Islam,Neglect andDiscovery 117
he came to a startlingly pragmatic conclusion. We should think, he says, of a quantity
not as a line, a surface, a volume or a time, but as a quantity which the mind abstracts from everything, and which
belongs to numbers, but not to absolute and true numbers, for the ratio ofAtoBmay often not be numerically
measurable, that is to say one may not be able to find two numbers whose ratio it is...This is how calculators and
surveyors proceed when they speak of a half or other fraction of a supposedly indivisible unit, or of a root of five or
ten etc. (Khayyam tr. Rozenfel’d pp. 105–6, cited Youschkevitch p. 88)
In other words, the calculators and surveyors are already using numbers on the assumption
that they are the same as ‘quantities’; that if you can construct a length, there is a number which
corresponds to it (at least well enough). What is interesting is Omar’s explicit suggestion that
mathematicians could learn something from them.
Exercise 6.Show that the equation given is equivalent to abu K ̄amil’s problem, and solve the equation.
Exercise 7.Use the formulasin( 3 x)=3 sinx−4 sin^3 x to find a cubic equation forsin 10◦.
7. Al-Samaw’al and al-K ̄ash ̄i
TheCalculator’s Keyis an excellent guide to elementary mathematics, written to answer to the needs of a large public.
Considering the richness of its subject-matter, and the clarity and elegance of its presentation, this work holds an
almost unique place in the whole literature of the Middle Ages. (Youschkevitch 1976, p. 71)
It would take much more space to discuss all the varieties of mathematical practice which were
undertaken in the Islamic world, their connexions, and interrelations; although we shall return
briefly to their views on Euclid in Chapter 8. However, in a specific attempt to investigate the themes
of innovation, tradition, and continuity, let us consider two later mathematicians whose approach
was very different, al-Samaw’al (1125–1180) and al-K ̄ash ̄i (d. 1429). In both cases, there are
acknowledgements of particular influences, neither is working in what we have called the Greek
tradition, and both raise interesting unsolved problems about the aim and scope of their work.
In particular, both present examples of what we might call excess, that is, calculation beyond
what is necessary or useful and here we would differ from Youschkevitch’s opinion above, with his
references to ‘elementary mathematics’ and ‘a large public’. In contrast to al-Uql ̄idis ̄i or abu-l-Waf ̄a,
they seem to be carriedaway bytheir subject. Why?
Al-Samaw’al appears the more straightforward case. His major work—of those which survive—
isal-B ̄ahir fi-l-jabr(‘The Shining Treatise on Algebra’). Written, it is said, when he was 19, it is
a conscious attempt to strengthen and deepen the results of his predecessor al-Karaj ̄i, a century
earlier. (In many respects al-Karaj ̄i had laid the foundation for al-Samaw’al’s work, so much is
undisputed; however, here we shall consider it in isolation.) The work is quite long and contains
a variety of results (on systems of linear equations, for example), but it is most celebrated for the
curious study of ‘polynomials’ (al-Samaw’al calls them ‘composed expressions’) in which:
- the primary aim is not to find the ‘thing’—it seems, in the main, to be treated as an abstract
entity to be manipulated; - the powers of the thing considered are not only positive (in principle, as large as you like) but
negative; what we would call 1/x,1/x^2 ,...