BOOK REVIEW

# The Math of Make-Believe

*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*

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