You are here

Al-Kāshī's Miftāḥ al-Ḥisab, Volume I: Arithmetic

Nuh Aydin, Lakhdar Hammoudi
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvea
, on

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.