122 A History ofMathematics
learners that al-K ̄ash ̄i wroteThe Calculator’s Key, a very diverse collection of arithmetic, algebra,
and geometry with results of the most various kinds. Unlike al-Samaw’al’s book, this became some-
thing of a best-seller; the British Library, which is not strong on mathematics, has four manuscripts,
two from the nineteenth century. His aims are stated at the outset, after a brief summary of his many
achievements:
Although some of these [methods] could not be discovered with the help of the six algebraic [forms] (i.e. al-Khw ̄arizm ̄i’s
six quadratic equations), yet in the course of this work I found numerous principles with whose help the groundwork
of arithmetic is developed by the simplest means, on the easiest road, with the greatest profit and with the clearest
exposition. I decided to write these principles and desired to clarify them so that they could be an instruction for
others and a guide for the learned. Therefore, I have written this book and collected in it all which calculators may
need, avoiding both the tedium of long-windedness and the excess of brevity. For the majority of methods I have
drawn up tables, so as to simplify examination by the geometer. All the tables established in this book have been
prepared by me and to me belongs all that is sweet and bitter in them, with the exception of seven tables...(Al-K ̄ash ̄i
1967, intro)Indeed, the tables are a notable contribution to the work. We may already see a heavy dependence
on the table for the exposition of complex calculations in al-Samaw’al; but in al-K ̄ash ̄i they are
everywhere, as he admits. There are the standard tables (multiplication, conversion from decimals
to sexagesimals and back; sines, and so on); tables of currency conversion, of the properties of
metals and other substances; tables of the areas of polygons, and more usefully (one might think),
of different kinds of arches used in architecture (see Fig. 1). Almost always the numerical results
are more accurate than they have any reason to be, and often they are given both in decimals and
sexagesimals. As can be seen from the quote, al-K ̄ash ̄i feels that they are an important contribution;
he asserts his intellectual property in them, as well as an emotional relation (the sweet and the
bitter). Most famously, beyond the ‘static’ tables, we have the ‘dynamic’ ones which show how
you do a calculation. The reader is shown how to construct them, told in detail where to draw
horizontal and vertical lines and make entries, so as (for example) to extract a root; and the often
quoted example in which he extracts the fifth root of 44,240,899,506,197 in decimals can serve
as a model.
This example (of a method which may be due to the Chinese, even if they did not carry it to such
lengths—see Chapter 4; and which al-Samaw’al worked, if with less explanation, in sexagesimals)
has been extensively discussed, in particular by Berggren (1986, pp. 53–63). When he comes to
doing the same and more in sexagesimals, it is more a summary:
In our treatise entitled ‘Treatise on the circumference’ [his calculation ofπ], we have found the roots of many
numbers with many digits and adapted them in different ways. Anyone who wishes to know more can turn to this
book. Furthermore, we present here an example of the extraction of a cube root and another example of the extraction
of a cubo-cube [sixth] root, but, so as to avoid long-windedness in this book, we shall not here give an explanation of
the process [as he did for the fifth root]. It is easy for anyone who knows how to do it with Indian numbers, as it was
explained in Book 1.At a certain point, we see, the tables, which are given, are a substitute for an explanation of
the method.
To see al-K ̄ash ̄i’s style of exposition in a different context, an extract from the geometrical section
of theCalculator’s Key, on the regular solids, is in Appendix C, with the inevitable table which
sets out all the measurements you may possibly need for them. Clearly considered an outstanding
mathematician by his circle and beyond, al-K ̄ash ̄i still appears something of an enigma. Given