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uses them simply and with facility. In some sense, his introduction of them seems to be a claim to
their invention—allowing that one does not always know who may have preceded one.
We divided the unit into ten parts, we then divided each tenth into ten parts, and then each of them into a further ten
parts, and then each of them into a further ten parts and so on, the first division being into tenths, and in the same
way the second into decimal seconds and the third into decimal thirds and so on, so that the orders of decimal fractions
and wholes are in the same relation as is the principle in astronomical numbering [i.e. sexagesimals].
We call this ‘decimal fractions’. (Al-K ̄ash ̄i 1967 book 3, chapter 6)
From this (rather late) stage in his book, al-K ̄ash ̄i sets out his results, where possible, in both forms,
both sexagesimal and decimal. Whether his work ‘diffused’ to Stevin, whose notation was different
and in some ways less user-friendly, is still unclear, though it appears increasingly a possibility.
But before al-K ̄ash ̄i, as Rashed pointed out, stands al-Samaw’al, who also can claim a place as
inventor; and before him there appears (according to Saidan) the still earlier tenth-century figure
of al-Uql ̄idis ̄i, who seems to be using decimals in at least two places in his book. And in between
these writers there may be many others of whom we know nothing. Rashed considers the claims
of al-Uql ̄idis ̄i unacceptable; there is no sign that he was following a practice which he understood
in a systematic way. On the other hand, he may have been one of a number of reckoners who had
realized the obvious fact, as al-K ̄ash ̄i states it: that, with Indian numbers as with sexagesimals,
you could continue on the right as well as on the left, with your number (e.g. ‘5’) having a smaller
meaning the farther you went. This is what al-Uql ̄idis ̄i seems to be doing when, in one of his crucial
passages, he performs a sequence of halvings on 19:
For example, we want to halve 19 five times. We say: one half of 9 is four and a half; we set the half as 5 before the
four; [remember that, Arabic being written right to left, ‘before’ means ‘to the right of ’] next, we halve the ten. We
mark the units place. That becomes 95.
Now we halve the five and the nine; we get 475. We halve that and get 2′375, the units place being thousands to
what is before it, for if we want to say what we have got, we say that halving has led to two and 375 of one thousand...
A great deal of ink has been spilled over that single dash between the 2 and the 375; is it a decimal
point, and why are there no others; and did al-Uql ̄idis ̄i understand the fact? A tentative conclusion
might be that he did, to some extent, but that he would not dream of ‘codifying’ the idea as al-K ̄ash ̄i
did five hundred years later; he was a calculator, not a mathematician. Indeed, the illustration
shows that the actual discovery of decimal fractions is not as much of a marvel as one might
suppose. If you want to show your skill in Indian numbers by halving repeatedly, then you fall upon
them almost naturally.
Having mentioned al-K ̄ash ̄i in the context of decimal fractions, let us now turn to a broader appre-
ciation of his work, and ofTheCalculator’sKeyin particular. In the long letter to his father published
by Edward Kennedy in (1983), al-K ̄ash ̄i gives a fascinating picture of the court of his patron Ulugh
Beg. This may have been the ‘end’ of Islamic mathematics as far as our official histories go, but the
society is far from being in decline; the atmosphere is one in which a sizeable and intensely com-
petitive community of scholars strive to obtain the king’s approval, primarily on the basis of their
mathematical ability. Al-K ̄ash ̄i, who was not given to false modesty (in Kennedy’s classic understate-
ment) makes it clear to his father that he has consistently come out best in all of these competitions,
partly because of his skill in combining theory, calculating ability, and knowledge of the construc-
tion of instruments. It was for this unusually mathematically literate community of teachers and