AL-NASAW?, ABU ‘L-H?ASAN, ‘AL? IBN AH?MAD

(fl. Baghdad, 1029–1044)

arithmetic, geometry.

Arabic biographers do not mention al-Nasaw?, who has been known to the scholarly world since 1863, when F. Woepcke made a brief study of his al-Muqni? fi ?l-His?b al-Hind? (Leiden, MS 1021). The intro- duction to this text shows that al-Nasaw? wrote, in Persian, a book on Indian arithmetic for presentation to Magd al-Dawla, the Buwayhid ruler in Khurasan who was dethroned in 1029 or 1030. The book was presented to Sharaf al-Mul?k, Vizier of Jal?l al-Dawla, ruler in Baghdad. The vizier ordered al-Nasaw? to write in Arabic in order to be more precise and concise, and the result was al-Muqni?. Al-Nasaw? seems to have settled in Baghdad; another book by him, Tajr?d Uqlidis (Salar-Jang, MS 3142) was dedicated in highly flattering words to al-Murtad? (965–1044), an influential Shi?ite leader in Baghdad. Nothing else can be said about his life except that al-Nasaw? refers to Nas?, in Khurasan, where he probably was born.

Al-Nasaw? 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 al-Nasaw? cannot be credited with priority in this respect. The two rules were known to earlier writers on Hindu-Arabic arithmetic. The first appeared in the Pat?ganita of?r?dh?r?c?rya (750–850). Like others, al-Nasaw? 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 al-Uql?dis? (tenth century), the discoverer of decimal fractions, retained some roots in the decimal form.

In al-Muqni?, al-Nasaw? 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 K?shv?r ibn Labb?n (ca. 971–1029) had written an arithmetic for astronomers, and Ab? Hanifa al-Dinawar? (d. 895) had written one for businessmen; but K?shy?r’s proved to be rather like a business arithmetic and Ab? Han?fa’s more like a book for astronomers. K?shy?r’s work, Us?l His?b al-Hind, which is extant, shows that al-Nasaw?’s remark was unfair. He adopted K?shy?r’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 K?shy?r and al-Nasaw? would subtract 4 from 53, obtain 49, subtract 8 from 95, and so on. Only finger-reckoners agree with them in this.

In discussing subtraction of fractional quantities, al-Nasaw? enunciated the rule (n₁ + f₁) - (n₂ + f₂) = (n₁ - n₂) + (f₁ - f₂, where n₁ and n₂ are integers and f₁ and f₂ are fractions. He did not notice the case when f₂ > f₁ and the principle of “borrowing”should be used.

Al-Nasaw? gave K?shy?r’s method of extracting the cube root and, like him, used the approximation where p³ is the greatest cube in n and r = n - p³. Arabic works of about the same period used the better rule

Later works called 3p² + 3p + 1 the conventional denominator.

Al-Mugni? differs from K?shy?r’s Us?l 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 al-Nasaw? could claim no priority for these features, since others, such as al-Uql?dis?, had already done the same thing.

Three other works by al-Nasaw?, all geometrical, are extant. One of them is al-Ishb??, 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 Nas?r al-D?n al-T?s?. The last is Tajr?d Uql?dis (“An Abstract From Euclid”). In the introduction, al-Nasaw? points out that Euclid’s Elements is necessary for one who wants to study geometry for its own sake, but his Tajr?d 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 Tajr?d with the Elements, however, shows that al-Nasaw?’s work is a copy of books I-VI, on plane geometry and geometrical algebra, and book XI, on solid geometry, with some constructions omitted and some proofs altered.

BIBLIOGRAPHY

I. Original Works. Al-Nasaw?’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 al-??s? in Ma?kh?dh?t Arshim?dis, no. 10 of his Ras?il, II (Hyderabad-Deccan, 1940); al-Ishb?, 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; Kit?b al=l?mi? f? amthilat al-Zij al-j?mi? (“Illustrative Examples of the Twenty-Five Chapters of the Zij al-j?mi? of K?shy?r), in H?jj? Khal?fa, Kashf (Istanbul, 1941), col. 970; and Ris?la f? ma?rifat al-taqwim wa60357;l-asturl?b (“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 el-Nasaw?,”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 Kushy?r ibn Labb?n, Us?l His?b al-Hind, in M. Levey and M. Petruck, Principles of Hindu Reckoning (Madison, Wis., 1965), 55–83.

A. S. Saidan