Background
Abu Nasr Mansur was born around 960. He was probably a native of Gilan, Persia. It is likely that he belonged to the family of Banu ‘Iraq who ruled Khwarizm until it fell to the Ma’muni dynasty in 995.
Abu Nasr Mansur was born around 960. He was probably a native of Gilan, Persia. It is likely that he belonged to the family of Banu ‘Iraq who ruled Khwarizm until it fell to the Ma’muni dynasty in 995.
Abu Nasr Mansur was a disciple of Abu’l Wafa’ al-Buzjani.
Abu Nasr Mansur was the teacher of al-Biruni. Abu Nasr passed most of his life in the court of the monarchs ‘All ibn Ma’mun and Abu’l-‘Abbas Ma’mun, who extended their patronage to a number of scientists, including al-Biruni and Ibn Sina. About 1016, the year in which Abu’l-lAbbas Ma’mun died, both Abu Nasr Mansur and al-Biruni left Khwarizm and went to the court of Sultan Mahmud al-Ghaznawi in Ghazna, where Abu Nasr spent the rest of his life.
Abu Nasr’s fame is due in large part to his collaboration with al-Biruni. Although this collaboration is generally considered to have begun in about 1008, the year in which al-Biruni returned to Khwarizm from the court of Jurjan, there is ample evidence for an earlier date. For example, in his Al-Athar al-baqiya (“Chronology”), finished in the year 1000, al-Biruni refers to Abu Nasr as Ustadhi - “my master,” while Abu Nasr dedicated his book on the azimuth, written sometime before 998, to his pupil.
This collaboration also presents grave difficulties in assigning the authorship of specific works. A case in point is some twelve works that al-Biruni lists as being written “in my name” (bismi), a phrase that has led scholars to consider them to be of his own composition. Nallino has, however, pointed out that bisml might also mean “addressed to me” or “dedicated to me by Abu Nasr - and there is considerable evidence in support of this interpretation. For instance, the phrase is used in this sense in both medieval texts (the Mafatih al-'ulum of Muhammad ibn Ahmad al-Khwarizmi of 977) and modern ones of which there is no doubt of the authorship. The incipits and explicits of the works in question make it clear, moreover, that they were written by Abu Nasr in response to al-Biruni’s request for solutions to specific problems that had arisen in the course of his more general researches. Indeed, in some of al-Biruni’s own books he mentioned Abu Nasr by name and stated that his book incorporates the results of some investigations that the older man carried out at his request. Al-Biruni gave Abu Nasr full credit for his discoveries - as, indeed, he gave full credit to each of his several collaborators, including Abu Sahl al-Masihi, a certain Abu ‘Ali al-Hasan ibn al-Jili (otherwise unidentified) and Ibn Sina, who wrote answers to philosophical questions submitted to him by al-Biruni.
Abu Nasr Mansur 's work on the Spherics is particularly noteworthy because the original Greek manuscript has been lost. In addition, Mansur contributed to the development of trigonometry by discovering the sine law a/sin A = b/sin B = c/sin C. He is also famous for the scientific upbringing of al-Biruni.
The extent of the collaboration between Abu Nasr and al-Biruni may be demonstrated by the latter’s work on the determination of the obliquity of the ecliptic. Al-Biruni carried out observations in Khwarizm in 997, and in Ghazna in 1016, 1019, and 1020. Employing the classical method of measuring the meridian height of the sun at the time of the solstices, he computed the angle of inclination as 23°35'. On the other hand, however, al-Biruni became acquainted with a work by Muhammad ibn al-Sabbah, in which the latter described a method for determining the position, ortive amplitude, and maximum declination of the sun. Since al-Biruni’s copy was full of apparent errors, he gave it to Abu Nasr and asked him to correct it and to prepare a critical report of Ibn al-Sabbah’s techniques.
Abu Nasr thus came to write his Risdla fi 'l-barahin 'ala lamal Muhammad ibn al-Sabbah (“A Treatise on the Demonstration of the Construction Devised by Muhammad Ibn al-Sabbah”), in which he took up Ibn al-Sabbah’s method in detail and demonstrated that it must be in error to the extent that it depended on the hypothesis of the uniform movement of the sun on the ecliptic. According to Ibn al-Sabbah, the ortive amplitude of the sun at the solstice (a,) may be obtained by making three observations of the solar ortive amplitude (a1, a2, a3) at thirty-day intervals within a single season of the year.
Al-Biruni then took up Abu Nasr’s clarification of Ibn al-Sabbah’s work, citing it in his own Al-Qanun al-Mas'udi and Tahdid. He remained, however, primarily interested in obtaining the angle of inclination, and simplified Ibn al-Sabbah’s methods to that end. He thus, within the two formulas, substituted three and two, respectively, observations of the declination of the sun for the three and two observations of solar ortive amplitude. By this method, he obtained values for the angle of inclination of 23°25T9" and 23°24'16", respectively. These values are clearly at odds with that then commonly held (23°35') and confirmed by al-Biruni’s own observations. Al-Biruni then returned to Abu Nasr’s work and explained the discrepancy as being due to Ibn al-Sabbah’s supposition of the uniform motion of the sun on the ecliptic, as well as to the continuous use of sines and square roots.
Abu Nasr’s contributions to trigonometry are more direct. He is one of the three authors (the others being Abu’l Wafa’ and Abu Mahmud al-Khujandi) to whom al-Tusi attributed the discovery of the sine law.
The question of which of these three mathematicians was actually the first to discover this law remains unresolved, however. Luckey has convincingly argued against al-Khujandi, pointing out that he was essentially a practical astronomer, unconcerned with theoretical problems. Both Abu Nasr and Abu’l Wafa’, on the other hand, claimed discovery of the law, and while it is impossible to determine who has the better right, two considerations would seem to corroborate Abu Nasr’s contention. First, he employed the law a number of times throughout his astronomical and geometrical writings; whether or not it was his own finding, he nevertheless dealt with it as a significant novelty. Second, Abu Nasr treated the demonstration of this law in two of his most important works, the Al-Majisti al-Shalii (“Almagest of the Shah") and the Kitab fi 'l-sumut (“Book of the Azimuth"), as well as in two lesser ones, Risala fi ma'rifat al-qisiyy al-fala- kiyya (“Treatise on the Determination of Spherical Arcs”) and Risala fi 'l-jawab 'an masa’il handasiyya su'ila*anha (“Treatise in Which Some Geometrical Questions Addressed to Him are Answered").
The Al-Majisti al-Shahi and the Kitab ft 'l-sumut have both been lost. It is known that the latter was written at the request of al-Biruni, as well as dedicated to him and that it was concerned with various procedures for calculating the direction of the qibla. Abu Nasr’s other significant work, the most complete Arabic version of the Spherics of Menelaus, is, however, still extant (although the original Greek text is lost). Of the twenty-two works that are known to have been written by Abu Nasr, a total of seventeen remain, of which sixteen have been published.
In addition to the books cited above, the remainder of Abu Nasr’s work consisted of short monographs on specific problems of geometry or astronomy. These lesser writings include Risala fi hall shubha 'aradat fi 'l-thalitha'ashar min Kitab al-Usul (“Treatise in Which a Difficulty in the Thirteenth Book of the Elements is Solved”); Maqdlafi ¡slab shakl min kitab Mana!incus fi ’l-kuriyyat 'adala fihi musallihu hadha ’l-kitdb (“On the Correction of a Proposition in the Spherics of Menelaus, in Which the Emendators of This Book Have Erred"); Risala fisan'at al-asturlab bi 'l-tariq al- sind'i (“Treatise on the Construction of the Astrolabe in the Artisan’s Manner”); Risala fi 'l-asturlab al-sartani al-muyannah fi haqiqatihi bi 'l-tariq al-sina'i (“Treatise on the True Winged Crab Astrolabe, According to the Artisan’s Method”); and Fas! min kitab fi kuriyyat al-sama’ (“A Chapter From a Book on the Sphericity of the Heavens”).
There is no information on whether Abu Nasr Mansur was ever married or had any children.
