110 A History ofMathematics
translated. In fact, with such disparate sources, the idea that the Islamic work could be simple
borrowing and transmission makes no sense; a synthesis was essential. This involved raising what
appeared to be unanswered questions, and writing new books in more useful forms for practical
ends (as the examples above illustrate).
In an article which we have already cited, which forms one of the most interesting theoretical
discussions of early Islamic mathematics, Høyrup claims that this new synthesis marked a radical
change in the use of mathematics comparable to the work of the Greeks discussed in Chapter 2.
[T]he break [which led to the acknowledgment of the practical implications of theory] took place earlier, in the
Islamic Middle Ages, which first came to regard it as a fundamental epistemological premise that problems of social
and technological practice can (and should) lead to scientific investigation, and that scientific investigation can (and
should) be applied in practice. Alongside the Greek miracle we shall hence have to reckon anIslamic miracle.(Høyrup
1994, pp. 92–3)You are referred to Høyrup’s article both for his detailed arguments in establishing the nature
of the new approach, and for his attempts to account for its origins. He considers and rejects
a number of suggestions, finally opting for a description of the nature of Islam which he calls
(perhaps unfortunately) ‘practical fundamentalism’.
We shall return to the role of Islam as religion, philosophy, and way of life later. Let us now look
at the interaction between new and old in the knowledge produced by the early mathematicians.5. Algebra—the origins
I have established, in my second book, proof of the authority and precedent in algebra andal-muq ̄abalaof Muh.ammad
ibn M ̄us ̄a al-Khw ̄arizm ̄i, and I have answered that impetuous man Ibn Barza on his attribution to ‘Abd al-H.amid,
whom he said was his grandfather. (Ab ̄uK ̄amil, cited Rashed 1994, p. 19, n. 3)
I have always been very anxious to investigate all types of theorems and to distinguish those that can be solved in
each species, giving proofs for my distinctions, because I know how urgently this is needed in the solution of difficult
problems. (Khayyam 1931, p. 44)The word, in its derivation (from Arabic ‘al-jabr’, usually rendered ‘restoring’), suggests that
what we call algebra begins with the Arabs. Like all other questions of origins, this can be disputed
on various grounds; we have seen that the Babylonians knew how to solve problems which were
equivalent to quadratic equations (Chapter 1). So what was so important and influential about the
Islamic contribution? There is no better place to start than the original textbook by al-Khw ̄arizm ̄i.
This was enormously influential both in the Islamic world, and in medieval Europe; ab ̄uK ̄amil,
as quoted above, illustrates the general agreement about al-Khw ̄arizm ̄i’s priority, and his method
and language survived with adaptations until the sixteenth century in Europe, when something
more like our modern notation was introduced. Part of the text of his book (1986) is reproduced in
Appendix A. This illustrates the core of the book, the treatment of quadratic equations, although
a very large part is in fact given over to ‘applications’ to practical situations (e.g. inheritance),
and to geometry. He defines‘roots’, ‘squares’, and ‘numbers’, the three objects which enter into
his algebra, in terms of what you will do with them; the definition is not so much conceptual as
operational, and this itself throws light on how he is thinking.
A root is any quantity which is to be multiplied by itself, consisting of units, or numbers ascending, or fractions
descending. (Fauvel and Gray 6.B.1, p. 229)