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Episodes in the Mathematics of Medieval Islam

J. L. Berggren
Springer Verlag
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Glen van Brummelen
, on

Teachers who use history to motivate mathematics are usually well aware of its power to transform the subject from alien symbol-pushing to an exciting human adventure. Judging by the popularity of "history in the classroom" sessions at mathematics meetings and the number of accessible books written on the subject in recent years, we are now witnessing a "historic" pedagogical shift. But while the sources are plentiful, they are not balanced — and material for non-Western cultures is often the most difficult to find. One popular history of mathematics textbook, in fact, relegates all of Islamic and Chinese mathematics to one section at the end of the chapter "Twilight of Greek Mathematics: Diophantus"!

This state of affairs is especially lamentable because much of the mathematics we teach has roots in and connections with these cultures. In the case of Islam there is extra cause for dismay, since Islam falls naturally between Greece and the Renaissance as part of our Western tradition. One admirable effort to make Islamic mathematics accessible to educators has been J. L. Berggren's Episodes in the Mathematics of Medieval Islam. (Full disclosure: the author was the reviewer's doctoral supervisor.) Written in 1986 and inspired by Asger Aaboe's classic Episodes in the Early History of Mathematics, this book contains a wealth of classroom-ready examples of much of the mathematics one finds in high school and early college: arithmetic, geometry, algebra, and trigonometry.

Why review an 18-year-old book? For one, Springer has just released a paperback edition, identical to the original (other than the plain cover) at about half the price. For another, Episodes remains the only book-length source of Arabic mathematics accessible to teachers, and the material has not been outdated. Springer has taken the right step by issuing a paperback edition to get the book into the hands of a more general readership.

Some episodes fare better than others in today's classroom; for instance, few modern students (alas) will have the opportunity and patience to gain the background required to understand spherical astronomy. However, most examples in this book, properly motivated, can enliven an otherwise pedestrian lecture. For instance, Jamshid al-Kashi's method of determining 5th roots, made obsolete by the likes of Texas Instruments, motivates the study of the binomial theorem by allowing the student to see the theorem work itself out in a practical numeric context. The uses of conic sections through sundials and applications of geometry to Islamic façades demonstrate that the world of geometry, so often portrayed as pure and unsullied by outside influences, interacted with culture in significant and diverse ways. In algebra, the descriptions of various computational and geometric approaches to the solutions of quadratic and cubic equations emphasize the validity of casting problems in different ways to achieve different insights. The arithmetization of algebra by al-Karaji and others is one of many ways in which Islam transformed our Greek mathematical heritage into something much more familiar to our students. The close historical partnership between trigonometry and astronomy is underscored well. In some of my course evaluations, students recall with amazement our attempt to reproduce al-Biruni's method for finding the circumference of the earth using nothing more than a yardstick and a large piece of cardboard.

The re-issue of this gem is significant and welcomed. It will enrich your classes and deepen your perspective on mathematics and culture.

Glen van Brummelen teaches at Bennington College and has written extensively about Islamic mathematical astronomy.


Preface; 1. Introduction; 2. Islamic Arithmetic; 3. Geometrical Constructions in the Islamic World; 4. Algebra in Islam; 5. Trigonometry in the Islamic World; 6. Spherics in the Islamic World; Index.