Abu Kamil Shuja'ibn Aslam ibn Muhammad ibn Shuja
Muḥammad ibn Mūsā al-Khwārizmī (c. 780 – c. 850), formerly Latinized as Algorithmi, was a Persian scholar who produced works in mathematics, astronomy, and geography under the patronage of the Caliph Al-Ma'mun of the Abbasid Caliphate. Around 820 AD he was appointed as the astronomer and head of the library of the House of Wisdom in Baghdad.
(Algebra of Abu Kamil in a Commentary by Mordecai Finzi: A...)
Algebra of Abu Kamil in a Commentary by Mordecai Finzi: Abukamil Shuja Ibn Aslam.
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أبو كامل شجاع بن أسلم بن محمد بن شجاع
Abū Kāmil Shujāʿ ibn Aslam was born in Egypt, in the same year in which Mohammad Al-Khwarizmi died, Abu-Kamil is often referred to as his successor. There is virtually no biographical material available on Abu Kamil. Since Abū Kāmil’s work is mentioned by Ibn al-Nadīm in his bio-bibliographical Fihrist (c. 379/990) (339), he may be presumed to have lived from about 235/850 to about 317/930.
Although we know nothing of Abu Kamil's life we do understand something of the role he plays in the development of algebra. Abu Kamil is first mentioned by al-Nadlm in a bibliographical work, The Fihrisl (987), where he is fisted 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 Khaldun (1322-1406) stated that Abu Kamil wrote his algebra after the first such work by al-Khwarizml, and Hajjl Khalifa (1608-1658) attributed to him a work supposedly concerned with algebraic solutions of inheritance problems.
Among the works of Abu Kamil extant in manuscripts is the Kilab al-tarcPif fi’l-hisab (“Book of Rare Things in the Art of Calculation”). According to H. Suter 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. Abu Kamil’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 Abu Kamil.
A work of both geometric and algebraic interest is the Kitdb . . . al-mukhammas wa’al-mtfashshar . . . (“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 geomelriae.
The outstanding advance of Abu Kamil over al-Khwarizm, as seen from these equations, is in the use of irrational coefficients. Another manuscript, which is independent of the Taraif mentioned above, is the most advanced work on indeterminate equations by Abu Kamil. 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 Abu Kamil gave all his problems rhetorically; in this text, his only mathematical notation was of integers. Many of the problems in Kildb fi’l-jabr wa’l- muqdbala had been previously solved by al-Khwarizml.
The Babylonians stressed the algebraic form of geometry as did al-Khwarizml. However, Abu Kamil 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-Khwarizml’s and more practical than Euclid’s. Thus Abu Kamil 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.
It is possible that Greek algebra was known to Abu Kamil through Heron of Alexandria, although a direct connection is difficult to prove. The influence of Heron is, however, definite in Abraham bar Hiyya’s work. That Abu Kamil influenced both al-Karajl and Leonardo Fibonacci may be demonstrated from the examples they copied from his work. Thus through Abu Kamil, mathematical abstraction, elaborated together with a more practical mathematical methodology, impelled the formal development of algebra.
Abū Kāmil is often called "Father of Algebra" and remembered for his work in the field of algebra, where he added practical elements to mathematics; for example, he used algebraic equations to solve problems of inheritance. He used Greek and Babylonian ideas in his work, and introduced powers greater than two to Arab mathematics.
His most notable achievements were in the treatises he published in these fields were among the best-selling science texts of that era. Some of them which were later translated to English include: the Book on Algebra, the Book on Survey and Geometry, the Book of Two Errors, the Book of Kernel, and the Book on Augmentation and Diminution. But above all else, Abu-Kamil is remembered today as the first person who employed irrational numbers as solutions, as well as coefficients to equations.
As the intellectual heir of Al-Khwarizmi, he followed-up by building on the works of his predecessor: gaining great fame and influence as a result. Abubakar Al-Karaji of Persia was attracted to his works; and so was Leonardo Fibonacci. In fact, Fibonacci used Abu-Kamil’s works as the bases for the treatises he later published and circulated in Europe. Thus, it could be said that (in addition to Fibonacci’s efforts) the greatest progress which Medieval Europe made in mathematics was due to Abu-Kamil Shuja: who harnessed, updated and provided the best Mid-Eastern works of Mohammad Al-Khwarizmi plus the best Asian curriculum masterminded by Liu Hui.
(Algebra of Abu Kamil in a Commentary by Mordecai Finzi: A...)
Abu Kamil lived during the era of Islamic Golden Age and in his religious affiliation belonged to Islam.
There is certainly no doubt that Abu Kamil considered that he was building on the foundations of algebra as set up by al-Khwarizmi and indeed he forms an important link in the development of algebra between al-Khwarizmi and al-Karaji. There is another reason for Abu Kamil's importance, however, which is that his work was the basis of Fibonacci's books. So not only is Abu Kamil important in the development of Arabic algebra, but, through Fibonacci, he is also of fundamental importance in the introduction of algebra into Europe. The author of presents a list of parallels between Abu Kamil's works on algebra and the works of Fibonacci, and he also discusses the influence of Abu Kamil on two algebra texts of al-Karaji.
Abu Kamil 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 Abu Kamil’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 Wafa1 (940-998) translated his work into Arabic (ca. 998).
Abu Kamil, following al-Khwarizml, 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-Khwarizmfs method and is to be found in the Mishnat ha-Middot, the oldest Hebrew geometry, which dates back to A.D. 150. This idea of root is related to the Egyptian khet (“cubit strip”).
Quotations:
He described al-Khwarizmi as:
"... the one who was first to succeed in a book of algebra and who pioneered and invented all the principles in it."
Again Abu Kamil wrote:-
"I have established, in my second book, proof of the authority and precedent in algebra of Muhammad ibn Musa al-Khwarizmi, and I have answered that impetuous man Ibn Barza on his attribution to Abd al-Hamid, whom he said was his grandfather."
