Abu al-Wafa' al-Buzjani

He was born in Buzhgan, (now Torbat-e Jam) in Khorasan (in today's Iran). At age 19, in 959, he moved to Baghdad and remained there until his death 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 such as Abu Nasr Mansur, Abu-Mahmud Khojandi, Kushyar Gilani and al-Biruni.^[6] In Baghdad, he received patronage from members of the Buyid court.^[7]

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 described a quadrant instrument in his Kitāb az-Zīj.^[6] 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.^[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]

He defined the tangent function, and he established several trigonometric identities in their modern form, where the ancient Greek mathematicians had expressed the equivalent identities in terms of chords.^[10] The trigonometric identities he introduced were:

${\displaystyle \sin(a\pm b)=\sin(a)\cos(b)\pm \cos(a)\sin(b)}$

${\displaystyle \cos(2a)=1-2\sin ^{2}(a)}$

${\displaystyle \sin(2a)=2\sin(a)\cos(a)}$

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

• 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]

• 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]

• O'Connor, John J.; Robertson, Edmund F., "Mohammad Abu'l-Wafa Al-Buzjani", MacTutor History of Mathematics Archive, University of St Andrews
• Hashemipour, 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. (PDF version)
• Raynaud, D. (2012), "Abū al-Wafāʾ Latinus? A Study of Method", Historia Mathematica, 39 (1): 34–83, doi:10.1016/j.hm.2011.09.001, S2CID 119600916 (PDF version)
• Youschkevitch, A.P. (1970). "Abū'l-Wafāʾ al-Būzjānī, Muḥammad Ibn Muḥammad Ibn Yaḥyā Ibn Ismāʿīl Ibn al-ʿAbbās". Dictionary of Scientific Biography. Vol. 1. New York: Charles Scribner's Sons. pp. 39–43. ISBN 0-684-10114-9.