116 A History ofMathematics
been vastly extended, without this ever being made explicit. The earlier author never said that
numbers could not be square roots, and the later one never said that they could, but all the same
the idea of what ‘numbers’ are allowed has changed.
It is easy, in the fairly open climate of discussion in Islamic mathematics, to find differences of
approach such as those described above; and they are not confined to algebra. There are explicit
arguments, for example, about the merits of those who were already being called (as they were by
Pappus) ‘the Ancients’ (al-qudam ̄a’):[Ab ̄u-l-Waf ̄a’] said how much he appreciated the book, which he considered of great value, although he regretted
that the author followed the way of the ancients in their use of the ‘cutting diagram’ and of compound ratios. He
said that he had obtained, to determine the azimuth, elegant methods which were more concise and better. (Al-B ̄ir ̄un ̄i
1985, p. 96)However, to use this particular quarrel (about who was the first to find a formula for trigonometry
on a sphere), or any other to divide Islamic mathematicians into ‘schools’ as has sometimes been
done seems premature, and probably misguided. Saidan in his introduction to al-Uql ̄idis ̄i (1978)
calls attention to attempts of earlier historians to distinguish those mathematicians who used
Indian numbers from those who used sexagesimals (or ‘astronomers’ numbers’ as they were called);
and points out that it was common, especially in teaching texts, to use both, since the student
might need both. As for Greek authority, it was universally recognized, and used as and when
necessary together with more ‘modern’ methods. The case of Omar Khayyam (eleventh century) is
particularly worth considering. In his algebra, he considers in detail the case of cubic equations. He
was the ‘eastern mathematician’ referred to by ibn Khald ̄un who had brought the number of types
to more than 20 by introducing the various types of cubics (cubes and things equal to numbers,
and so on). Besides being the natural next step after the well-understood quadratics, these had
arisen in a number of special problems which he lists; a problem of Archimedes on cutting the
sphere, trigonometric problems such as finding sin 10◦given that one knows sin 30◦, and so on.
As has often been noted, he acknowledges that it would be desirable to find a solution in terms
of a numerical procedure (what we would call a formula), as had been done for quadratics and as
Tartaglia and Cardano were to do in the sixteenth century.
When, however, the object of the problem is an absolute number, neither we, nor any of those who are concerned
with algebra, have been able to prove this equation—perhaps others who follow us will be able to fill the gap—except
when it contains only the three first degrees, namely, the number, the thing, and the square. (Khayyam 1931, p. 49)Unable to achieve this, he followed the very Greek practice of drawing intersecting conic sections,
just as Menaechmus had done for the simplest casex^3 =2.^10
On the whole, such a solution would have been acceptable to a Greek (supposing the problem to
have been posed in the first place). Omar was in some ways particularly close to the Greek geometers
in his outlook; he criticized ibn al-Haytham for using motion to prove the parallel postulate, and
the algebraists in general for using the ‘ungeometrical’ powers of the unknown above the third.
However, it may have occurred to him to ask a question which fits much better into the framework
of the algebra we have been discussing above: namely, if you have constructed a solution (e.g. to
x^3 +x^2 =3) geometrically, what kind of a number have you found, and what can you do with it?
There is a clue; when, in a different work, he considered the difficulties in Euclid’s theory of ratios,
- For an extract from Omar’s work see Fauvel and Gray.