Abū al-Wafāʾ Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī or Abū al-Wafā Būzhjānī (, ; 10 June 940 – 15 July 998)
was a
Persian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
and
astronomer
An astronomer is a scientist in the field of astronomy who focuses on a specific question or field outside the scope of Earth. Astronomers observe astronomical objects, such as stars, planets, natural satellite, moons, comets and galaxy, galax ...
who worked in
Baghdad
Baghdad ( or ; , ) is the capital and List of largest cities of Iraq, largest city of Iraq, located along the Tigris in the central part of the country. With a population exceeding 7 million, it ranks among the List of largest cities in the A ...
. He made important innovations in
spherical trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the edge (geometry), sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, ge ...
, and his work on
arithmetic
Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms.
...
for businessmen contains the first instance of using
negative number
In mathematics, a negative number is the opposite (mathematics), opposite of a positive real number. Equivalently, a negative number is a real number that is inequality (mathematics), less than 0, zero. Negative numbers are often used to represe ...
s in a
medieval Islamic text.
He is also credited with compiling the tables of
sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite th ...
s and
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
s 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, he moved to
Baghdad
Baghdad ( or ; , ) is the capital and List of largest cities of Iraq, largest city of Iraq, located along the Tigris in the central part of the country. With a population exceeding 7 million, it ranks among the List of largest cities in the A ...
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
Baghdad ( or ; , ) is the capital and List of largest cities of Iraq, largest city of Iraq, located along the Tigris in the central part of the country. With a population exceeding 7 million, it ranks among the List of largest cities in the A ...
at the time and others such as
Abu Nasr Mansur,
Abu-Mahmud Khojandi,
Kushyar Gilani and
al-Biruni.
In Baghdad, he received patronage from 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 described a quadrant instrument in his ''Kitāb az-Zīj''.
His use of the concept of the
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
helped solve problems involving right-angled
spherical triangles. He developed a new technique to calculate
sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite th ...
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
, image_flag = Flag of Uzbekistan.svg
, image_coat = Emblem of Uzbekistan.svg
, symbol_type = Emblem of Uzbekistan, Emblem
, national_anthem = "State Anthem of Uzbekistan, State Anthem of the Republ ...
). 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
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the edge (geometry), sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, ge ...
, planetary theory, and solutions to determine the direction of
Qibla
The qibla () is the direction towards the Kaaba in the Great Mosque of Mecca, Sacred Mosque in Mecca, which is used by Muslims in various religious contexts, particularly the direction of prayer for the salah. In Islam, the Kaaba is believed to ...
.
Mathematics
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.
[ The trigonometric identities he introduced were:
:
:
:
He has discovered the law of tangents and may have discovered the ]law of sines
In trigonometry, the law of sines (sometimes called the sine formula or sine rule) is a mathematical equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law,
\frac \,=\, \frac \,=\, \frac \,=\ ...
for spherical triangles, however, other scholars like Abu-Mahmud Khojandi have been credited with the latter achievement:
:
where are the sides of the triangle (measured in radians on the unit sphere) and are the opposing angles.[Jacques Sesiano, "Islamic mathematics", p. 157, in ]
Some sources suggest that he introduced the tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
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
Diophantus of Alexandria () (; ) was a Greek mathematician who was the author of the '' Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations.
Although Jose ...
, al-Khwārizmī, and Euclid
Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
's ''Elements''.
Legacy
* The crater Abul Wáfa on the Moon is named after him.
* On 10 June 2015, Google
Google LLC (, ) is an American multinational corporation and technology company focusing on online advertising, search engine technology, cloud computing, computer software, quantum computing, e-commerce, consumer electronics, and artificial ...
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