When I first took a copy of Nahin's book off the shelf, I expected a history book operating under the usual rules that seem to dominate easy reading books on science today - no equations. What I found instead was an unexpected surprise that immediately cemented my decision to purchase the book - it is chuck full of equations. But then, how do you write a book about mathematics without using equations? I'm glad that for this one, at least, the publishers listened to reason.
Of course, the book isn't all equations. There is some downright interesting history in it as well. For the most part, however, this is a book that illustrates the equations (or at least their modern counter parts) that led mathematicians to develop the concept of the square root of a negative number, eventually leading to the branch of mathematics we call today complex analysis. Having said that, I should point out that this is not a mathematics book on complex analysis [for that, a better choice is "Complex Variables," by Mark J. Ablowitz and Athanassios S. Fokas, Cambridge University Press, 1997]. The author does not develop theorems or proofs, and many of the demonstrations stretch the notion of mathematical proofs - but they are not intended to be mathematical proofs at all, but just that - demonstrations. Think of this book as a mathematicians leisurely romp through the mathematical history of root negative one, with an average of at least two or three equations on every page. The mathematics isn't advanced by any means. If you are reasonably grounded in algebra, geometry, trigonometry (and lots of it), and a little calculus (including a few differential equations) you should have no trouble at all. Plan on working through the equations, though, step by step. You won't want to miss a single "aaaahhh."
I really have only two complaints about Nahin's book, both of which are really pretty minor. The first complaint is that none of the equations are numbered. This means the author is constantly saying things like "now go back to the first equation in the last section and notice ...." I found this sometimes hard to follow, and would have appreciated a few key equations having numbers (and a box) associated with them. Another complaint is that the book has some typographical errors in some of the equations that can sometimes interfere with following the derivations.
Don't misunderstand, though. This is one of the best leisure books on mathematics I've read in a long time. The author writes clearly, has an incredible breadth of knowledge, and presents some really beautiful mathematics. It was a real let down when I finally finished, and realized how tough it was going to be finding another book to which I would look with such yearning at the end of the day for a relaxing evening of intellectual entertainment.
The book begins with the story of cubics, and how their solutions involved the square root of negative numbers. From there the book moves toward early work, or the "first try" at understanding complex numbers. There is some interesting history about Rene Descartes and John Wallis, as well as stories about Casper Wessel, Gauss, Argand, Warren, Mourey, and, of course, De Moivre.
The books first three chapters have the most history. The last four chapters offer more examples of how complex analysis has played a pivotal role in science and technology. The author offers some interesting uses of complex analysis in the solving of integrals, trigonometric identities, Kepler's laws of satellite orbits, and, of course, circuit analysis in electrical engineering.
My favorite chapter by far is chapter six, titled "wizard mathematics." It seems there was a "aaaahhh" on at least every other page. This chapter is devoted to illuminating some of the mathematical prowess of wizards such as Euler, Bernoulli, Fagnano, Cotes, Riemann, and Schellback. Plan on using up at least one highlighter on this chapter alone.
Nahin ends with a chapter on complex analysis in the nineteenth century, and Cauchy's integral formulas (there is also a brief discussion and derivation of Green's theorem). Then, as with the other chapters, Nahin gives lots of examples of what you can do with these mathematical tools, and where they can take you.
Easily one of the best books I've ever read. If you love mathematics, your library really cannot be considered complete unless this book, tattered and worn with lots of dog-eared pages and scribbles all over the margins, is on the shelf.
Duwayne Anderson September 22, 1999
