Abu al-Wafa' Buzjani
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 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 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. Section::::"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 formula_1 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 formula_7 are the sides (measured in radians on the unit sphere) and formula_8 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. He also wrote translations and commentaries on the algebraic works of Diophantus, al-Khwārizmī, and Euclid's "Elements".