Abu Kamil Shuja ibn Aslam ibn Muhammad ibn Shuja

(about 850 in (possibly) Egypt - about 930)

Abu Kamil Shuja is sometimes known as al-Hasib al-Misri, meaning the calculator from Egypt. Very little is known about Abu Kamil's life - perhaps even this is an exaggeration and it would be more honest to say that we have no biographical details at all except that he came from Egypt and we know his dates with a fair degree of certainty.
The Fihrist (Index) was a work compiled by the bookseller Ibn an-Nadim around 988. It gives a full account of the Arabic literature which was available in the 10th century and it describes briefly some of the authors of this literature. The Fihrist includes a reference to Abu Kamil and among his works listed there are: (i) Book of fortune, (ii) Book of the key to fortune, (iii) Book on algebra, (vi) Book on surveying and geometry, (v) Book of the adequate, (vi) Book on omens, (vii) Book of the kernel, (viii) Book of the two errors, and (ix) Book on augmentation and diminution. Works by Abu Kamil which have survived, and will be discussed below, include Book on algebra, Book of rare things in the art of calculation, and Book on surveying and geometry.
Although we know nothing of Abu Kamil's life we do understand something of the role he plays in the development of algebra.

Beforeal-Khwarizmi we have no information of how algebra developed in Arabic countries, but relatively recent work by a number of historians of mathematics as given a reasonable picture of how the subject developed afteral-Khwarizmi. The role of Abu Kamil is important here as he was one ofal-Khwarizmi's immediate successors. In fact Abu Kamil himself stressesal-Khwarizmi's role as the "inventor of algebra". He describedal-Khwarizmi as (see for example [1] or [2]):-

... 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 Musaal-Khwarizmi, and I have answered that impetuous man Ibn Barza on his attribution to Abd al-Hamid, whom he said was his grandfather.
There is certainly no doubt that Abu Kamil considered that he was building on the foundations of algebra as set up byal-Khwarizmi and indeed he forms an important link in the development of algebra betweenal-Khwarizmi andal-Karaji. There is another reason for Abu Kamil's importance, however, which is that his work was the basis ofFibonacci's books. So not only is Abu Kamil important in the development of Arabic algebra, but, throughFibonacci, he is also of fundamental importance in the introduction of algebra into Europe. The author of [3] presents a list of parallels between Abu Kamil's works on algebra and the works ofFibonacci, and he also discusses the influence of Abu Kamil on two algebra texts ofal-Karaji.
The Book on algebra by Abu Kamil is in three parts: (i) On the solution ofquadratic equations[4], (ii) On applications of algebra to the regular pentagon and decagon, and (iii) OnDiophantine equations [5]and problems of recreational mathematics. The part on the regular pentagon and decagon is studied in detail in [6], while the remainder of the work is described in [7]. The content of the work is the application of algebra to geometrical problems. It is the combination of the geometric methods developed by the Greeks together with the practical methods developed byal-Khwarizmi mixed with Babylonian methods.

An important step forward in Abu Kamil's algebra is his ability to work with higher powers of the unknown than x2. These powers are not given in symbols but are written in words, yet the naming of the powers tell us that Abu Kamil had begun to understand what we would write in symbols as xnxm= xn+m. For example he uses the expression "square square root" for x5 (i.e. x2.x2.x), "cube cube" for x6 (i.e. x3.x3), "square square square square" for x8 (i.e. x2.x2.x2.x2). In fact Abu Kamil works easily with the powers up to x8 which appear in the text. The algebra contains 69 problems which include many of the 40 problems considered byal-Khwarizmi, but with a rather different approach to them.
The Book on surveying and geometry is studied in detail in [8]. It was written by Abu Kamil, not for mathematicians, but rather for government land surveyors. Because of the people that it was aimed at, the work contains no proofs. Rather it presents a number of rules, some of which are far from easy, each given for the numerical solution of a geometric problem. Each rule is illustrated with a worked numerical example. Mainly the rules are for calculating the area, perimeter, diagonals etc. of figures such as squares, rectangles, and various different types of triangle. Abu Kamil also gives rules to calculate the volume and surface area of various solids such as rectangular parallelepipeds, right circular prisms, square pyramids, and circular cones.
The work also deals with circles and here Abu Kamil takes  =22/7. A whole section is devoted to calculating the area of the segment of a circle. The final part of the work gives rules for calculating the side of regular polygons of 3, 4, 5, 6, 8, and 10 sides eitherinscribed[9] in, orcircumscribed [10]about, a circle of given diameter. For the pentagon and decagon the rules which Abu Kamil gives, although without proof in this work, were fully proved in his algebra book.
The Book of rare things in the art of calculation is concerned with solutions to indeterminate equations. Sesiano in [11] discusses Abu Kamil's work on indeterminate equations and he argues that his methods are very interesting for three reasons. Firstly Abu Kamil is the first Arabic mathematician who we know solved indeterminate problems of the type found inDiophantus's work. Secondly, as far as we know, Abu Kamil wrote beforeDiophantus's Arithmetica had been studied in depth by the Arabs. Thirdly, Abu Kamil explains certain methods which are not found in the known books of the Arithmetica.

Article by: J J O'Connor and E F Robertson

Notes:

1-R Rashed, The development of Arabic mathematics : between arithmetic and algebra (London, 1994).
2-R Rashed, Entre arithmétique et algèbre: Recherches sur l'histoire des mathématiques arabes (Paris, 1984).
3-S Shalhub, The calculations and algebra of abu Kamil Shuja ibn Aslam and his effects on the work of al-Karaji and on the work of Leonardo Fibonacci (Arabic), in Deuxième Colloque Maghrebin sur l'Histoire des Mathématiques Arabes (Tunis, 1990), A23-A39.
4-A quadratic equation is an equation with a second order term: ax2 + bx + c = 0 in general.
5-A Diophantine equation is one which is to be solved for integer solutions only.
6-R Lorch, Abu Kamil on the pentagon and decagon, Vestigia mathematica (1993), 215-252.
7-J Sesiano, La version latine médiévale de 'l'Algèbre' d'Abu Kamil, in Vestigia mathematica (Amsterdam, 1993), 315-452.
8-J Sesiano, Le Kitab al-Misaha d'Abu Kamil, Centaurus 38 (1996), 1-21.
9-A circle is said to be inscribed to a triangle or other polygon if the edges are tangents to the circle.
The polygon is than said to becircumscribed to the circle.
The (unique) circle inscribed to a triangle is called the incircle and its centre is the incentre.

10-A circle is said to be circumscribed to a triangle or other polygon if the vertices of the polygon lie on the circle.
The polygon is than said to beinscribed in the circle.

The (unique) circle circumscribed to a triangle is called the circumcircle and its centre is the circumcentre.

11-J Sesiano, Les méthodes d'analyse indéterminée chez abu Kamil, Centaurus 21 (2) (1977), 89-105.