Abu al-Wafa' Buzjani

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.

Abu al-Wafa' al-Buzjani
Buzjani, the Persian
BornJune 10, 940
Buzhgan, Iran
DiedJuly 15, 998 (aged 58)
Academic background
Academic work
EraIslamic Golden Age
Main interestsMathematics and Astronomy
Notable worksAlmagest of Abū al-Wafā'
Notable ideas
InfluencedAl-Biruni, Abu Nasr Mansur


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]


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]


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]


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]


  • 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]


  • 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.


  1. ^ "بوزجانی". Encyclopaediaislamica.com. Archived from the original on 2008-10-25. Retrieved 2009-08-30.
  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. ^ a b O'Connor, John J.; Robertson, Edmund F., "Mohammad Abu'l-Wafa Al-Buzjani", MacTutor History of Mathematics archive, University of St Andrews.
  5. ^ 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.
  6. ^ a b c d e f g h Hashemipour 2007.
  7. ^ Kennedy, E. S. (1956). Survey of Islamic Astronomical Tables. American Philosophical Society. p. 12.
  8. ^ 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
  9. ^ a b 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.



Year 940 (CMXL) was a leap year starting on Wednesday (link will display the full calendar) of the Julian calendar.


Year 998 (CMXCVIII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar.

Abul Wafa (crater)

Abul Wafa is an impact crater located near the lunar equator on the far side of the Moon, named after the Persian mathematician and astronomer Abu al-Wafa' Buzjani. To the east are the crater pair Ctesibius and Heron. In the northeast lies the larger crater King, and to the southwest is Vesalius.

The perimeter of this crater somewhat resembles a rounded diamond shape. The rim and inner walls are rounded from impact erosion, and have lost some definition. There are ledges around most of the inner wall that may have once been terraces or slumped piles of scree.

A small but notable crater lies on the inner surface of the north rim of Abul Wáfa, and there is a small crater formation attached to the exterior southwest wall. The outer rim is relatively free of impacts, and the interior floor is marked only by a few small craterlets.


The Almagest () is a 2nd-century Greek-language mathematical and astronomical treatise on the apparent motions of the stars and planetary paths, written by Claudius Ptolemy (c. AD 100 – c. 170). One of the most influential scientific texts of all time, its geocentric model was accepted for more than 1200 years from its origin in Hellenistic Alexandria, in the medieval Byzantine and Islamic worlds, and in Western Europe through the Middle Ages and early Renaissance until Copernicus.

The Almagest is the critical source of information on ancient Greek astronomy. It has also been valuable to students of mathematics because it documents the ancient Greek mathematician Hipparchus's work, which has been lost. Hipparchus wrote about trigonometry, but because his works appear to have been lost, mathematicians use Ptolemy's book as their source for Hipparchus's work and ancient Greek trigonometry in general.

Ptolemy set up a public inscription at Canopus, Egypt, in 147 or 148. N. T. Hamilton found that the version of Ptolemy's models set out in the Canopic Inscription was earlier than the version in the Almagest. Hence it cannot have been completed before about 150, a quarter-century after Ptolemy began observing.

Bujgan District

Bujgan District (Persian: بخش بوژگان‎) is a district (bakhsh) in Torbat-e Jam County, Razavi Khorasan Province, Iran. At the 2006 census, its population was 18,400, in 3,916 families. The District has two cities: Nilshahr & Ahmadabad-e Sowlat. The District has two rural districts (dehestan): Dasht-e Jam Rural District and Harirud Rural District.

Astronomer Abu al-Wafa' Buzjani was born in the district.


Būzghān (Persian: بوژگان‎) (also Puchkan, Buzjan) is a village in Torbat-e-Jam County in Iran's Khorasan-e Razavi province. Historically Buzghan was a city and was the seat of government in the historic Persian province of Jam (Zam).

Buzghan is the birthplace of one of the most important Persian astronomers and mathematicians, Abu al-Wafa' Buzjani.


Girih (Persian: گره‎, "knot") is a decorative Islamic geometric artform used in architecture and handicraft objects, consisting of angled lines that form an interlaced strapwork pattern.

Girih decoration is believed to have been inspired by Syrian Roman knotwork patterns from the 2nd century AD. The earliest girih dates from around 1000 AD, and the artform flourished until the 15th century. Girih patterns can be created in a variety of ways, including the traditional compass and straightedge; the construction of a grid of polygons; and the use of a set of girih tiles with lines drawn on them: the lines form the pattern. Patterns may be elaborated by the use of two levels of design, as at the 1453 Darb-e Imam shrine. Square repeating units of known patterns can be copied as templates, and historic pattern books may have been intended for use in this way.

The 15th century Topkapı Scroll explicitly shows girih patterns together with the tilings used to create them. A set of tiles consisting of a dart and a kite shape can be used to create aperiodic Penrose tilings, though there is no evidence that such a set was used in medieval times. Girih patterns have been used to decorate varied materials including stone screens, as at Fatehpur Sikri; plasterwork, as at mosques and madrasas such as Hunat Hatun, Kayseri; metal, as at Sultan Hassan mosque, Cairo; and in wood, as at the Great Mosque of Cordoba.

Jabir ibn Aflah

Abū Muḥammad Jābir ibn Aflaḥ (Arabic: أبو محمد جابر بن أفلح‎, Latin: Geber/Gebir; 1100–1150) was an Arab Muslim astronomer and mathematician from Seville, who was active in 12th century al-Andalus. His work Iṣlāḥ al-Majisṭi (Correction of the Almagest) influenced Islamic, Jewish, and Christian astronomers.

July 15

July 15 is the 196th day of the year (197th in leap years) in the Gregorian calendar. There are 169 days remaining until the end of the year.

June 10

June 10 is the 161st day of the year (162nd in leap years) in the Gregorian calendar. There are 204 days remaining until the end of the year.

Law of sines

In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of a triangle (any shape) to the sines of its angles. According to the law,

where a, b, and c are the lengths of the sides of a triangle, and A, B, and C are the opposite angles (see the figure to the right), while d is the diameter of the triangle's circumcircle. When the last part of the equation is not used, the law is sometimes stated using the reciprocals;

The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the triangle is not uniquely determined by this data (called the ambiguous case) and the technique gives two possible values for the enclosed angle.

The law of sines is one of two trigonometric equations commonly applied to find lengths and angles in scalene triangles, with the other being the law of cosines.

The law of sines can be generalized to higher dimensions on surfaces with constant curvature.

List of Iranian mathematicians

The following is a list of Iranian mathematicians including ethnic Iranian mathematicians.

Mathematics in medieval Islam

Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important progress was made, such as the full development of the decimal place-value system to include decimal fractions, the first systematised study of algebra (named for The Compendious Book on Calculation by Completion and Balancing by scholar Al-Khwarizmi), and advances in geometry and trigonometry.Arabic works also played an important role in the transmission of mathematics to Europe during the 10th to 12th centuries.


Nishapur or Nishabur (pronunciation ; Persian: نیشابور‎, also Romanized as Nīshāpūr, Nišâpur, Nişapur, Nīshābūr, Neyshābūr, and Neeshapoor, from Middle Persian: New-Shabuhr, meaning "New City of Shapur", "Fair Shapur", or "Perfect built of Shapur") is a city in Razavi Khorasan Province, capital of the Nishapur County and former capital of Province Khorasan, in northeastern Iran, situated in a fertile plain at the foot of the Binalud Mountains. It had an estimated population of 239,185 as of 2011 and its county 433,105. Nearby are the turquoise mines that have supplied the world with turquoise for at least two millennia.

The city was founded in the 3rd century by Shapur I as a Sasanian satrapy capital. Nishapur later became the capital of Tahirid dynasty and was reformed by Abdullah Tahir in 830, and was later selected as the capital of Seljuq dynasty by Tughril in 1037. From the Abbasid era to the Mongol invasion of Khwarezmia and Eastern Iran, the city evolved into a significant cultural, commercial, and intellectual center within the Islamic world. Nishapur, along with Merv, Herat and Balkh were one of the four great cities of Greater Khorasan and one of the greatest cities in the middle ages, a seat of governmental power in eastern of caliphate, a dwelling place for diverse ethnic and religious groups, a trading stop on commercial routes from Transoxiana and China, Iraq and Egypt.

Nishapur reached the height of its prosperity under the Samanids in the 10th century, but was destroyed and the entire population slaughtered by Mongols in 1221. This massacre, combined with subsequent earthquakes and other invasions are believed to have destroyed the pottery industry the city was known for.

Poncelet–Steiner theorem

In Euclidean geometry, the Poncelet–Steiner theorem is one of several results concerning compass and straightedge constructions with additional restrictions. This result states that whatever can be constructed by straightedge and compass together can be constructed by straightedge alone, provided that a single circle and its centre are given.


Pythagoreanism originated in the 6th century BC, based on the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in Crotone, Italy. Early-Pythagorean communities lived throughout Magna Graecia. Espousing a rigorous life of the intellect and strict rules on diet, clothing and behavior comprised a cult of following Pythagorean's Code. Peculiar, the Code's diet, prohibits the consumption or even touching any sort of bean or legume. Pythagoras’ death and disputes about his teachings led to the development of two philosophical traditions within Pythagoreanism. The practitioners of akousmatikoi were superseded in the 4th century BC as a significant mendicant school of philosophy by the Cynics. The Pythagorean mathēmatikoi philosophers were in the 4th century BC absorbed into the Platonic school.

Following the political instability in the Magna Graecia, some Pythagorean philosophers fled to mainland Greece while others regrouped in Rhegium. By about 400 BC the majority of Pythagorean philosophers had left Italy. Pythagorean ideas exercised a marked influence on Plato and through him, on all of Western philosophy. Many of the surviving sources on Pythagoras originate with Aristotle and the philosophers of the Peripatetic school.

As a philosophic tradition, Pythagoreanism was revived in the 1st century BC, giving rise to Neopythagoreanism. The worship of Pythagoras continued in Italy and as a religious community Pythagoreans appear to have survived as part of, or deeply influenced, the Bacchic cults and Orphism.


The Qibla (Arabic: قِـبْـلَـة‎, "Direction", also transliterated as Qiblah, Qibleh, Kiblah, Kıble or Kibla), is the direction that should be faced when a Muslim prays during Ṣalāṫ (Arabic: صَـلَاة‎). It is fixed as the direction of the Kaaba in the Hejazi city of Mecca. Most mosques contain a wall niche that indicates the Qiblah, which is known as a miḥrâb (Arabic: مِـحْـرَاب‎). Most multifaith prayer rooms will also contain a Qibla, although usually less standardized in appearance than one would find within a mosque.Muslims all praying towards the same point is traditionally considered to symbolize the unity of the Ummah (Arabic: اُمَّـة‎, the community Muslims worldwide), under the Sharī‘ah (Arabic: شَـرِيْـعَـة‎, Law of God). The Qiblah also has importance beyond Salah, and plays a part in various ceremonies. The head of an animal that is slaughtered using Ḥalāl (Arabic: حَـلَال‎, 'Allowed') methods is usually aligned with the Qiblah. After death, Muslims are usually buried with the body at right angles to the Qibla and the face turned right towards the direction of the Qiblah.

Spherical polyhedron

In mathematics, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most conveniently derived in this way.

The most familiar spherical polyhedron is the soccer ball (outside the United States, Canada, and Australia, a football), thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the beach ball, thought of as a hosohedron.

Some "improper" polyhedra, such as the hosohedra and their duals the dihedra, exist as spherical polyhedra but have no flat-faced analogue. In the examples below, {2, 6} is a hosohedron and {6, 2} is the dual dihedron.

Torbat-e Jam

Torbat-e Jam (Persian: تربت جام‎, also Romanized as Torbat-e Jām; also known as Torbat-e Sheykh Jām and Turbat-i-Shaikh Jam) is a city and capital of Torbat-e Jam County, in Khorasan Province, Iran. At the 2016 census, its population was 100,449. Torbat-e Jam is one of the ancient cities of Greater Khorasan.

Torbat-e Jām is an ancient city. It is about 160 kilometres (99 mi) southwest of Mashhad, about 60 kilometres (37 mi) north of Taybad, and about 40 kilometres (25 mi) west of the Afghanistan border. There are many ancient places there, like the mazar (tomb) of Sheikh Ahmad Jami and Prince Qasem-e Anvar. The county includes many villages, such as Bezd, Mahmoodabad, Nilshahr.

Mathematical works
People of Khorasan
Islamic scholars
Poets and artists
Historians and
political scientists

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