**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.

## 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]}

- {\displaystyle \sin(\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta }
- {\displaystyle \sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)}
- {\displaystyle \cos(2a)=1-2\sin ^{2}(a)}
- {\displaystyle \sin(2a)=2\sin(a)\cos(a)}

He also discovered the law of sines for spherical triangles:

- {\displaystyle {\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.

### Born: 10 June 940 in Buzjan (near Jam), Khorasan region (now in Iran)

Died: 15 July 998 in Baghdad (now in Iraq)

**Abu’l-Wafa** was brought up during the period that a new dynasty was being established which would rule over Iran. The Buyid Islamic dynasty ruled in western Iran and Iraq from 945 to 1055 in the period between the Arab and Turkish conquests. The period began in 945 when Ahmad Buyeh occupied the ‘Abbasid capital of Baghdad. The high point of the Buyid dynasty was during the reign of ‘Adud ad-Dawlah from 949 to 983. He ruled from Baghdad over all southern Iran and most of what is now Iraq. A great patron of science and the arts, ‘Adud ad-Dawlah supported a number of mathematicians and Abu’l-Wafa moved to ‘Adud ad-Dawlah’s court in Baghdad in 959. Abu’l-Wafa was not the only distinguished scientist at the Caliph’s court in Baghdad, for outstanding mathematicians such as al-Quhi and al-Sijzi also worked there.

Sharaf ad-Dawlah was ‘Adud ad-Dawlah’s son and he became Caliph in 983. He continued to support mathematics and astronomy and Abu’l-Wafa and al-Quhi remained at the court in Baghdad working for the new Caliph. Sharaf ad-Dawlah required an observatory to be set up, and it was built in the garden of the palace in Baghdad. The observatory was officially opened in June 988 with a number of famous scientists present such as al-Quhi and Abu’l-Wafa.

The instruments in the observatory included a quadrant over 6 metres long and a stone sextant of 18 metres. Abu’l-Wafa is said to have been the first to build a wall quadrant to observe the stars. However, the caliph Sharaf ad-Dawlah died in the following year and the observatory was closed.

Like many scientist of his period, Abu’l-Wafa translated and wrote commentaries, which have since been lost, on the works of Euclid, Diophantus and al-Khwarizmi. Some time between 961 and 976 he wrote *Kitab fi ma yahtaj ilayh al-kuttab wa’l-ummal min ‘ilm al-hisab* Ⓣ. In the introduction to this book Abu’l-Wafa writes that it ([3] or [4]):-

… comprises all that an experienced or novice, subordinate or chief in arithmetic needs to know, the art of civil servants, the employment of land taxes and all kinds of business needed in administrations, proportions, multiplication, division, measurements, land taxes, distribution, exchange and all other practices used by various categories of men for doing business and which are useful to them in their daily life.

It is interesting that during this period there were two types of arithmetic books written, those using Indian symbols and those of finger-reckoning type. Abu’l-Wafa’s text is of this second type with no numerals; all the numbers are written in words and all calculations are performed mentally. Early historians such as Moritz Cantor believed that there were opposing schools of authors, one committed to Indian methods, the other to Greek methods. However, this has since been disproved (see for example [9]), and it is now believed that mathematicians wrote for two differing types of readers. Abu’l-Wafa himself was an expert in the use of Indian numerals but these [1]:-

… did not find application in business circles and among the population of the Eastern Caliphate for a long time.

Hence he wrote his text using finger-reckoning arithmetic since this was the system used for by the business community. The work is in seven parts, each part containing seven chapters

^{1}/

_{2},

^{1}/

_{3},

^{1}/

_{4}, … ,

^{1}/

_{10}).Part II: On multiplication and division (arithmetical operations with integers and fractions).

Part III: Mensuration (area of figures, volume of solids and finding distances).

Part IV: On taxes (different kinds of taxes and problems of tax calculations).

Part V: On exchange and shares (types of crops, and problems relating to their value and exchange).

Part VI: Miscellaneous topics (units of money, payment of soldiers, the granting and withholding of permits for ships on the river, merchants on the roads).

Part VII: Further business topics.

This work is studied in detail in [12] (see also [10]). Of particular interest is the reference to negative numbers in Part II of Abu’l-Wafa’s treatise, and this particular aspect is studied in detail in [11] and [12] (see also [1]). This seems to be the only place that negative numbers have been found in medieval Arabic mathematics. Abu’l-Wafa gives a general rule and gives a special case of this where subtraction of 5 from 3 gives a “debt” of 2. He then multiples this by 10 to obtain a “debt” of 20, which when added to (10 – 3)(10 – 5) = 35 gives the product of 3 and 5, namely 15.

Another text written by Abu’l-Wafa for practical use was *A book on those geometric constructions which are necessary for a craftsman*. This was written much later than his arithmetic text, certainly after 990. The book is in thirteen chapters and it considered the design and testing of drafting instruments, the construction of right angles, approximate angle trisections, constructions of parabolas, regular polygons and methods of inscribing them in and circumscribing them about given circles, inscribing of various polygons in given polygons, the division of figures such as plane polygons, and the division of spherical surfaces into regular spherical polygons.

Another interesting aspect of this particular work of Abu’l-Wafa’s is that he tries where possible to solve his problems with ruler and compass constructions. When this is not possible he uses approximate methods. However, there are a whole collection of problems which he solves using a ruler and fixed compass, that is one where the angle between the legs of the compass is fixed. It is suggested in [1] that:-

Interest in these constructions was probably aroused by the fact that in practice they give more exact results than can be obtained by changing the compass opening.

Abu’l-Wafa is best known for the first use of the tan function and compiling tables of sines and tangents at 15′ intervals. This work was done as part of an investigation into the orbit of the Moon, written down in *Theories of the Moon*. He also introduced the sec and cosec and studied the interrelations between the six trigonometric lines associated with an arc.

Abu’l-Wafa devised a new method of calculating sine tables. His trigonometric tables are accurate to 8 decimal places (converted to decimal notation) while Ptolemy’s were only accurate to 3 places.

His other works include *Kitab al-Kamil* Ⓣ, a simplified version of Ptolemy’s *Almagest* Ⓣ. Although there seems to have been little of novel theoretical interest in this work, the observational data in it seem to have been used by many later astronomers.