A00039 - Abu al-Wafa (Abul Wefa), Tenth Century Persian Mathematician and Astronomer

 

Abu al-Wafa (Abul Wefa) Abul Wefa (Abul Wafa' Buzjani) (Abu al-Wafa' Muhammad ibn Muhammad ibn Yahya ibn Isma'il ibn al-'Abbas al-Buzjani) (June 10, 940 - July 1, 998). Mathematician and astronomer who played a major role in the development of sines and cosines as they apply to the field of trignonometry. These he used to correct astronomical calculations carried forward from classical into Islamic times. Born on June 10, 940, in either Buzshan, Khorasan Province, or Buzadhan, Kuhistan Province, Iran, during the reign of the ‘Abbasid caliph al-Mutaqqi, Abul Wefa lived during a period of extraordinary cultural and intellectual productivity. His own fields of accomplishment, mathematics and astronomy, were already widely recognized as essential elements of high Islamic civilization. Very little is known about Abul Wefa’s early life. Apparently, his early education in mathematics occurred under the tutelage of two uncles, one of whom (Abu Amr al-Mughazili) had received formal training from the famous geometricians Abu Yahya al-Marwazi and Abu ’l Ala ibn Karnib. Whatever the possible source of patronage for the young man’s further education may have been, his decision to move to Baghdad at the age of nineteen (in 959) greatly benefitted the ‘Abbasid court. Baghdad at this time was politically troubled, following the seizure of de facto control by a military clique headed by the Persian Buyid emirs. Thereafter, the Buyids dominated the house of the caliphs until their fall from power in 1055. The Buyids were inclined to favor talented Persians who were drawn toward scholarly circles in the center of the empire. It is reported, for example, that it was Abul Wefa, himself then forty years of age and well established (circa 980), who introduced the Persian scholar and philosopher Abu Hayyan al-Tawhidi into the Baghdad entourage of the vizier Ibn Sa’dan. Abu Hayyan soon became famous under the vizier’s patronage, composing a major work, al-Imta’ wa’l mu’anasa (a collection of notes drawn from philosophical and literary “salon” meetings), with a dedication to Ibn Sa’dan. Patronage for Abul Wefa’s work in courtly circles, however, must have come from a different milieu, that of the so-called Baghdad School. This scientific assembly flourished in the‘Abbasid capital in the last century before its conquest by the Seljuk Turks in 1055. According to some historians, patronage for the natural sciences in particular came precisely during the period in which Abul Wefa passed into the main stages of his scholarly career. The Buyid emir Adud al-Dawlah (978-983) had nurtured an interest in astronomy through his own studies. He passed this interest on to his son, Saraf al-Dawlah, who built an observatory next to his palace and called scholars from all regions of the empire to glorify the reputation of his reign by carrying out scientific experiments. Abul Wefa was among this group. The environment for learning in the Baghdad School, with its circle of eminent Islamic scientists, may explain how the young Persian scholar mastered so many technical fields in such a limited period of time. Beyond mere speculation regarding Abul Wefa’s early personal contacts, however, one must consider the importance of translation work in the Baghdad School. Abul Wefa himself translated the work of the Greek algebraist Diophantus (fl. c. 250), who had explored the field of indeterminate algebraic equations. Abul Wefa was also known for his studies of, and commentaries on, Euclid. There are, however, no surviving texts to indicate what use he made of the work of these two forerunners of the classical pre-Islamic period. By contrast, Abul Wefa’s attention to the work of the second century Greek astronomer Ptolemy not only contributed to the preservation and transmission to the medieval West of the classical knowledge contained in Ptolemy’s Mathematike suntaxis (c. 150; Almagest) but also earned for him an original and lasting reputation as an Islamic mathematician. The Almagest examined the field of trigonometry, which proposed mathematical relationships in terms of the angles and sides of right triangles. This called for the development of sines, or systematic relationships defined in a right triangle working from one of the acute angles, symbolically represented as A. Modern trigonometry expresses this relationship as sin A = a/c, or sin A is equal to the ratio of the length of the side opposite that angle (a) to the length of the hypotenuse (c). Ptolemy, in pioneering the field of spherical trigonometry, had laid down an approximate method for calculating sines (which he described as “chords”). Abul Wefa, however, drew on his studies of Indian precedents in the field of trigonometry that were unknown to Ptolemy, as well as models provided by Abul Wefa’s predecessor al-Battani (858-929), to perfect Ptolemy’s chords. This was done by applying algebraic, instead of geometric, methods of systematizing the sines. In particular, Abul Wefa’s development of the “half-chord” made it possible to achieve much more precise measurements that would eventually be used in surveying and navigation. The most immediate application of his tables of sines, however, was in the field of astronomy. One of Abul Wefa’s contributions which left a legacy that lasted for many centuries involved the study of evection, or irregularity, in the longitude of the moon. Later European commentators looked at the Islamic astronomer’s work and concluded that he, not Tycho Brahe (1546-1601), had been the first scientist to posit the theory of the “third inequality of the moon.” Although this theory was later proved to be erroneous, the debate at least drew attention to the importance of Abul Wefa’s originality in the field. Abul Wefa himself compiled, in addition to his well-known tables of sines, a book of astronomical tables entitled Zif al-wadith (that which is clear). Like his earlier work on sines, this text is not extant in the original. Scholars tend to agree, however, that certain anonymous manuscripts preserved in European libraries, such as the Zij al-shamil, are taken from Abul Wefa’s work. Works that have survived and that have been at least partially translated include a book of arithmetic entitled Kitab fi ma yahtaj ilayh al-kuttab wa l-‘ummal min ‘ilm al-hisab (Book on What is Necessary from the Science -- of Arithmetic for Scribes and Businessmen [961-976]), the Kitab fi ma yahtaj ilayh al-sani ‘min al-a’mal al-handasiyha (Book on What is Necessary from Geometric Construction for the Artisan [after 990]), and a book entitled Kitab al-kamil. It is thought that Abul Wefa may have still been living in Baghdad at the time of his death in 998 (July 1, 998). The crater Abul Wafa on the Moon is named after Abul Wefa. Wefa, Abul see Abul Wefa Abul Wafa' Buzjani see Abul Wefa Buzjani, Abul Wafa' see Abul Wefa Abu al-Wafa' Muhammad ibn Muhammad ibn Yahya ibn Isma'il ibn al-'Abbas al-Buzjani see Abul Wefa

Abū al-Wafāʾ, Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī (Persian: ابوالوفای بوزجانی‎)  (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 sec and cosec 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.

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 Al-Quhi and Al-Sijzi who were in Baghdad at the time and others like Abu Nasr ibn Iraq, Abu-Mahmud Khojandi, Kushyar ibn Labban and Al-Biruni. In Baghdad, he received patronage by members of the Buyid court.

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 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 al-Kuhi, 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.
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.
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:

where A, B, C are the sides 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.

• 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 yaḥtaj ilayh al-ṣāniʿ min al-aʿmāl al-handasiyya).

• "A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen", (Kitāb fī mā yaḥtaj 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.

The crater Abul Wáfa on the Moon is named after him.