This monograph is a translation by Anna Pierrehumbert of the original 1999 French work by Jacques Sesiano. In the author’s words, its goal is not to be an exhaustive survey of the history of algebra, but “merely to present some significant steps in solving equations and, wherever applicable, to link these developments to the extension of the number system.” The author and his translator succeed admirably in accomplishing this goal. In a mere 140 pages (there are 30 pages of original texts in Mesopotamian (translated), Greek, Latin, Arabic, Hebrew, French, German, and Provencal as appendices), Sesiano and Pierrehumbert give a very enjoyable overview of the development of algebra and number systems.

Sesiano begins with linear and quadratic equations solved in Mesopotamia and the difficulties introduced by the lack of a proper symbol for zero and the need to know 1711 products to carry out multiplication in a base-60 system with 59 non-zero digits. A distinctive aspect of this monograph is its use of actual problems and solutions found in historical documents to illustrate the development of algebra. The author annotates these examples, often rephrasing the steps in more convenient modern notation, and indicates the underlying goals and motivation of the original solvers. He carefully notes the difference between their methods and the methods we would use today.

In successive chapters, Sesiano details algebra’s development. In Chapter 2, he discusses the different types of algebra in ancient Greece, and the use of an unknown variable. In Chapter 3, the development of algebra in the Islamic world, and their approach to geometrically solving linear, quadratic, and cubic equations is covered. In Chapter 4, Sesiano vividly describes the state of algebra in Medieval Europe at the beginning of the twelfth century, the reintroduction of the mathematics of antiquity, and the introduction of Arabian mathematics. Fibonacci and his *Liber abaci *play the central role. The gradual acceptance of negative numbers and the beginning of algebraic attempts to solve the cubic equation conclude the chaper. In the final fifth chapter, Sesiano covers algebra in the Renaissance. The birth of algebraic symbolism, and the subsequent general solution of the cubic and quartic equations by dal Ferro, Tartaglia, and Cardano in the early sixteenth century are described in depth. The introduction and gradual acceptance of complex numbers is discussed. The chapter concludes with a brief discussion of how algebra developed after the Renaissance and the work leading to the proof of the insolvability of the quintic by Abel and Galois in the early nineteenth century.

Both Sesiano and Pierrehumbert deserve credit for creating a book that is a pleasure to read. It is written in a lively style that draws the reader in. While the book is not a comprehensive history, it provides a good overview of the highlights in the development of algebra and will whet the appetite of many readers to learn more. It is written at a level appropriate for use by undergraduate students at all levels and would be profitably used in a history of mathematics course or as supplement in an algebra course.

Tom Hagedorn is Associate Professor of Mathematics at The College of New Jersey.