Abu al-Wafa' Buzjani

Abū al-Wafāʾ, Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī or Abū al-Wafā Būzhjānī ( fa, ابوالوفا بوزجانی or بوژگانی) (10 June 940 – 15 July 998) was a Persian 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. 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 until his death, in 998. 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. In Baghdad, he received patronage by members of the Buyid court.

Astronomy

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

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 Abū Sahl al-Qūhī, 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 al-Biruni.

''Almagest''

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

Mathematics

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

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

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

:$\frac = \frac = \frac$

where $A, B, C$ are the sides (measured in radians on the unit sphere) and $a, b, c$ are the opposing angles.Jacques Sesiano, "Islamic mathematics", p. 157, in

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

Works

* ''Almagest'' ( ''Kitāb al-Majisṭī'').
* A book of zij called ''Zīj al‐wāḍiḥ'' (), no longer extant.
* "A Book on Those Geometric Constructions Which Are Necessary for a Craftsman", ( ''Kitāb fī mā yaḥtāj ilayh al-ṣāniʿ min al-aʿmāl al-handasiyya''). 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.
* "A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen", ( ''Kitāb fī mā yaḥtāj ilayh al-kuttāb wa’l-ʿummāl min ʾilm al-ḥisāb''). 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, al-Khwārizmī, and Euclid's ''Elements''.

Legacy

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

Notes

References

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External links

{{DEFAULTSORT:Buzjani, Abu Al-Wafa
10th-century Iranian mathematicians
People from Torbat-e Jam
940 births
998 deaths
Scientists who worked on qibla determination
10th-century Iranian astronomers
Mathematicians from Nishapur
Scholars under the Buyid dynasty