Islam,Neglect andDiscovery 111
This may seem less than clear to us, but it enables a description—the first—of what a general
quadratic equation is. Note that the ‘root’, or solution, is allowed to be a fraction although not
worse.^6 You will still find this language, stretched to its limits, used in Tartaglia’s rule for solving
the cubic in the 1540s (see Chapter 6). There are six forms of the quadratic equation—this is
dictated by the need for all numbers which are used to be positive. A typical one reads: ‘Roots and
squares equal to numbers’; somexs (as we would say) added to somex^2 s equal some number.
Al-Khw ̄arizm ̄i does not wish, like the Babylonians, to list particular cases and assume that you can
deduce the general rule; he wants his statement to be general, but he does not have our version
of a general symbolic language (which dates from the seventeenth century) ‘aroots +bsquares
equalcnumbers’. (Interestingly, although Diophantus’sArithmetic, which did use a more abstract
notation, was translated relatively early into Arabic—ninth century, later than al-Khw ̄arizm ̄i—his
methods were not adopted, any more than they were in the Greek world.)
If, in a parenthesis, we consider how one is taught to solve such an equation today, the commonest
method is to give a simple literal formula, whether it is proved or not. Writing the equation
ax+bx^2 =c, we deduce:
x=−a±√
a^2 + 4 bc
2 bwhich ‘always applies’. The reason we can do this is because we can explain how to deal with
several problems raised by the formula.
First, one, or both of the values we find may be negative numbers, which were first considered as
possible solutions in India by Bhaskara in the eleventh century, and were still being argued about
400 years later; as we have seen (Chapter 4) this was found easy by the Chinese, but their attitude
seems not to have been transmitted to the West.
Second, we have to be prepared to take the square root of any number we like. This raises two
levels of problems; a ‘naming’ problem if the number is positive but not a square (say 5), which
we shall see dealt with below; and a worse one—what are we talking about at all?—if it is negative
(say−3). These were coped with at different times in more or less satisfactory ways, and a school
mathematics course will similarly try to steer the student through them progressively.
Until the sixteenth century or later, though, no such general formula was considered, since even
negative roots had to be dealt with separately if they were allowed at all. Hence the pattern which
al-Khw ̄arizm ̄i set for dealing with equations case by case, as set out above. After describing the
different cases, he moves on to the case ‘roots and squares equal to numbers’ mentioned above,
and deals with the problem of abstraction by alternating the general statement with its application
to the particular example ‘one square and ten roots equal to thirty-nine dirhems’.^7 The solution
goes back to Babylon (‘you halve the number of roots, which gives five’), but has suddenly become
general as well as particular. It is easy to see the reasons for the long popularity of al-Khw ̄arizm ̄i’s
text: he has grasped the idea of explaining the method through an example, as al-Uql ̄idis ̄i was to do
in his arithmetic (and as was to become common practice in Islamic texts, and the European ones
which derived from them).
- Note also that despite al-Khw ̄arizm ̄i’s role in introducing Indian numbers, they are not used in the algebra text, where numbers
are always written out as words (‘thirty-nine’). - Why dirhems—a unit of currency? As one might say, ‘x^2 + 10 x=39 euros’. The implication is that there is a practical use for
the sum; and there is some attempt to justify this later.