Abū alWafāʾ, Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn alʿAbbās alBūzjānī or Abū alWafā Būzhjānī (Persian: ابوالوفا بوزجانی or بوژگانی)^{[1]} (10 June 940 – 15 July 998) was a Persian^{[2]} 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.^{[3]} 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.
Contents

Life 1

Astronomy 2

Mathematics 3

Works 4

Legacy 5

Notes 6

References 7

External links 8
Life
He was born in Buzhgan, (now Torbate 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.^{[3]} He was a contemporary of the distinguished scientists Abū Sahl alQūhī and AlSijzi who were in Baghdad at the time and others like Abu Nasr ibn Iraq, AbuMahmud Khojandi, Kushyar ibn Labban and AlBiruni.^{[4]} In Baghdad, he received patronage by members of the Buyid court.
Astronomy
Abu AlWafa' was the first to build a wall quadrant to observe the sky.^{[4]} It has been suggested that he was influenced by the works of AlBattani as the latter describes a quadrant instrument in his Kitāb azZīj.^{[4]} His use of tangent helped to solve problems involving rightangled spherical triangles, and 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 and that of alBiruni (who was living in Kath, now a part of Uzbekistan). The result was very close to presentday calculations, showing a difference of approximately 1 hour between the two longitudes. Abu alWafa is also known to have worked with Abū Sahl alQūhī, 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 alBiruni's.
Almagest
Among his works on astronomy, only the first seven treatises of his Almagest (Kitāb alMajisṭī) are now extant.^{[6]} The work covers numerous topics in the fields of plane and spherical trigonometry, planetary theory, and solutions to determine the direction of Qibla.^{[4]}
Mathematics
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.^{[7]}
\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta
He also discovered the law of sines for spherical triangles:

\frac{\sin A}{\sin a} = \frac{\sin B}{\sin b} = \frac{\sin C}{\sin c}
where A, B, C are the sides (measured in radians on the unit sphere) and a, b, c are the opposing angles.^{[7]}
Some sources suggest that he introduced the tangent function, although other sources give the credit for this innovation to alMarwazi.^{[7]}
Works

Almagest (كتاب المجسطي Kitāb alMajisṭī).

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

"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 alṣāniʿ min alaʿmāl alhandasiyya). This text contains over one hundred geometric constructions 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 from the Science of Arithmetic for Scribes and Businessmen", (كتاب في ما يحتاج إليه الكتاب والعمال من علم الحساب Kitāb fī mā yaḥtāj ilayh alkuttāb wa’lʿummāl min ʾilm alḥisāb). This is the first book where negative numbers have been used in the medieval Islamic texts.
He also wrote translations and commentaries on the algebraic works of Diophantus, alKhwārizmī, and Euclid's Elements.
Legacy
The crater Abul Wáfa on the Moon is named after him.
Notes

^ "بوزجانی". Encyclopaediaislamica.com. Retrieved 20090830.

^ "Iran" in USECO History of Humanity, ed. by M.A. Bakhit, Volume 4 of History of humanity : scientific and cultural development,UNESCO, 2000 pg 375: ""The science of trigonometry as known today was established by Islamic mathematicians. One of the most important of these was the Muslim Abu'l Wafa Buzjani (d. 997 or 998), who wrote a work called the Almagest dealing mostly with trigonometry"" [2]

^ ^{a} ^{b} .

^ ^{a} ^{b} ^{c} ^{d} Moussa, Ali (2011). "Mathematical Methods in Abū alWafāʾ's Almagest and the Qibla Determinations". Arabic Sciences and Philosophy (

^ Kennedy, E. S. (1956). Survey of Islamic Astronomical Tables. American Philosophical Society. p. 12.

^ ^{a} ^{b} ^{c} Jacques Sesiano, "Islamic mathematics", p. 157, in
References

.

Hashemipour, Behnaz (2007). "Būzjānī: Abū al‐Wafāʾ Muḥammad ibn Muḥammad ibn Yaḥyā al‐Būzjānī". In Thomas Hockey; et al. The Biographical Encyclopedia of Astronomers. New York: Springer. pp. 188–9. (PDF version)

Raynaud, D. (2012), "Abū alWafāʾ Latinus? A Study of Method", Historia Mathematica 39 (1): 34–83, (PDF version)

Youschkevitch, A.P. (1970). "Abū’lWafāʾ AlBūzjānī, Muḥammad Ibn Muḥammad Ibn Yaḥyā Ibn Ismāʿīl Ibn AlʿAbbās".
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