American Scientist

An Imaginary Tale: The Story of √-1. Paul J. Nahin. 251 pp. Princeton University Press, 1998. $24.95.

Trigonometric Delights. Eli Maor. 236 pp. Princeton, 1998. $24.95.

As a teenager, Richard Feynman recorded in a math notebook his wonder at Euler's formula e^iπ + 1 = 0. This mathematical identity links the five most important constants—e (the exponential), i = √-1 (sometimes written as j), Π , 1 and 0—with the three most important operations (power, + and =). A history of the exponential has been published recently in Eli Maor's e: The Story of a Number (Princeton, 1994; in paperback this year), whereas a potpourri of pi and its properties is presented in David Blattner's The Joy of Π (Walker and Co., 1997). Now, in books that can be read for fun and profit by anyone who has taken courses in introductory calculus, plane geometry and trigonometry, Paul Nahin (an electrical engineering professor) and Eli Maor (a mathematics historian) provide further conceptual and historical insights into these mathematical constants that so fascinated the young Feynman.

Nahin's An Imaginary Tale: The Story of √-1 traces the development of ideas about complex numbers, that is, numbers of the form x + iy, with x and y conventional "real" numbers and iy called "imaginary." The puzzling aspects of i appeared in the 15th century when there were problems with finding roots of cubic equations. In the 17th century, Descartes, the inventor of Cartesian coordinates, denied the possibility of a geometrical construction for imaginary numbers. However, in 1797 the Norwegian surveyor Caspar Wessel first introduced the complex plane, in which x forms one axis and y the other. This plane was soon rediscovered by Buée and Argand in France and eventually by Gauss in Germany. There soon followed an abundance of properties of complex numbers uniting analysis, geometry and trigonometry, such as Euler's formula given above, although this result was essentially first obtained by Roger Cotes, a colleague of Isaac Newton. All of this, up to the time of Cauchy and the theory of functions of complex variables in the mid-19th century, is clearly described by Nahin with many pretty examples.

Maor's Trigonometric Delights starts the story much earlier than Nahin’s history, since trigonometric ideas can be traced back to the building of the pyramids of Egypt. Maor clearly describes the interplay between trigonometry, astronomy, geometry, cartography and analysis. Following several of the chapters are interesting biographical sketches, such as that of Maria Agnesi (1718–99), one of the first female mathematicians. Maor writes that the "witch of Agnesi" curve first described by her shows up only rarely in applications, noting that it occurs in probability theory as the Cauchy distribution. However, as physicists and electrical engineers know, this curve is very common in descriptions of resonance phenomena as the Lorentzian curve, after the 19th-century physicist, H. A. Lorentz. Generally, Maor's book is more descriptive than Nahin's and develops the mathematics in less detail. Both books make interesting reading and will help readers of American Scientist to appreciate the development of these branches of mathematics.—William Thompson, Physics and Astronomy, University of North Carolina, Chapel Hill