A History of Mathematics From Mesopotamia to Modernity

112 A History ofMathematics

a

b

a

b

c

e

f d

Fig. 3Al-Khw ̄arizm ̄i’s first picture for the quadratic equation.

‘It is necessary’, he continues, ‘that we should demonstrate geometrically the truth of the same

problems which we have explained in numbers.’ Why is it necessary? There appear to be three

requirements for the author:

1. to state what to do in general;


2. to illustrate it in particular;


3. to prove that it works.

Is it the weight of the Greek heritage which implies that ‘proof ’ means geometry? One might
suppose so, since the Greek texts were being translated when al-Khw ̄arizm ̄i wrote. In any case, the
geometry looks nothing like Euclid, or even his more practically minded followers such as Heron.
The picture (Fig. 3) compared with later proofs of the same method, is completely transparent; it is
a good exercise to follow the proof through and see how verbal explanation and picture connect to
give a convincing account of why the solution is the right one.
There has been considerable discussion of how ‘good’ a mathematician al-Khw ̄arizm ̄i was (the
article in theDictionary of Scientific Biographyis dismissive). As already stated, the method which
he set out was ancient, wherever he derived it; and his exposition, his examples, and his proof
were (as the extract shows) at a fairly low mathematical level. However, this seems to miss an
important point; such arguments assume that mathematicians deserve study only insofar as the
work which they do is hard, while often this is not at all the case. (While Descartes was capable
of hard work in mathematics, he disliked it, and his outstanding contribution, the coordinate
representation of curves, is simple in the extreme.) What al-Khw ̄arizm ̄i did was to introduce a
new way of thinking about the problem which brought together solution and proof in a major
synthesis, involving both generalization and simplification. That the mathematics involved wasnot
very difficult was an essential reason for the method’s survival more or less unchanged over the next
600 years.
About 50 years later, Th ̄abit ibn Qurra—who by general agreementwasan able and interesting
mathematician—wrote a text on quadratic equations. In contrast to al-Khw ̄arizm ̄i’s treatise, it is a
mere six pages. It was translated into German during the Second World War, and later into Russian;