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# BOOK REVIEW : A Special Text Presents Math as an Art Form : JOURNEY THROUGH GENIUS: THE GREAT THEOREMS OF MATHEMATICS by William Dunham John Wiley & Sons \$19.95, 300 pages

May 15, 1990|LEE DEMBART

Reviewing a book a week doesn't allow much time to stop and dawdle, but "Journey Through Genius" is one that I wanted to savor for a while. It's a very special math book, as good as any popular book in the subject sincT. Bell wrote "Men of Mathematics" in 1937.

There's probably no such thing as a math book for people who don't like mathematics, but "Journey Through Genius" comes close. William Dunham, a mathematician at Hanover College in Indiana, has written an inspired piece of intellectual history that explains a dozen of the greatest theorems in the history of mathematics, placing them in their historical contexts in the larger world as well as in the narrower world of pure thought.

For example, his chapter on Euclid's proof of the Pythagorean theorem ("In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides") gives a rounded synopsis of the history of mathematics and the history of Greece before turning to Euclid, geometry and right triangles.

To get to the Pythagorean theorem, which is the 47th proposition in the first book of Euclid's "Elements," Dunham marches straight through the first 46 propositions, arguing that the 47th is the emotional climax, the theorem to which all the others point.

Similarly, the chapter on Isaac Newton, which includes a terrific description of Newton's approximation of pi , begins by placing Newton in the same century as Cervantes and Shakespeare, as well as Galileo, Descartes, Pascal and Fermat. Logarithms were perfected and introduced just before Newton was born in 1642.

(The book has a welcome paragraph on logarithms. We who take pocket calculators and computers for granted should recall this early and very successful method for speeding up arithmetic. A century later, Pierre-Simon Laplace observed that logarithms "by shortening the labors doubled the life of the astronomer." By what factor have computers lengthened scientists' lives today?)

Dunham's project is to treat mathematics like an art form: Just as great novels, great symphonies and great paintings are the objects of study in appreciating literature, music and art, theorems are the appropriate objects of study in appreciating mathematics.

When he describes the actual proof of the theorem, the timeless logical insight that solved a problem in pure thought, Dunham's descriptions are crystal clear. Not all of them are easy (Cardano and the solution of the cube and Cantor's investigation of the transfinite realm, for example).

But they all reward effort. They are written for interested, non-specialist readers--preferably those who enjoyed high school math and remember some of it.

Dunham notes that mathematics is built on the twin pillars of arithmetic and geometry--numbers and space--and they have alternately held sway as long as people have thought about such things. As a result, the book embraces geometry (Archimedes' "On the Sphere and the Cylinder") and numbers (Euler's number theory). In each case, Dunham lucidly describes the problem and then equally lucidly presents the ingenious solution.

Unlike any other sphere of human inquiry, the truths of mathematics, once proved, are proved forever. They do not rest on experiment or judgment or opinion. They rest on naked logic.

In general, we think of Greek science as quaint, if anything, but we read Euclid with awe. Twenty-three centuries later, its beauty is still compelling.

Dunham's book captures and conveys the power behind mathematical proof. Mathematics is the science of pure thought. I shudder to think it.

The book contains a fair amount about the characters who do mathematics. It takes a special kind of person to muck about in naked logic, and these peculiarities sometimes manifest themselves in other aspects of mathematicians' lives. They tend to be odd people.

But most important is their work. Having opened up for examination a dozen great theorems, Dunham wonders what about them makes them great and concludes, with G. H. Hardy, that they possess "economy, inevitability and unexpectedness."

In that sense, too, mathematics is like art. The combination of inevitability and unexpectedness packs unbeatable emotional appeal.

OK. So this isn't a book for everyone. I wish it were. But for those who like this sort of thing, this book is a gem.

Next: Jonathan Kirsch reviews "Shark Tank: Greed, Politics and the Collapse of Finley Kumble" by Kim Isaac Eisler.