* An Imaginary Tale: The Story of the Square Root of -1*, Paul J. Nahin, 1998. 257+xvi pp.$29.95 cloth. ISBN 0-691-02795-1. Princeton University Press, 41 William Street, Princeton, NJ 08540, http://www.pup.princeton.edu/ In the United Kingdom: Princeton University Press, Chichester, West Sussex.

The author of this book, Paul Nahin, is a Professor of electrical engineering at the University of New Hampshire , who became fascinated with the mysterious role of √-1 in electronics. Turning to history for insight, he brings the perspectives of both a practical man and a scholar in conveying the enchantment of mathematical ideas for a non-mathematician.

Nahin provides an engaging account that weaves together historical insights, mathematical examples, paradoxes, anecdotes and digressions to capture the essence of this convoluted tale.The solution of cubic equations in the sixteenth century forced mathematicians to deal with the square root of -1, and in the first major chapter of the book, the author handles this episode judiciously and leisurely. After a brief look at geometrical attempts to represent √-1, there is a study of nineteenth century representations, in particular by Caspar Wessel (1745-1818), whose work remained unknown to his contemporaries.

A chapter on various applications, such as the derivation of Kepler's Laws of planetary motion, should convince the reader of the utility of complex numbers. Then follows a pot pourri of neat mathematical gambits going back to the eighteenth century that caught Nahin's fancy, and the book concludes with a sketch of the rudiments of analytic functions and the calculus of residues.

I enjoyed the book. There is good balance between mathematical examples and narrative.The mathematics is handled with dispatch, although the perspective is more technical than structural.

This volume is heartily recommended as ancillary reading, particularly to college students in mathematics and engineering, as well as to talented high school students with facility in algebra and geometry. The material is well-paced and imbued with verve and wit. It will whet their appetites for sophisticated and elegant mathematics.

Ed Barbeau

Professor Emeritus

University of Toronto .