Died: 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.
Before al-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 after al-Khwarizmi.
The role of Abu Kamil is important here as he was one of al-Khwarizmi's
immediate successors. In fact Abu Kamil himself stresses al-Khwarizmi's
role as the "inventor of algebra". He described al-Khwarizmi
as (see for example [4] or [5]):-

... 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 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 [12] 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.

The *Book on algebra* by Abu Kamil is in three parts: (i)
On the solution of quadratic equations, (ii)
On applications of algebra to the regular pentagon and decagon, and (iii) On Diophantine equations and
problems of recreational mathematics. The part on the regular pentagon and
decagon is studied in detail in [7], while the remainder of the work is
described in [10]. 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 by al-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 *x*^{2}. 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 *x*^{n}*x*^{m} =
*x*^{n+m}. For example he uses the expression "square
square root" for *x*^{5} (i.e.
*x*^{2}.*x*^{2}.*x*), "cube cube" for
*x*^{6} (i.e. *x*^{3}.*x*^{3}), "square
square square square" for *x*^{8} (i.e.
*x*^{2}.*x*^{2}.*x*^{2}.*x*^{2}).
In fact Abu Kamil works easily with the powers up to *x*^{8} which
appear in the text. The algebra contains 69 problems which include many of the
40 problems considered by al-Khwarizmi,
but with a rather different approach to them.

The *Book on surveying and geometry* is studied in detail
in [9]. 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 p = ^{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 either inscribed in, or circumscribed 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 in Diophantus's
work. Secondly, as far as we know, Abu Kamil wrote before Diophantus'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.*