Born: June 10, 940, Buzhgan
Died: 998 CE, Baghdad
Era: Islamic Golden Age
Region: Baghdad
Main Interest: Mathematics and Astronomy
Notable Ideas: Tangent Functions, Law of Sines, and other Trigonometrical identities
Major Work: Almagest of Abu Al-Wafa
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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) (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 for the next forty years, and died there 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 ibn
Iraq, Abu-Mahmud Khojandi, Kushyar ibn Labbanand 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 ofAl-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 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's.
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
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.
He
also discovered the law of sines for spherical triangles:
Failed
to parse (unknown function "\A"): \frac{\A}{\sin a} = \frac{\B}{\sin
b} = \frac{\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.
Some
sources suggest that he introduced the tangent function, although other
sources give the credit for this innovation to al-Marwazi.
Works
·
Almagest (كتاب المجسطي Kitāb al-Majisṭī).
·
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 al-ṣāniʿ min al-aʿmāl al-handasiyya). 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 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
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