**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)^{[1]} (10 June 940 – 15 July 998) was a Persian^{[2]}^{[3]} 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.

Abu al-Wafa' al-Buzjani | |
---|---|

Born | June 10, 940 Buzhgan, Iran |

Died | July 15, 998 | (aged 58)

Residence | Baghdad |

Academic background | |

Influences | Al-Battani |

Academic work | |

Era | Islamic Golden Age |

Main interests | Mathematics and Astronomy |

Notable works | Almagest of Abū al-Wafā' |

Notable ideas | |

Influenced | Al-Biruni, Abu Nasr Mansur |

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.^{[4]} 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.

## Contents

## LifeEdit

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.^{[4]} 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 Labban and Al-Biruni.^{[5]} In Baghdad, he received patronage by members of the Buyid court.^{[6]}

## AstronomyEdit

Abu Al-Wafa' was the first to build a wall quadrant to observe the sky.^{[5]} 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*.^{[5]}
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.^{[6]}

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.^{[6]} While what is extant from his works lacks theoretical innovation, his observational data were used by many later astronomers, including al-Biruni.^{[6]}

*Almagest*Edit

Among his works on astronomy, only the first seven treatises of his *Almagest* (*Kitāb al-Majisṭī*) are now extant.^{[7]} The work covers numerous topics in the fields of plane and spherical trigonometry, planetary theory, and solutions to determine the direction of Qibla.^{[5]}^{[6]}

## MathematicsEdit

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.^{[8]}

He also discovered the law of sines for spherical triangles:

where *A*, *B*, *C* are the sides (measured in radians on the unit sphere) and *a*, *b*, *c* are the opposing angles.^{[8]}

Some sources suggest that he introduced the tangent function, although other sources give the credit for this innovation to al-Marwazi.^{[8]}

## WorksEdit

*Almagest*(كتاب المجسطي*Kitāb al-Majisṭī*).- A book of zij called
*Zīj al‐wāḍiḥ*(زيج الواضح), no longer extant.^{[6]} - "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*).^{[9]}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.^{[10]}^{[11]} - "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*).^{[9]}This is the first book where negative numbers have been used in the medieval Islamic texts.^{[6]}

He also wrote translations and commentaries on the algebraic works of Diophantus, al-Khwārizmī, and Euclid's *Elements*.^{[6]}

## LegacyEdit

## NotesEdit

**^**"بوزجانی". Encyclopaediaislamica.com. Archived from the original on 2008-10-25. Retrieved 2009-08-30.**^**Ben-Menahem, A. (2009).*Historical encyclopedia of natural and mathematical sciences*(1st ed.). Berlin: Springer. p. 559. ISBN 978-3-540-68831-0.970 CE Abu al-Wafa al-Buzjani (940–998, Baghdad). Persian astronomer and mathematician.

**^**Sigfried J. de Laet (1994).*History of Humanity: From the seventh to the sixteenth century*. UNESCO. p. 931. ISBN 978-92-3-102813-7.The science of trigonometry as known today was established by Islamic mathematicians. One of the most important of these was the Persian Abu' l-Wafa' Buzjani (d. 997 or 998), who wrote a work called the Almagest dealing mostly with trigonometry

- ^
^{a}^{b}O'Connor, John J.; Robertson, Edmund F., "Mohammad Abu'l-Wafa Al-Buzjani",*MacTutor History of Mathematics archive*, University of St Andrews. - ^
^{a}^{b}^{c}^{d}Moussa, Ali (2011). "Mathematical Methods in Abū al-Wafāʾ's Almagest and the Qibla Determinations".*Arabic Sciences and Philosophy*. Cambridge University Press.**21**(1). doi:10.1017/S095742391000007X. - ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}Hashemipour 2007. **^**Kennedy, E. S. (1956).*Survey of Islamic Astronomical Tables*. American Philosophical Society. p. 12.- ^
^{a}^{b}^{c}Jacques Sesiano, "Islamic mathematics", p. 157, in Selin, Helaine; D'Ambrosio, Ubiratan, eds. (2000),*Mathematics Across Cultures: The History of Non-western Mathematics*, Springer, ISBN 1-4020-0260-2 - ^
^{a}^{b}Youschkevitch 1970. **^**Raynaud 2012.**^**Gamwell, Lynn (2 December 2015). "Why the history of maths is also the history of art". The Guardian. Retrieved 3 December 2015.

## ReferencesEdit

- O'Connor, John J.; Robertson, Edmund F., "Mohammad Abu'l-Wafa Al-Buzjani",
*MacTutor History of Mathematics archive*, University of St Andrews. - Hashemipour, Behnaz (2007). "Būzjānī: Abū al‐Wafāʾ Muḥammad ibn Muḥammad ibn Yaḥyā al‐Būzjānī". In Thomas Hockey; et al.
*The Biographical Encyclopedia of Astronomers*. New York: Springer. pp. 188–9. ISBN 978-0-387-31022-0. (PDF version) - Raynaud, D. (2012), "Abū al-Wafāʾ Latinus? A Study of Method",
*Historia Mathematica*,**39**(1): 34–83, doi:10.1016/j.hm.2011.09.001 (PDF version) - Youschkevitch, A.P. (1970). "Abū'l-Wafāʾ Al-Būzjānī, Muḥammad Ibn Muḥammad Ibn Yaḥyā Ibn Ismāʿīl Ibn Al-ʿAbbās".
*Dictionary of Scientific Biography*.**1**. New York: Charles Scribner's Sons. pp. 39–43. ISBN 0-684-10114-9.