118 A History ofMathematics
Fig. 5Table from al-Samaw’al.In al-Samaw’al’s famous phrase, from his introduction, his aim is to proceed ‘by operating on the
unknowns by using all the arithmetical tools which the arithmetician uses to operate on known
numbers’. In other words, you must—at least—be able to add, subtract, multiply, and divide your
‘things’ (xs andys) any number of times. This leads to expressions which are complicated in our
terms, let alone in twelfth-century notation, where ‘1/x’ is designated by ‘part of thing’ and so on.
Al-Samaw’al wastes no time; by his fourth page he is giving a table of powers of the thing up to
the ninth, which we would callx^9 and he calls ‘cube cube cube’ in the positive direction, and down
to 1/x^9 , or ‘part of cube cube cube’ in the negative. The table (reproduced in Fig. 5) is an interesting
mixture of notations. While the second row describes the powers in words (‘square cube’ etc.), the
first row keeps track in a more rational way by using numbers going in both directions (expressed
by letters of the Arabic alphabet),includingzero. Underneath he gives the examples of powers,
positive and negative, for the numbers 2 and 3. And here another notational problem; while the
Indian numbers do very well to express 2, 4, 8,...,2^9 =512, the corresponding fractions have
to be written in words starting with ‘half ’ and ending with ‘an eighth of an eighth of an eighth’.
(In parenthesis, one notes that the ease with which the Egyptians, and the Greeks following them
wrote unit fractions seems to have disappeared; changes in notation are not always for the better.)
The power zero is correctly assigned to 1.
One has a sense, in the chapter on polynomials which follows, that al-Samaw’al is working at
the limits of the notational possibilities which were then available, and trying to expand them