**Abu Nasri Mansur ibn Ali ibn Iraq** (Persian: أبو نصر منصور بن علی بن عراق; c. 960 – 1036) was a Persian^{[1]} Muslim mathematician and astronomer. He is well known for his work with the spherical sine law.^{[2]}^{[3]}

Abu Nasr Mansur was born in Gilan, Persia, to the ruling family of Khwarezm, the Afrighids.^{[4]} He was thus a prince within the political sphere. He was a student of Abu’l-Wafa and a teacher of and also an important colleague of the mathematician, Al-Biruni. Together, they were responsible for great discoveries in mathematics and dedicated many works to one another.

Most of Abu Nasri’s work focused on math, but some of his writings were on astronomy. In mathematics, he had many important writings on trigonometry, which were developed from the writings of Ptolemy. He also preserved the writings of Menelaus of Alexandria and reworked many of the Greeks theorems.

He died in the Ghaznavid Empire (modern-day Afghanistan) near the city of Ghazna.

**Abu Nasr Mansur** was a native of Gilan which is mentioned in *The Regions of the World*, a Persian geography book of 982. His family, the Banu Iraq, were rulers of Khwarazm the region adjoining the Aral Sea, and it was in this region that Abu Nasr Mansur studied and became a disciple of Abu’l-Wafa. Abu Nasr Mansur was teaching in this area when he first began his association with al-Biruni whom he taught from about 990. This began an important collaboration which was to go on for many years.

The end of the 10^{th} century and beginning of the 11^{th} century was a period of great unrest in the Islamic world and there were civil wars in the region in which Abu Nasr Mansur was living. Khwarazm was at this time part of the Samanid Empire which ruled from Bukhara. Other states in the region were the Ziyarid state with its capital at Gurgan on the Caspian sea. Further west the Buwayhid dynasty ruled over the area between the Caspian sea and the Persian Gulf, and over Mesopotamia. Another kingdom which was rapidly rising in influence was the Ghaznavids whose capital was at Ghazna in Afganistan.

In 995 the Banu Iraq, of which Abu Nasr Mansur was a prince, was overthrown in a coup. It is not clear what happened to Abu Nasr Mansur at this stage but certainly his pupil al-Birunifled at the outbreak of the civil war. It seems that Abu Nasr Mansur was, some little time later, employed at the court of Ali ibn Ma’mun and remained at the court when his brother Abu’l Abbas Ma’mun succeeded him. Both these brother married sisters of the ruler Mahmud from the powerful state at Ghazna which would eventually take control of Ma’mun’s kingdom.

Both Ali ibn Ma’mun and Abu’l Abbas Ma’mun were patrons of the sciences and supported a number of top scientists at their court. Not only did Abu Nasr Mansur work there but from about 1004 al-Biruni also worked there, renewing the collaboration between him and his teacher. The wars in the region, however, were to disrupt the scientific work of Abu Nasr Mansur and eventually he and al-Biruni left in about 1017. Mahmud was extending his influence over the region from his base in Ghazna and made a demand of Abu’l Abbas Ma’mun in 1014 to have his name inserted into the Friday prayers. This was a signal that he wanted an end to Ma’mun’s rule and for the region to come under his control. After Ma’mun had at least partially agreeded to Mahmud’s demands, he was killed by his own army for what they considered to be an act of treachery. Following this Mahmud marched his army into the region and gained control. Both Abu Nasr Mansur and al-Biruni seem to have left with the victorious Mahmud. It seems that Abu Nasr Mansur spent most of the rest of his life at Mahmud’s court in Ghazna.

Abu Nasr Mansur is perhaps most famous for his collaboration with al-Biruni. Certainly Abu Nasr Mansur worked on many topics as a result of requests from al-Biruni and a total of twenty-five works are known to have been written by him. It is possible that in this list of twenty-five two names of works that have come down to us are different titles for the same work while another title may refer to a fragment of a larger work (in which case there are only twenty-three). Seventeen works have survived and they show that Abu Nasr Mansur was an extremely able astronomer and mathematician. Of Abu Nasr Mansur’s works seven are on mathematics, the rest are on astronomy. All the surviving works have been published, most have been translated into at least one European language, and this gives some indication of the importance attached to his work.

Many of Abu Nasr Mansur’s works were dedicated to his student al-Biruni. In fact al-Biruni himself lists twelve works which he says Abu Nasr Mansur dedicated to him (although some historians read al-Biruni’s words as meaning that he wrote the works himself, but this interpretation seems highly unlikely). The first such work which Abu Nasr Mansur dedicated to al-Biruni was written around 997, soon after the civil war had disrupted their work. In his own writings al-Biruni sometimes quotes results due to Abu Nasr Mansur which he says he worked on at al-Biruni’s request. Certainly both men seem keen to give full credit to the other’s contributions.

Abu Nasr Mansur’s main achievements are his commentry on the *Spherics* of Menelaus, his role in the development of trigonometry from Ptolemy’s calculation with chords towards the trigonometric functions used today, and his development of a set of tables which give easy numerical solutions to typical problems of spherical astronomy.

Abu Nasr Mansur’s reworking of the *Spherics* of Menelaus is particularly important since the Greek original of Menelaus work has been lost, although there are several Arabic versions. Menelaus’s work formed the basis for Ptolemy’s numerical solutions of spherical astronomy problems in the *Almagest* Ⓣ. The work is in three books: the first book studies properties of spherical triangles, the second book investigates properties of systems of parallel circles on a sphere as they intersect great circles, while the third book gives a proof of Menelaus’s theorem.

In his work on trigonometry Abu Nasr Mansur discovered the sine law

*a*/sin

*A*=

*b*/sin

*B*=

*c*/sin

*C*.

Abu’l-Wafa may have discovered this law first and Abu Nasr Mansur may have learnt it from him. Certainly which of the two has priority is hard to determine and will almost certainly never be known with certainty. A third person who is sometimes credited with this same discovery is al-Khujandi but it seems less likely that he was the discoverer since, as Samso writes in [1]:-

… he was essentially a practical astronomer, unconcerned with theoretical problems.

The claims of al-Khujandi to be the discoverer of the sine law are further discussed in his biography.

The article [4] is a description and study of *The table of minutes* of Abu Nasr Mansur. The author of [4] traces the origins of the work to the 10^{th} century *Damascene tables* by Habash. Abu Nasr Mansur’s treatise discusses the five trigonometric functions which are used to solve problems in spherical astronomy. The article shows the improvement achieved by Abu Nasr Mansur in using 1 as the value of the radius, instead of 60 as was done by most Arabic astronomers.

Other work by Abu Nasr Mansur on astronomical topics included four works on the construction and application of the astrolabe. His proof of the sine law appears a number of times in his works, for example in *Almagest of the Shah, Book of the azimuth, Treatise on the determination of spherical arcs*, and *Treatise in which some geometrical questions addressed to him are answered*. The questions referred to in the title of this last work were addressed to him by al-Biruni. Others works include *Treatise in which a difficulty in the thirteenth book of the Elements is solved* and *A chapter from a book on the sphericity of the heavens*.