Ghiyāth Al-Dīn Jamshīd al-Kāshī was
born in Kāshān, a city in what is now Iran, in the late
14th century. He seems to have been trained in his home town and
began his career there as well. The earliest firm date we have is
1406, when he observed a lunar eclipse. Sometime in the mid-1410s he
moved to Samarkand (now in Uzbekistan), where he was perhaps the
most prominent of the many scholars working for Ulugh Beg. He was
involved in the design and construction of the Samarkand
observatory. He is known both for his astronomical work and for his
prowess as a calculator, most famously his approximations of
\(\pi\) and \(\sin(1^\circ)\).
The Miftāḥ al-Ḥisab, whose title
translates as “The Key to Calculation,” was written
around 1427. Its goal is to provide practical rules for solving
problems — “all that professional calculators
need.” The rules are presented without justification.
This is the first of three volumes giving a translation of (one of
the manuscripts of) the Miftāḥ. The manuscript
that was used is at the Süleymaniye Library in Istanbul, and is
dated “Ramandan 854,” which translates to October,
1450. That date puts it very close to the original; this may well be
the earliest extant manuscript. In this book, the manuscript is
reproduced photographically on the even-numbered pages, with a
translation on the facing page. Thus, this is not a critical
edition, in which we would expect several manuscripts to be compared
and textual issues to be discussed. Similarly, there are some notes
to the translation, but no detailed analysis of the contents.
The authors are neither professional historians nor Arabists, but
rather mathematicians interested in preserving and calling attention
to a significant part of their heritage. This is clear in their
introduction, for example: authorities are quoted for this or that
opinion about al-Kāshī, but their opinions are often not
assessed and the authors do not argue for their own
interpretation.
The translation seems to be fairly literal, resulting occasionally
in fairly strange English:
The procedure in this is to multiply whatever is in the first
positions of the multiplicand, I mean the digits from the right side
by each digit of the multiplier going from right to left, and we put
down the first result. If there is no tens in the result, then we
write a zero in its place… (p. 55)
Part of the difficulty here comes from the fact that mathematical
operations are being described in words, but the English could have
been better.
While the text of the Miftāḥ includes a list of its
contents (pages 25–33 of the translation), the book itself has
no table of contents for the translation. All the contents list is
“The Miftāḥ Translation” beginning on page
15, followed by a “Glossary” on page 237. A reader
looking, say, for the beginning of the discussion of the second
treatise (on fractions) will have to page through the translation
looking for it. Indeed, there is no indication of which part of the
Miftāḥ is included in this first volume. (It contains
first three treatises described on pages 25–27; I assume that
the next two volumes of the translation will correspond to the
fourth and fifth treatises, on measurement and algebra,
respectively).
Scholars who study the mathematics of Islamicate societies often
point out that there are many untranslated texts and probably many
yet-to-be-discovered manuscripts as well. There are just not enough
specialists in this period of the history of mathematics. Thus, it
is useful to have this translation of the Miftāḥ;
perhaps it will spur the professionals to give us a proper critical
edition, translation, and analysis.
Fernando Q. Gouvêa is Carter Professor of Mathematics at
Colby College.