Islam,Neglect andDiscovery 119
where he can. Sometimes an example (such as( 10 /a^3 )(a^2 +a)=( 10 /a)+( 10 /a^2 ))is set out
and explained in words; sometimes a more general formula (such asa(b/c)=b(a/c)is described by
using a series of Arabic lettersa,b,c,...to denote the ‘unknowns’ and the results of multiplying and
dividing them. This in itself is not new—the use of letters to denote general numbers or quantities
can be paralleled in Euclid—but in combination with the traditional algebraic language it gives
the feeling (which Rashed expresses strongly) that we have something near to a ‘new’ abstract
algebra.
The real coup is achieved when, after another 24 pages, al-Samaw’al sets out to divide two
expressions (polynomials) according to the schema shown in Fig. 6. Before the table is set up,
the problem is set out in words (with a few figures interspersed). Translated into our notation, it
amounts to dividing
20 x^6 + 2 x^5 + 58 x^4 + 75 x^3 + 125 x^2 + 96 x+ 94 +140
x+
50
x^2+
90
x^3+
20
x^4Fig. 6Table from al-Samaw’al showing division.