*Number from Ahmes to Cantor,* Midhat Gazalé, 2000. 297 pp., illustrations, $35 cloth. ISBN 0-691-00515-X. Princeton University Press, 41 William Street, Princeton, NJ 08540. (800) 777-4726, www.pup.princeton.edu

While the title of this book may seem to indicate that it is primarily a history of the concept and development of number, it is not. For those who seek such a reference, go elsewhere. But for readers who would like to explore number theory and acquire some fascinating insights into mathematical thinking, it is a gem. The first fifty-eight pages provide an historical introduction to number systems and numeration, the symbols and the words. This material spans developments from Mesopotamian tally tokens through Leibniz’ metaphysical interpretations of binary arithmetic. The remainder of *Number from Ahmes to Cantor* then takes a conceptual leap into number theory considering such topics as: the division algorithm; mixed base positional systems; periodic bases; congruence’s; Diophantine equations and continued fractions. A final chapter leads the reader on a tour of infinity beginning with infinite series and ending with Cantor’s transfinite arithmetic.

Midhat Gazalé writes from his perspective in the telecommunication industry. Former president of AT&T-France and a Professor of Telecommunication and Computer Management at the University of Paris, he provides a glimpse of modern uses for number theory. His mathematical insights and lucid articulation of mathematical ideas and procedures should inspire teachers. Historical anecdotes, quotations and graphical illustrations enliven the presentation. The demonstrations given for the irrationality of √2 (pp 173-176) and "*e*" (pp 183-184) are pedagogically appealing and the discussion on exploring the concept of infinity (Chapter 7) is excellent. In summary, this is a well-crafted book, an example of good mathematical exposition. For highly motivated students, it can serve as a source of ideas for number theory projects.

I highly recommend this book for personal reading and university library acquisition.

Frank Swetz, Professor Emeritus, The Pennsylvania State University