AlNasawī, Abu
ALNASAWī, ABU ‘LḤASAN, ‘ALī IBN AḤMAD
(fl. Baghdad, 1029–1044)
arithmetic, geometry.
Arabic biographers do not mention alNasawī, who has been known to the scholarly world since 1863, when F. Woepcke made a brief study of his alMuqni͑ fi ͗lHisāb alHindī (Leiden, MS 1021). The intro duction to this text shows that alNasawī wrote, in Persian, a book on Indian arithmetic for presentation to Magd alDawla, the Buwayhid ruler in Khurasan who was dethroned in 1029 or 1030. The book was presented to Sharaf alMulūk, Vizier of Jalāl alDawla, ruler in Baghdad. The vizier ordered alNasawī to write in Arabic in order to be more precise and concise, and the result was alMuqni͑. AlNasawī seems to have settled in Baghdad; another book by him, Tajrīd Uqlidis (SalarJang, MS 3142) was dedicated in highly flattering words to alMurtadā (965–1044), an influential Shi͑ite leader in Baghdad. Nothing else can be said about his life except that alNasawī refers to Nasā, in Khurasan, where he probably was born.
AlNasawī has been considered a forerunner in the use of the decimal concept because he used the rules and where k is taken as a power of 10. If K is taken as 10 or 100, the root is found correct to one or two decimal places. There is now reason to believe that alNasawī cannot be credited with priority in this respect. The two rules were known to earlier writers on HinduArabic arithmetic. The first appeared in the Patīganita ofŚrīdhārācārya (750–850). Like others, alNasawī rather mechanically converted the decimal part of the root thus obtained to the sexagesimal scale and suggested taking K as a power of sixty, without showing signs of understanding the decimal value of the fraction. Their concern was simply to transform the fractional part of the root to minutes, seconds, and thirds. Only alUqlīdisī (tenth century), the discoverer of decimal fractions, retained some roots in the decimal form.
In alMuqni͑, alNasawī presents Indian arithmetic of integers and common fractions and applies its schemes to the sexagesimal scale. In the introduction he criticizes earlier works as too brief or too long. He states that Kūshvār ibn Labbān (ca. 971–1029) had written an arithmetic for astronomers, and Abū Hanifa alDinawarī (d. 895) had written one for businessmen; but Kūshyār’s proved to be rather like a business arithmetic and Abū Hanīfa’s more like a book for astronomers. Kūshyār’s work, Usūl Hisāb alHind, which is extant, shows that alNasawī’s remark was unfair. He adopted Kūshyār’s schemes on integers and, like him, failed to understand the principle of “borrowing”in subtraction. To subtract 4,859 from 53,536, the Indian scheme goes as follows: Arrange the two numbers as 53536 4859.
Subtract 4 from the digit above it; since 3 is less than 4, borrow 1 from 5, to turn 3 into 13, and subtract. And so on. Both Kūshyār and alNasawī would subtract 4 from 53, obtain 49, subtract 8 from 95, and so on. Only fingerreckoners agree with them in this.
In discussing subtraction of fractional quantities, alNasawī enunciated the rule (n_{1} + f_{1})  (n_{2} + f_{2}) = (n_{1}  n_{2}) + (f_{1}  f_{2}, where n_{1} and n_{2} are integers and f_{1} and f_{2} are fractions. He did not notice the case when f_{2} > f_{1} and the principle of “borrowing”should be used.
AlNasawī gave Kūshyār’s method of extracting the cube root and, like him, used the approximation where p^{3} is the greatest cube in n and r = n  p^{3}. Arabic works of about the same period used the better rule
Later works called 3p^{2} + 3p + 1 the conventional denominator.
AlMugni͑ differs from Kūshyār’s Usūl in that it explains the Indian system of common fractions, expresses the sexagesimal scale in Indian numerals, and applies the Indian schemes of operation to numbers expressed in this scale. But alNasawī could claim no priority for these features, since others, such as alUqlīdisī, had already done the same thing.
Three other works by alNasawī, all geometrical, are extant. One of them is alIshbā͑, in which he discusses the theorem of Menelaus. One is a corrected version of Archimedes’Lemmata as translated into Arabic by Thahit ibn Qurra, which was later revised by Nasīr alDīn alTūsī. The last is Tajrīd Uqlīdis (“An Abstract From Euclid”). In the introduction, alNasawī points out that Euclid’s Elements is necessary for one who wants to study geometry for its own sake, but his Tajrīd is written to serve two purposes: it will be enough for those who want to learn geometry in order to be able to understand Ptolemy’s Almagest, and it will serve as an introduction to Euclid’s Elements. A comparison of the Tajrīd with the Elements, however, shows that alNasawī’s work is a copy of books IVI, on plane geometry and geometrical algebra, and book XI, on solid geometry, with some constructions omitted and some proofs altered.
BIBLIOGRAPHY
I. Original Works. AlNasawī’s writings include “On the Construction of a Circle That Bears a Given Ratio to Another Given Circle, and on the Construction of All Rectilinear Figures and the Way in Which Artisans Use Them,”cited by alṮūsī in Ma͗khūdhāt Arshimīdis, no. 10 of his Rasāil, II (HyderabadDeccan, 1940); alIshb͑, trans. by E. Wiedemann in his Studien zur Astronomie der Araber (Erlangen, 1926), 80–85—see also H. Burger and K. Kohl, Geschichte des Transversalensätze (Erlangen, 1924), 53–55; Kitāb al=lāmi͑ fī amthilat alZij aljāmi͑ (“Illustrative Examples of the TwentyFive Chapters of the Zij aljāmi͑ of Kūshyār), in Hājjī Khalīfa, Kashf (Istanbul, 1941), col. 970; and Risāla fī ma͑rifat altaqwim wa60357;lasturlāb (“A Treatise on Chronology and the Astrolabe”), Columbia University Library, MS Or. 45, op. 7.
II. Secondary Literature. See H. Suter, “Über des Rechenbuch des Ali ben Ahmed elNasawī,”in Bibliotheca mathematica, 2nd ser., 7 (1906), 113–119; and F. Woepcke, “Mémoires sur la propagation des chiffres indiens,”in Journal asiatique, 6th ser., 1 (1863), 492 ff.
See also Kushyār ibn Labbān, Usūl Hisāb alHind, in M. Levey and M. Petruck, Principles of Hindu Reckoning (Madison, Wis., 1965), 55–83.
A. S. Saidan
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