At about the same time as he began to study medicine, al-Samawal also began to study mathematics. He was about thirteen years old when he began serious study, beginning with Hindu methods of calculation and a study of astronomical tables. Baghdad at this time was not a great centre for mathematical learning, and al-Samawal had soon mastered all the mathematics which his teachers knew. These teachers had covered topics including an introduction to 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 eighteen years old he had read almost all the available mathematical literature. The work which most impressed him was that of al-Karaji, yet he found himself less than completely satisfied with it and began to work out improvements for himself. His most famous treatise al-Bahir fi'l-jabr, meaning The brilliant in algebra, was written when al-Samawal was only nineteen years old. 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 al-Bahir was published in an edition with notes and an introduction in . Details of the work are also given in  and . The treatise consists of four books: (1) On premises, multiplication, division and extraction of roots, (2) On extraction of unknown quantities, (3) On irrational magnitudes, and (4) On classification of problems.
Al-Samawal's predecessors had begun to develop what has been called by historians today the "arithmetisation of algebra". In fact al-Samawal was the first to give this development a precise description when he wrote that it was concerned (see for example  or ):-
... with operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known.This will strongly suggest to mathematicians today that al-Samawal was developing the study of polynomial rings and indeed this is a fair description of the work he was undertaking. In the first book of the al-Bahir he defines powers x, x2, x3, ... , x-1, x-2, x-3, ... . 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.
Al-Samawal could not have described arithmetic operations on powers of the unknown without having developed an understanding of negative numbers. He had refined the ideas of his predecessors into a form which would not be given by European mathematicians until many centuries later. He also used 0 in his calculations writing :-
If we subtract a positive number from an empty power, the same negative number remains.By this al-Samawal meant, in modern notation,
0 - a = -a.He continued:-
... if we subtract the negative number from an empty power, the same positive number remains.Again in modern notation this is 0 - (-a) = a.
Multiplication of negative numbers was also completely understood by al-Samawal. He wrote :-
... the product of a negative number by a positive number is negative, and by a negative number is positive.In Book 2 of al-Bahir 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 a xn is a square, and finding x so that axn + bxn-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. The method is attributed by al-Samawal to al-Karaji and provides the only surviving account of this remarkable work.
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, then prove the case n = 3 based on his result for n = 2, and carry on to around n = 5 before remarking that one can continue the process indefinitely. Although this is not induction proper, it is a major step towards understanding inductive proofs. We should also comment that he was not the first to use this form of recursive reasoning, since al-Karaji had used similar methods. The result in Book 2 which al-Samawal himself was most proud of (and rightly so) is
12 + 22 + 32 + ... + n2 = n(n + 1)(2n + 1)/6which does not appear in earlier texts.
Book 3 contains a description of how to carry out arithmetic with irrational numbers. It follows Book X of Euclid's Elements and, although a very fine exposition of these ideas, contains little that is original. One result here which again particularly pleased al-Samawal was his calculation of how to rationalise √30/(√2 + √5 + √6). Al-Karaji had failed to solve this problem, which explains why al-Samawal was particularly pleased to solve it. [My computer algebra package saves me having to do the thinking al-Samawal had to do and gives (5√6 + 2√5 + 6√15 - 20√2)/13.]
The final book of al-Bahir contains an interesting example of a problem in combinatorics, namely to find ten 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. In Book 4 al-Samawal 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 which :-
... if one could assume the existence of their solution, this existence would lead to an absurdity.After writing the al-Bahir al-Samawal travelled in many countries including Iraq, Syria, Kohistan (a mountainous area in Pakistan and Afghanistan) and Azerbaijan (northwestern of Iran). We know from his own writings that he was in Maragheh in Azerbaijan on 8 November 1163, for on that date al-Samawal made a commitment to the faith of Islam. This decision was not taken without a great deal of thought by al-Samawal. He had put much effort into testing the validity of the claims made by the major religions and he reports that on 8 November 1163 he decided that Islam was the most satisfactory. He wrote a work Decisive refutation of the Christians and Jews which has survived.
Of course al-Samawal's father, being Jewish, would have found his son's conversion to Islam a painful experience and al-Samawal, not wishing to hurt his father, delayed his conversion for four years. After this time al-Samawal wrote to his father setting out his reasons for changing his religion from the Jewish faith to Islam. At this time the much travelled al-Samawal was in Aleppo, in northern Syria, and his father set out at once to see him on receiving the letter. However, al-Samawal's father died on the journey before seeing his son.
We mentioned that al-Samawal was trained in medicine in his youth. In fact he practised 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. Unfortunately, no details of these have survived.
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. In particular al-Samawal shows that he is interested in the psychological aspects of disease. His remedy for depression is :-
... well-lighted houses, the sight of running water and verdure, warm baths and music.Most of the works of al-Samawal have not survived, but he is reported to have written 85 books or articles. Some other of al-Samawal's mathematical writings have survived but these are elementary works of relatively little importance. They contain work on fractions with examples such as showing how to express 80/(3 × 7 × 9 × 10) as the sum of fractions with numerators 1. He gives
80/(3 × 7 × 9 × 10) = 1/(3 × 10) + 1/(3 × 7 × 9) + 1/(3 × 9 × 10)The work here is carried out using the sexagesimal system, showing that, although mathematicians of this period favoured the decimal system, commercial use still favoured the sexagesimal system. Al-Samawal's elementary texts were clearly teaching books.
Another of al-Samawal's surviving works is The exposure of the errors of the astrologers which, as the title suggests, argues against the scientific value of astrology. The introduction to this work has been translated into English in .
Article by: J J O'Connor and E F Robertson
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