Abu al-Wafa' Buzjani




"Abul Wáfa" redirects here. For the lunar crater, see Abul Wáfa (crater).












































Abu al-Wafa' al-Buzjani
Buzjani, the Persian.jpg
Born June 10, 940

Buzhgan, Iran
Died July 15, 998(998-07-15) (aged 58)
Baghdad
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

  • Tangent function

  • Law of sines

  • Several trigonometric identities
Influenced
Al-Biruni, Abu Nasr Mansur


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.


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






  • 1 Life


  • 2 Astronomy


    • 2.1 Almagest




  • 3 Mathematics


  • 4 Works


  • 5 Legacy


  • 6 Notes


  • 7 References


  • 8 External links





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.[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]



Astronomy


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


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]



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.[8]



sin⁡(α±β)=sin⁡αcos⁡β±cos⁡αsin⁡β{displaystyle sin(alpha pm beta )=sin alpha cos beta pm cos alpha sin beta }sin(alpha pm beta )=sin alpha cos beta pm cos alpha sin beta

sin⁡(a+b)=sin⁡(a)cos⁡(b)+cos⁡(a)sin⁡(b){displaystyle sin(a+b)=sin(a)cos(b)+cos(a)sin(b)}{displaystyle sin(a+b)=sin(a)cos(b)+cos(a)sin(b)}

cos⁡(2a)=1−2sin2⁡(a){displaystyle cos(2a)=1-2sin ^{2}(a)}{displaystyle cos(2a)=1-2sin ^{2}(a)}

sin⁡(2a)=2sin⁡(a)cos⁡(a){displaystyle sin(2a)=2sin(a)cos(a)}{displaystyle sin(2a)=2sin(a)cos(a)}


He also discovered the law of sines for spherical triangles:


sin⁡Asin⁡a=sin⁡Bsin⁡b=sin⁡Csin⁡c{displaystyle {frac {sin A}{sin a}}={frac {sin B}{sin b}}={frac {sin C}{sin c}}}{frac {sin A}{sin a}}={frac {sin B}{sin b}}={frac {sin 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.[8]


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



Works




  • 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]



Legacy



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

  • on June 2015 Google has changed its logo in memory of Abu al-Wafa' Buzjani.



Notes





  1. ^ "بوزجانی". Encyclopaediaislamica.com. Archived from the original on 2008-10-25. Retrieved 2009-08-30..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  2. ^ 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.


  3. ^ 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


  4. ^ ab O'Connor, John J.; Robertson, Edmund F., "Mohammad Abu'l-Wafa Al-Buzjani", MacTutor History of Mathematics archive, University of St Andrews.


  5. ^ abcd 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.


  6. ^ abcdefgh Hashemipour 2007.


  7. ^ Kennedy, E. S. (1956). Survey of Islamic Astronomical Tables. American Philosophical Society. p. 12.


  8. ^ abc 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


  9. ^ ab Youschkevitch 1970.


  10. ^ Raynaud 2012.


  11. ^ Gamwell, Lynn (2 December 2015). "Why the history of maths is also the history of art". The Guardian. Retrieved 3 December 2015.




References




  • 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&ndash, 43. ISBN 0-684-10114-9.



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