120 A History ofMathematics
by 2x^3 + 5 x+ 5 +( 10 /x). This is far from being the hardest such sum which will be tackled; in
particular:- all the signs are positive;
- the division has an exact result.
At this point the simple-minded reader might reasonably ask: what on earth did al-Samaw’al
have in mind? The calculations which he undertook seem to be an end in themselves, a display
of technical virtuosity on a theme which could have had no practical application, and which led
nowhere. The example shown above, by the way, is by no means the end of the story; later, a division
by ‘six squares and twelve units’( 6 x^2 + 12 )has no exact result. He therefore simply continues
as far as possible, noting finally that any future coefficient can be determined by a formula. He is
clearly on his way to understanding a particular form of infinite series. (The calculation is discussed
in Berggren 1986, pp. 117–18.) Who the audience of his book could have been, and what they
made of his work, remains at present a mystery; no subsequent algebraist refers to it. And on the
face of it, such preoccupations give the lie to any easy characterization of Islamic mathematics as
practical or down-to-earth.
A clue could be provided by a still more obscure work of al-Samaw’al, his recently discovered
unpublished arithmetic. This is discussed at some length by Rashed (1994), where an extract is
provided (untranslated), with a promise of future publication of the whole. In this text, accord-
ing to Rashed, al-Samaw’al effectively introduced decimal fractions, using a schema very much
like the one in Fig. 6; with ascending and descending powers of 10 (successive figures in the
decimal expansion) taking the place of the powers of the unknown ‘thing’. This of course has
a much more useful appearance from our present-day viewpoint, although as Rashed concedes
by writing the numbers in a table al-Samaw’al had not yet arrived at a simple and efficient
notation.
Once the phrase ‘decimal fractions’ is mentioned, we have to deal with a long-running
controversy over who was their originator. The question is interesting, but not because it really
matters much any more. In textbooks from the 1950s or before, it was claimed that the invention
was due to Simon Stevin (Netherlands, 1574), despite the fact that al-K ̄ash ̄i’s much earlierCalcu-
lator’s Key, which used them extensively, was already known widely enough. There was no obvious
line of influence from al-K ̄ash ̄i to Stevin, and Stevin’s was the first European discovery; it followed
that he was the inventor.
Besides the obvious Eurocentrism of such a judgement, and the increasing evidence that
al-K ̄ash ̄i’s work did influence western Europe via Constantinople and Venice,^11 this illustrates
the whole problem of how one ascribes priority. The main interest in a mathematics textbook
(medieval or modern), is to explain how you use a technique, not where the author obtained it;
and this seems true even of Islamic writers who worked in a culture where citation of sources
could be quite careful. Hence even where work is original, such originality may not be claimed,
and this leaves the field wide open for historians (who may care more than is necessary) to argue
about who is copying whom, and whether a writer really understands the method he is explaining.
Al-K ̄ash ̄i certainly did know what decimal fractions were; he has a technical term for them, and- This issue is discussed by Youschkevitch (1976, p. 75) and Rashed (1994, pp. 131–2).