Source : http://www.stetson.edu/~efriedma/periodictable/html/Sb.html

Ibn Yahya al-Maghribi Al-Samawal

1130?-1180?

Al-Samawal was brought up in a family where learning was highly
valued, and the first topic which interested him was medicine, perhaps because
he had an uncle who was medical doctor. At about the same time as he began to
study medicine, al-Samawal also began to study mathematics. He was about 13 when
he began serious study, beginning with Hindu methods of calculation and a study
of astronomical tables. His tutors also taught him surveying, elementary
algebra, and the geometry of the first few books of Euclid's *Elements*.

In order to take his mathematical studies further, al-Samawal had to study on his own. He read the works of Abu Kamil, al-Karaji and others so that by the time he was 18 he had read almost all the available mathematical literature. His most famous treatise was written when al-Samawal was only 19. It is a work of great importance both for the original ideas which it contains and also for the information that it records concerning works by al-Karaji which are now lost.

The treatise consists of 4 books. In Book 1, he defines integer powers of variables. After defining polynomials, al-Samawal describes addition, subtraction, multiplication and division of polynomials. He also gave methods for the extraction of the roots of polynomials. He even used negative numbers and zero in his calculations, and understood multiplication of negative numbers.

In Book 2, al-Samawal describes the theory of quadratic equations but, rather
surprisingly, he gave geometric solutions to these equations despite algebraic
methods having been fully described by al-Khwarizmi, al-Karaji, and others.
Al-Samawal also described the solution of indeterminate equations such as
finding x so that ax^{n} or ax^{n} + bx^{n-1} is a
square. Also in this book is al-Samawal's description of the binomial theorem
where the coefficients are given by the Pascal triangle.

Perhaps one of the most remarkable achievements appearing in Book 2 is
al-Samawal's use of an early form of induction. What he does is to demonstrate
an argument for n = 1, then prove the case n = 2 based on his result for n = 1,
and so on to n = 5 before remarking that one can continue the process
indefinitely. The result in Book 2 which al-Samawal himself was most proud of is
1^{2} + 2^{2} + 3^{2} + ... + n^{2} =
n(n+1)(2n+1)/6.

Book 3 contains a description of how to carry out arithmetic with irrational numbers. Although it is a very fine exposition of these ideas, it contains little that is original. One result here which again particularly pleased al-Samawal was his calculation of how to rationalize a fraction with the sum of three different square roots in the denominator.

The final book contains an interesting example of a problem in combinatorics, namely to find 10 unknowns given the 210 equations which give their sums taken 6 at a time. Of course such a system of 210 equations need not be consistent and al-Samawal gave the 504 conditions which are necessary for the system to be consistent. He also classifies problems into necessary problems, namely ones which can be solved; possible problems, namely ones where it is not known whether a solution can be found or not, and impossible problems, namely problems whose solutions would lead to an absurdity.

Al-Samawal travelled a great deal. Although he was raised Jewish, he made a
commitment to Islam after testing the validity of the claims made by the major
religions, and he wrote a work *Decisive refutation of the Christians and
Jews*.

Al-Samawal practiced his medical skills on his journeys, and became quite famous for this expertise in this area. Several rulers, always keen to have the best possible doctors, became patients of al-Samawal. He relates in his writings that he developed some medicines which were almost miraculous cures. The only medical work by al-Samawal which has survived is essentially a sex manual which includes many erotic stories. The work exhibits the fact that al-Samawal was a good scientific observer in his descriptions of various diseases.

Most of the works of al-Samawal have not survived, but he is reported to have written 85 books or articles. Other al-Samawal's mathematical writings have survived but these are elementary works of relatively little importance. Al-Samawal's elementary texts were clearly teaching books. Another of his surviving works argues against the scientific value of astrology.