114 A History ofMathematics
What Th ̄abit does next is equally interesting; he goes through his method and shows, stage by
stage, that it is the same as the method used ‘in algebra’. The ‘algebra’ is the method described by
al-Khw ̄arizm ̄i—without his geometric proof —and it seems reasonable on various grounds (short
separation in time, the fact that they were near-colleagues, al-Khw ̄arizm ̄i’s acknowledged status
as ‘founder’) to suppose that it is precisely his book which is referred to. This ‘dialogue’ casts some
light on different ways of thinking about geometry, numbers, and algebra in the earliest period of
Islamic mathematics. It would seem that Th ̄abit is saying: ‘Anything which you can do by algebra,
I can do by Euclid book II’. If so, there is some misunderstanding both of al-Khw ̄arizm ̄i’s algebra
(which is about numerical recipes for solving practical problems) and of Euclid (which is about
something more abstract and quite different). More positively, we could see it as an attempt to
harmonize the down-to-earth practice of algebra with Greek theory. Whether misunderstanding
or harmonization, such a tension between theory and practice was to be of immense value in the
further development of the Islamic tradition.
We have already entered the domain of (reasonable) conjecture about what the text means, in
terms of the various ways tenth-century mathematicians thought about numbers and geometry.
The problem is what Th ̄abit means by ‘is known’—the argument being that to say that the side (or
its square) is known is to solve the quadratic equation. There are two competing interpretations of
this. Ingeometricterms, it means simply that the line which represents the side can be constructed,
which is certainly true. But what has been passed over is thenumericalquestion of what happens
when your answer is not a whole number, as it was in al-Khw ̄arizm ̄i’s version. If the equation is
‘square and two roots equal one’, then the answer, whichever method you use to arrive at it, is (as
we would say)
√
2 −1. Because Th ̄abit is avoiding using numerical examples, he gives us no idea
about whether such numbers are allowedas numbers, not as geometrically constructed lines. They
have no name.
There is a useful word for ‘having no name’ in Arabic, which was variously applied: it is ‘as.amm’,
or ‘deaf ’. This was initially aplied to certain fractions; you can say the fractions up to one-tenth
using one word, but after that you have to use phrases like ‘one part of thirteen’, and such fractions
were ‘as.amm’. But in al-Uql ̄idis ̄i’s arithmetic, the same word was applied to squares which have
no roots; by extension (since if you are thinking for example, of a square of area 5, you are also
thinking of its side) it denoted their inexpressible roots. This word translated, when the Arabic
arithmetics were put into Latin, into the Latin word for ‘deaf ’, ‘surdus’, used in the form ‘surd’
as recently as 50 years ago to refer to roots like√
- At some point a linguistic concept about
numbers whose names you could speak translated into a way round their unspeakability. They
are still numbers, but numbers which need phrases rather than a single word to express them.
Al-Uql ̄idis ̄i devoted some space to finding approximations for such square roots, in chapters which
follow on the exact root extraction quoted above. His formulae were not new, but the use of Indian
numbers makes the procedure more transparent. (A great deal has by now been written on this
subject. A detailed and careful summary is Karine Chemla 1994.)
Exercise 3.What kinds of combination of roots, squares, and numbers make an allowable equation, in
the terms set out by al-Khw ̄arizm ̄i?Exercise 4.Why is the algebraic formula given equivalent to Euclid’s proposition II.6?Exercise 5.How would (a) al-Khw ̄arizm ̄i’s method and (b) Th ̄abit’s construction approach the equation
‘square and two roots equal to one’?