Ab? K
Ab? K?mil Shuj??Ibn Aslam Ibn Mu?ammad Ibn Shuj??
(b. ca. 850, d. ca. 930), mathematics.
Often called al-??sib al-Mi?r? (“the reckoner of Egypt”) Ab? K?mil was one of Islam’s greatest algebraists in the period following the earliest Muslim algebraist, al-Khw?rizm? (fl. ca. 825). In the Arab world this was a period of intellectual ferment, paric ularl in mathematics and the sciences
There is virtually no biographical material available on Ab? K?mil. He is first mentioned by al-Nad?m in a bibliographical work, The Fihrist (987), where he listed with other mathematicians under “The New Reckoners and Arithmeticians,” which refers to those mathematicians who concerned themselves with the practical algorisms, citizens’ arithmetic, and practical geometry (see Bibliography). Ibn khald?n (1322–1406) state that Ab? K?mil wrote his algebra after the first such work by al-Khw?rizm?, and ?ajj? Khal?fa (1608–1658) attributed to him a work supposedly concerned with algebraic solutions of inheritance problems.
Among the works of Ab? K?mil extant in manuscripts is the Kit?b al-?ar??if fi’l-?is?b (“Book of Rare Things in the Art of Calculation”). According to H. Suter1 this text is concerned with integral solutions of some indeterminate equations; much earlier, Diophantus (ca. first century a.d.) had concerned himself with rational, not exclusively integral, solutions. Ab? K?mil’s solutions are found by an ordered and very systematic procedure. Although indeterminate equations with integral solutions had been well known in ancient Mesopotamia, it was not until about 1150 that they appeared well developed in India. Aryabhata (b. a.d. 476) had used continued fractions in solutions, but there is uncertain evidence that this knowledge had been passed on in any ordered form to the Arabs by the time of Ab? K?mil.
A work of both geometric and algebraic interest is the Kit?b … al-mukharnmas wa’al-mu?ashshar … (“On the Pentagon and Decagon”). The text is algebraic in treatment and contains solutions for a fourth-degree equation and for mixed quadratics with irrational coefficients. Much of the text was utilized by Leonardo Fibonacci (1175-ca. 1250) in his Practica geometriae.2 Some of the equations solved by Ab? K?mil in this work read as follows:
with s the side of a regular polygon inscribed in a circle. Also
with S the side of a regular polygon (pentagon here) which has an inscribed circle.3
The outstanding advance of Ab? K?mil over al-Khw?rizm?, as seen from these equations, is in the use of irrational coefficients.4 Another manuscript, which is independent of the ?ar?’if, mentioned above, is the most advanced work on indeterminate equations by Ab? K?mil. The solutions are not restricted to integers; in fact, most are in rational form. Four of the more mathematically interesting problems are given below in modern notation. It must be remembered that Ab? K?mil gave all his problems rhetorically; in this text, his only mathematical notation was of integers.
Many of the problems in Kit?b fi’l-jabr wa’lmuq?bala had been previously solved by al-Khw?rizm?. In Ab? K?mil’s work, a solution5 for x2 was worked out directly instead of first solving for x. Euclid had taken account of the condition x less than p/2 in x2 + q = px, whereas Ab? K?mil also solved the case of x greater than p/2 in this equation.
Ab? K?mil was the first Muslim to use powers greater than x2 with ease. He used x8 (called “square square square square”), x6 (called “cube cube”), x5 (called “square square root”), and x3 (called “cube”), as well as x2 (called “square”). From this, it appears that Ab? K?mil’s nomenclature indicates that he added “exponents.” In the Indian nomenclature a “square cube” is x6, in contradistinction. Diophantus (ca. a.d. 86) also added “powers,” but his work was probably unknown to the Arabs until Abu’l Waf?’ (940–998) translated his work into Arabic (ca. 998).
Ab? K?mil, following al-Khw?rizm?, when using jadhr (“root”) as the side of a square, multiplied it by the square unit to get the area (x ? 12). This method is older than al-Khw?rizm?’s method and is to be found in the Mishnat ha-Middot, the oldest Hebrew geometry, which dates back to a.d. 150.6 This idea of root is related to the Egyptian khet (“cubit strip”).7
The Babylonians stressed the algebraic form of geometry as did al-Khw?rizm?. However, Ab? K?mil not only drew heavily on the latter but he also derived much from Heron of Alexandria and Euclid. Thus he was in a position to put together a sophisticated algebra with an elaborated geometry. In actuality, the resulting work was more abstract than al-Khw?rizm?’s and more practical than Euclid’s. Thus Ab? K?mil effected the integration of ancient Mesopotamian practice and Greek theory to yield a wider approach to algebra.
Some of the more interesting problems to be found in the Algebra, in modern notation, are:8
It is possible that Greek algebra was known to Ab? K?mil through Heron of Alexandria, although a direct connection is difficult to prove. The influence of Heron is, however, definite in Abraham bar ?iyy’s work.9 That Ab? K?mil influenced both al-Karaj? and Leonardo Fibonacci may be demonstrated from the examples they copied from his work. Thus through Ab? K?mil, mathematical abstraction, elaborated together with a more practical mathematical methodology, impelled the formal development of algebra.
NOTES
1. “Das Buch der Seltenheiten.”
2. See also Suter, “Die Abhandlung des Ab? K?mil,”
3.Ibid., p. 37. Levey will soon publish the Arabic text of “On the Pentagon and Decagon,” discovered by him.
4. At least twenty of these problems from this text may be found in Leonardo Fibonacci, Scritti, Vol. 1, sect. 15: Vol. II.
5. Tropfke, Geschichte der Elementar-Mathematik, pp. 74–76; 80–82; Weinberg, “Die Algebra des abu Kamil.”
6. S. Gandz, “On the Origin of the Term ‘Root.’”
7. M. Levey, The Algebra of Ab? K?mil, pp. 19–20. P. Schub and M. Levey will soon publish Ab? K?mil’s advanced work on indeterminate equations, newly discovered in Istanbul.
8.Ibid., pp. 178, 184, 186, 202.
9. M. Levey, “The Encyclopedia of Abraham Savasorda” and “Abraham Savasorda and His Algorism.”
BIBLIOGRAPHY
1. Original Works. The following manuscripts of Ab? K?mil are available; Kit?b fi’l -jabr wa’l muq?bala (“Book on Algebra,” Paris BN MS Lat. 7377A; Munich Cod. MS Heb.225; Istanbul-MS Kara Mustafa 379), trans. into Hebrew by Mordecai Finzi ca. 1460. See also Kit?b al-?ar?? if fi’s ?is?b (“Book of Rare Things in the Art of Calculation,” Leiden, MS Arabic 1003, ff. 50r-58r; translations are found in Munich Cod. MS Heb. 225 and in Paris BN MS Lat. 7377A); Kit?b... al-mukhammas wa’l-mu?ashshar... (“On the Pentagon and Decagon,” Paris BN MS Lat.7377A; Munich Cod. MS Heb. 225; Istanbul-MS Kara Mustafa 379, ff.67r-75r); Al-w???y? bi’l-judh?r (MS Mosul 294) discusses the ordering of roots. Works of Ab? K?mil listed in The Fihrist of al-Nad?m (p.281) include Kit?b al-fal?? (“Book of Fortune”) Kit?b mift?? al-fal?? (“Book of the Key to Fortune”). Kit?b fi’l-jabr wa’l-muq?bala (“Book on Algebra”), Kit?b al-mis??a wa’l-handasa (“Book on Surveying and Geometry”), Kit?b al-kif?ya (“Book of the Adequate”), Kit?b al-?ayr (“Book on Omens”), Kit?b al-?as?r (“Book of the Kernel”), Kit?b al-kha?a?ayn (“Book of the Two Errors”), Kit?b al-jam? wa’l-tafr?q (“Book on Augmentation and Diminution”).
II. Secondary Literature. For works on both Arab mathematics and Ab? K?mil see the following; H.T. Colebrooke, Algebra with Arithmetic and Mensuration from the Sanskrit (London, 1817); G. Fluegel, ed. and trans., Lexicon bibliographicum et encyclopedicum a Haji Khalfa compositum (Leipzig, 1835–1858); W. Hartner, “Ab? K?mil Shudj??,” in the Encyclopedia of Islam, 2nd ed., I (Leiden, 1960), 132–133; II (Leiden, 1962), 360–362; Ibn Khald?n The Muqaddimah, Franz Rosenthal, trans., 3 vols. (New York, 1958); Leonardo Fibonacci, Scritti di Leonardo Pisano, 2 vols; Vol. 1, Liber abaci; Vol. II, Practica geometriae; S. Gandz, “On the Origin of the Term ‘Root,’” in Amerian Math, Monthly, 35 (1928), 67–75; M. Levey, The Algebra of Ab? K?mil (Kit?b fi’l-jabr wa’l-muq?bala) in a Commentary by Mordecai Finzi (Madison, Wisc., 1966), “The Encyclopedia of Abraham Savasorda: A Departure in Mathematical Methodology,” in Isis35 (1952), 257–264; and “Abraham Savasorda and His Algorism: A Study in Early European Logistic,” in Osiris, 11 (1954), 50–63.
For additional material see G. Libri, Histoire des sciences mathématiques en lltalie (Paris, 1938), pp.253–297; 2nd ed. (Paris, 1865), pp. 304–369; al-Nad?m, Fihrist al-?ul?m, G. Fluegel, ed. (Leipzig, 1871–1872); M. Steinschneider,Die Hebraeischen Uebersetzungen des Mittelalters und die Juden als Dolmetscher, a reprint (Graz, 1956), pp. 584–588; and H. Suter, “Die Abhandlung des Ab? K?mil šoja? b. Aslam über das Fünfeck und Zehneck,” in Bib. Math., 10 (1909–1910), 15–42; “Das Buch der Setenheiten der Rechenkunst von Ab? K?mil el-Misr?,” in Bib. Math., Ser. 3, 11 (1910–1911), 100–120; “Die Mathematiker und Astronomen der Araber und ihre Werke,” in Abhandlungen z. Gesch. d. Math. Wissenschaften, 10 (1900). See also J. Tropfke, Geschichte der Elementar-Mathematik, vol. III (Berlin, 1937); J. Weinberg, “Die Algebra des ab? K?mil Š??? ben Aslam” (doctoral diss., Munich, 1935); A. P. Youschkevitch. Geschichte der Mathematik im Mittelalter (Basel, 1964).
Martin Levey