# Abu al-Wafa' Buzjani

Abu al-Wafa' al-Buzjani
BornJune 10, 940
Buzhgan, Iran
DiedJuly 15, 998 (aged 58)
InfluencesAl-Battani
EraIslamic Golden Age
Main interestsMathematics and Astronomy
Notable worksAlmagest of Abū al-Wafā'
Notable ideas
InfluencedAl-Biruni, Abu Nasr Mansur

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)[2] was a Persian[3][4][5] 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.

## Life

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 Abū Sahl al-Qūhī and Al-Sijzi who were in Baghdad at the time and others like Abu Nasr Mansur, Abu-Mahmud Khojandi, Kushyar Gilani and Al-Biruni.[6] In Baghdad, he received patronage by members of the Buyid court.[7]

## Astronomy

Abu Al-Wafa' was the first to build a wall quadrant to observe the sky.[6] 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.[6] 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.[7]

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).[8] 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.[7] While what is extant from his works lacks theoretical innovation, his observational data were used by many later astronomers, including al-Biruni.[7]

### Almagest

Among his works on astronomy, only the first seven treatises of his Almagest (Kitāb al-Majisṭī) are now extant.[9] The work covers numerous topics in the fields of plane and spherical trigonometry, planetary theory, and solutions to determine the direction of Qibla.[6][7]

## Mathematics

He defined the tangent function, and he established several trigonometric identities such as ${\displaystyle \sin(a\pm b)}$ in their modern form, where the Ancient Greek mathematicians had expressed the equivalent identities in terms of chords.[10]

${\displaystyle \sin(\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta }$
${\displaystyle \sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)}$
${\displaystyle \cos(2a)=1-2\sin ^{2}(a)}$
${\displaystyle \sin(2a)=2\sin(a)\cos(a)}$

He may have introduced the law of sines for spherical triangles, though others like Abu-Mahmud Khojandi have been credited with the same achievement:[11]

${\displaystyle {\frac {\sin A}{\sin a}}={\frac {\sin B}{\sin b}}={\frac {\sin C}{\sin c}}}$

where ${\displaystyle A,B,C}$ are the sides (measured in radians on the unit sphere) and ${\displaystyle a,b,c}$ are the opposing angles.[10]

Some sources suggest that he introduced the tangent function, although other sources give the credit for this innovation to al-Marwazi.[10]

## Works

• Almagest (كتاب المجسطي Kitāb al-Majisṭī).
• A book of zij called Zīj al‐wāḍiḥ (زيج الواضح), no longer extant.[7]
• "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).[12] 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.[13][14]
• "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).[12] This is the first book where negative numbers have been used in the medieval Islamic texts.[7]

He also wrote translations and commentaries on the algebraic works of Diophantus, al-Khwārizmī, and Euclid's Elements.[7]

## Legacy

• The crater Abul Wáfa on the Moon is named after him.[15][16]
• On 10 June 2015, Google changed its logo in memory of Abu al-Wafa' Buzjani.[17]

## Notes

1. ^ "بوزجانی". Encyclopaediaislamica.com. Archived from the original on 2008-10-25. Retrieved 2009-08-30.
2. ^ a b c
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. ^ Subtelny, Maria E. (2007). Timurids in Transition. BRILL. p. 144. ISBN 9789004160316. Persian mathematician Abu al-Wafa Muhammad al-Buzjani
6. ^ a b c d Moussa, Ali (2011). "Mathematical Methods in Abū al-Wafāʾ's Almagest and the Qibla Determinations". Arabic Sciences and Philosophy. Cambridge University Press. 21 (1): 1–56. doi:10.1017/S095742391000007X.
7. ^ Stowasser, Barbara Freyer (2014-05-09). The Day Begins at Sunset: Perceptions of Time in the Islamic World. Bloomsbury Publishing. p. 83. ISBN 978-0-85772-536-3.
8. ^ Kennedy, E. S. (1956). Survey of Islamic Astronomical Tables. American Philosophical Society. p. 12.
9. ^ 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
10. ^ S. Frederick Starr (2015). Lost Enlightenment: Central Asia's Golden Age from the Arab Conquest to Tamerlane. Princeton University Press. p. 177. ISBN 9780691165851.
11. ^ a b
12. ^
13. ^ Gamwell, Lynn (2 December 2015). "Why the history of maths is also the history of art". The Guardian. Retrieved 3 December 2015.
14. ^ "Abul Wáfa". Gazetteer of Planetary Nomenclature. USGS Astrogeology Research Program.
15. ^ D. H. Menzel; M. Minnaert; B. Levin; A. Dollfus; B. Bell (1971). "Report on Lunar Nomenclature by The Working Group of Commission 17 of the IAU". Space Science Reviews. 12 (2): 136. Bibcode:1971SSRv...12..136M. doi:10.1007/BF00171763.
16. ^ "Abu al-Wafa' al-Buzjani's 1075th Birthday". Google. 10 June 2015.