Sahl al-Qūhī, 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 (Kitāb al-Majisṭī) 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 (كتاب المجسطي Kitāb al-Majisṭī).

  • A book of zij called Zīj al‐wāḍiḥ (زيج الواضح), no longer extant.[6]

  • "A Book on Those Geometric Constructions Which Are Necessary for a Craftsman", (كتاب في ما یحتاج إليه الصانع من الأعمال الهندسية Kitāb fī mā yaḥtāj ilayh al-ṣāniʿ min al-aʿmāl 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", (كتاب في ما يحتاج إليه الكتاب والعمال من علم الحساب Kitāb fī mā yaḥtāj ilayh al-kuttāb wa’l-ʿummāl min ʾilm al-ḥisāb).[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-Khwārizmī, and Euclid's Elements.[6]


  • The crater Abul Wáfa on the Moon is named after him.

  • On 10 June 2015, Google has changed its logo in memory of Abu al-Wafa' Buzjani.[12]


Citation Linkwww.encyclopaediaislamica.com"بوزجانی". Encyclopaediaislamica.com. Archived from the original on 2008-10-25. Retrieved 2009-08-30.
Sep 30, 2019, 6:24 AM
Citation Linkopenlibrary.orgBen-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.
Sep 30, 2019, 6:24 AM
Citation Linkbooks.google.comSigfried 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
Sep 30, 2019, 6:24 AM
Citation Linkwww-history.mcs.st-andrews.ac.ukO'Connor, John J.; Robertson, Edmund F., "Mohammad Abu'l-Wafa Al-Buzjani", MacTutor History of Mathematics archive, University of St Andrews.
Sep 30, 2019, 6:24 AM
Citation Link//doi.org/10.1017%2FS095742391000007XMoussa, Ali (2011). "Mathematical Methods in Abū al-Wafāʾ's Almagest and the Qibla Determinations". Arabic Sciences and Philosophy. Cambridge University Press. 21 (1). doi:10.1017/S095742391000007X.
Sep 30, 2019, 6:24 AM
Citation Linkopenlibrary.orgHashemipour, Behnaz (2007). "Būzjānī: Abū al‐Wafāʾ Muḥammad ibn Muḥammad ibn Yaḥyā al‐Būzjānī". In Thomas Hockey; et al. (eds.). The Biographical Encyclopedia of Astronomers. New York: Springer. pp. 188–9. ISBN 978-0-387-31022-0..
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Citation Linkopenlibrary.orgKennedy, E. S. (1956). Survey of Islamic Astronomical Tables. American Philosophical Society. p. 12.
Sep 30, 2019, 6:24 AM
Citation Linkopenlibrary.orgJacques 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
Sep 30, 2019, 6:24 AM
Citation Linkopenlibrary.orgYouschkevitch, A.P. (1970). "Abū'l-Wafāʾ Al-Būzjānī, Muḥammad Ibn Muḥammad Ibn Yaḥyā Ibn Ismāʿīl Ibn Al-ʿAbbās". Dictionary of Scientific Biography. 1. New York: Charles Scribner's Sons. pp. 39–43. ISBN 0-684-10114-9..
Sep 30, 2019, 6:24 AM
Citation Linkopenlibrary.orgRaynaud, D. (2012), "Abū al-Wafāʾ Latinus? A Study of Method", Historia Mathematica, 39 (1): 34–83, doi:10.1016/j.hm.2011.09.001.
Sep 30, 2019, 6:24 AM
Citation Linkwww.theguardian.comGamwell, Lynn (2 December 2015). "Why the history of maths is also the history of art". The Guardian. Retrieved 3 December 2015.
Sep 30, 2019, 6:24 AM
Citation Linkwww.google.com"Abu al-Wafa' al-Buzjani's 1075th Birthday". Google. 10 June 2015.
Sep 30, 2019, 6:24 AM
Citation Linkwww-history.mcs.st-andrews.ac.uk"Mohammad Abu'l-Wafa Al-Buzjani"
Sep 30, 2019, 6:24 AM
Citation Linkislamsci.mcgill.caPDF version
Sep 30, 2019, 6:24 AM
Citation Linkhalshs.archives-ouvertes.frPDF version
Sep 30, 2019, 6:24 AM
Citation Linkweb.archive.org"بوزجانی"
Sep 30, 2019, 6:24 AM
Citation Linkwww.encyclopaediaislamica.comthe original
Sep 30, 2019, 6:24 AM
Citation Linkbooks.google.comHistory of Humanity: From the seventh to the sixteenth century
Sep 30, 2019, 6:24 AM
Citation Linkwww-history.mcs.st-andrews.ac.uk"Mohammad Abu'l-Wafa Al-Buzjani"
Sep 30, 2019, 6:24 AM
Citation Linkdoi.org10.1017/S095742391000007X
Sep 30, 2019, 6:24 AM