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MANS??R IBN ‘AL? IBN ‘IR?Q, AB? NAS?R

(fl.. Knowarizm [now Kara-Kalpakskaya, A.S.S.R.]; d. Ghazna [?] [now Ghazni, Afghanistan], ca. 1036) mathematics, astronomy.

Ab? Nas?r was probably a native of G?l?n (Persia); it is likely that he belonged to the family of Ban? ‘Ir?q who ruled Khwarizm until it fell to the Ma’m?n? dynasty in a.d. 995. He was a disciple of Abu’l Waf?’ al-B?zj?n? and the teacher of al-B?r?n?. Ab? Na?sr passed most of his life in the court of the monarchs ‘Al? ibn Ma’m?n and Abu’l-‘Abb?s Ma’m?n, who extended their patronage to a number of scientists, including al-B?r?n? and Ibn S?n?. About 1016, the year in which Abu’l-‘Abb?s Ma’m?n, died, both Ab? Na?sr and al-B?r?n? left Khwarizm and went to the court of Sultan Mahm?d al-Ghaznaw? in Ghazna, where Ab? Na?sr spent the rest of his life.

Ab? Na?sr’s fame is due in large part to his collaboration with al-B?r?n?. Although this collaboration is generally considered to have begun in about 1008, the year in which al-B?r?n? returned to Khwarizm from the court of Jurj?n (now Kunya-Urgench, Turkmen S.S.R.), there is ample evidence for an earlier date. For example, in his Al-?th?r al-b?qiya (“Chronology”), finished in the year 1000, al-Bir?n? refers to Ab? Na?sr as Ust?dhi—“my master,” while Ab? Na?sr 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-B?r?n? lists as being written “in my name” (bism?), a phrase that has led scholars to consider them to be of his own composition. Nallino has, however, pointed out that bism? might also mean “addressed to me” or “dedicated to me”—by Ab? Nas?r—and there is considerable evidence in support of this interpretation. For instance, the phrase is used in this sense in both medieval texts (the Maf?t?? al-’ul?m of Mu?ammad ibn A?mad al-Khw?rizm? 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 Ab? Nas?r in reponse to al-b?r?n?’s request for solutions to specific problems that had arisen in the course of his more general researches, indeed, in some of al-B?r?n?l’s own books he mentioned Ab? Nas?r by name and stated that his book incorporates the results of some investigations that the older man carried out at his request. Al-B?r?n? gave Ab? Nas?r full credit for his discoveries—as, indeed, he gave full credit to each of his several collaborators, including Ab? Sahl al-Masi?h?, a certain Ab? ‘Ali al-H?asan ibn al-J?l? (otherwise unidentified) and Ibn Sin?, who wrote answers to philosophical questions submitted to him by al-Bir?n?.

The extent of the collaboration between Ab? Nas?r and al-Bir?n? may be demonstrated by the latter’s work on the determination of the obliquity of the ecliptic. Al-B?r?n? 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-B?r?n? became acquainted with a work by Muhammad ibn al-S?abb??, in which the latter described a method for determining the position, ortive amplitude, and maximum declination of the sun. Since al-B?r?ni’s copy was full of apparent errors, he gave it to Ab? Nas?r and asked him to correct it and to prepare a critical report of Ibn al-S?abb??’s techniques.

Ab? Nas?r thus came to write his Ris?la fi ’l-bar?hin ‘al? ‘amal Muhammad ibn al-S?abb?? (“A Treatise on the Demonstration of the Construction Devised by Mu?ammad Ibn l-S?abb??”), in which he took up Ibn al-Sabb?h’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-S?abb??, the ortive amplitude of the sun at solstice (a₁) 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. He thus reached the formula:

The same result may also be obtained from only two observations (a₁, a₂) if the distance (d) covered by the sun on the eliptic over the period between the two observations is known:

The value of a_t is thus extractable in two ways, and the value of the maximum declination can then be discovered by applying the formula of al-Batt?n? and H?abash:

Al-B?r?n? then took up Ab? Nas?r’s clarification of Ibn al-S?abb??’ work, citing it in his own Al-Q?n?n al-Mas’?d? and Ta?d?d. He remained, however, primarily interested in obtaining the angle of inclination, and simplified Ibn al-S?abb??’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°25'19" and 23°24’16", respectively. These values are clearly at odds with that then commonly held (23°35') and confirmed by al-B?r?n?’s own observations. Al-B?r?n? then returned to Ab? Nas?r’s work, and explained the discrepancy as being due to Ibn al-S?abb??’s supposition of the uniform motion of the sun on the ecliptic, as well as to the continuous use of sines and square roots.

Ab? Nas?r’s contributions to trigonometry are more direct. He is one of the three authors (the others being Abu’l Waf?’ and Ab? Mahm?d al-Khujand?) to whom al-T?s? attributed the discovery of the sine law whereby in a spherical triangle the sines of the sides are in relationship to the sines of the opposite angles as

or, in a plane triangle, the sides are in relationship to the sines of the opposite angles as

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-Khujand?, pointing out that he was essentially a practical astronomer, unconcerned with theoretical problems. Both Ab? Nas?r and Abu’l Waf?’, 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 Ab? Nas?r’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, Ab? Nas?r treated the demonstration of this law in two of his most important works, the Al-Majist?? al-Sh?h? (“Almagest of the Shah”) and the Kit?b fi ’l-sum?t (“Book of the Azimuth”), as well as in two lesser ones, Ris?la fi ma’rifat al-qisiyy al-fala-kiyya (“Treatise on the Determination of Spherical Arcs”) and Ris?la fi ’l-jaw?b ‘an mas?’ il handasiyya su? ila anh? (“Treatise in Which Some Geometrical Questions Addressed to Him are Answered”).

The Al-Majist?? al-Sh?h? and the Kit?b fi ’l-sum?t have both been lost. It is known that the latter was written at the request of al-B?r?n?, as well as dedicated to him, and that it was concerned with various procedures for calculating the direction of the qibla. Ab? Nas?r’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 Ab? Nas?r, a total of seventeen remain, of which sixteen have been published.

In addition to the books cited above, the remainder of Ab? Nas?r’s work consisted of short monographs on specific problems of geometry or astronomy. These lesser writings include Ris?la fi ?all shubha ‘arad?at fi ’l-th?litha ‘ashar min Kit?b al-Us??l (“Treatise in Which a Difficulty in the Thirteenth Book of the Elements is Solved”); Maq?la fi is?l?? shakl min kit?b M?n?l?wus fi ’-kuriyy?t ’adala fihi mu?salli?h? h?dha ’l-kit?b (“On the Correction of a Proposition in the Spherics of Menelaus, in Which the Emendators of This Book Have Erred”); Ris?la fi s?an’at al-ast?url?b bi ’l-t?ariq al-s?in?’i (“Treatise on the Construction of the Astrolabe in the Artisan’s Manner”);Ris?la fi ’l-asturl?n? al-mu?annath fi haq?qatihi bi ’l-asturl?b al-sin?‘i(“Treatise on the True Winged Crab Astrolabe, According to the Artisan’s Method”); and Fasl min kit?b fi kuriyyat al-sam?’ (“A Chapter From a Book on the Sphericity of the Heavens”).

BIBLIOGRAPHY

I. Original Works. Ab? Nas?r’s version of the Spherics of Menelaus exists in an excellent critical edition, with German trans., by Max Krause, “Die Sphärik von Menelaos aus Alexandrien in der Verbesserung von Ab? Nas?r Man?sur ibn ’Ali ibn ’Ir?q. Mit Untersuchungen zur Geschichte des Textes bei den islamischen Mathematikern,” in Abhandiungen der K. Gesellschaft der Wissenschaften zu Göttingen, Phil.-hist. Kl., no. 17 (Berlin, 1936). Most of the rest of his extant work has been badly edited as Ras?’il Ab? Nas?r Mans?r il?’-Bir?ni. D?’irat al-Ma’?rif al-’Uthm?niyya (Hyderabad, 1948); six of the same treatises are trans. into Spanish in Julio samó, Estudios sobre Ab? Nas?r Mans?r (Barcelona, 1969).

II. Secondary Literature. On Ab? Nas?r and his work, see D. J. Boilot, “L’oceuvre d’al-Baruni:essai bibliographique,” in Mélanges de l’Institut dominicain d’éudes orientales, 2 (1955), 161–256; “Bibliographie d’al-Beruni. Corrigenda et addenda,” ibid,., 3 (1956), 391–396; E. S. kennedy and H. Sharkas, “Two Medieval Methods for Determining the Obliquity of the Ecliptic,” in Mathematical Teacher, 55 (1962), 286–290; julio Samsó, Estudios sobre Ab? Nas?r Mans?r B. ’ Ali b. ’Ir?q (Barcelona, 1969); “Contribution a un an´lisis de la terminologia matem´ticoastronómica de Ab? Nas?r Mans?r b. ’Ali b. ’Ir?q,” in Pensamiento, 25 (1969), 235–248; Paul Luckey, “Zur Entstehung der Kugeldreiectsreshnung,” in Deutsche Mathematik, 5 (1940–1941), 405–446; Muhammad Shafi, “Ab? Nas?r ibn ’r?q aur us k? sanhah waf?t” (“Ab? Nas?r ibn ’ir?q and the Date of his Death”), in Urdu with English summary, in 60 ?gum münasebetyle Zeki Velidi Togan’a arma?gan. Symbolae in honorem Z. V. Togan (Istanbul, 1954–1955), 484–492; Heinrich Suter, “Zur Trigonometrie der Araber,” in Bibliotheca Mathematica, 3rd ser., X (1910), 156–160; and K. Vogel and Max Krause, “Die Sphärik von Menelaus aus Alexandrien in der Verbesserung von Ab? Nas?r b. ’Ali ibn ’Ir?q,” in Gnomon, 15 (1939), 343–395.