Gauss supposedly once said that the three greatest mathematicians in history were Archimedes, Newton and Eisenstein. Eisenstein, of course, gave us the Eisenstein criterion for the irreducibility of polynomials. Newton gave us calculus and gravity, but what about Archimedes? Do his mathematical discoveries really outshine those of Euler, Cauchy, Descartes and all the others Gauss left off his list? Is he really in a league with Newton and Eisenstein?

You should, of course, decide for yourself. Unless you read Greek, you could read Heath's widely available translations of the works of Archimedes, or the newer and less literal translations of Dijksterhuis, or you could read Sherman Stein's new little book on Archimedes. Some readers will already know Stein as the author of a well-known calculus text. The Archimedes book is considerably shorter.

The most intriguing legends about Archimedes are probably false. He probably didn't threaten to move the earth with a lever, or race naked through the streets crying "Eureka!" (despite the title of this book). He probably didn't burn ships with mirrors. Perhaps he didn't even get murdered by a Roman soldier after scolding him "Do not disturb my circles." Stein warns us how many of the legends about Archimedes are doubtful.

What we do have, though, a dozen surviving mathematical works, another four suspected works, and at least one work known to have been lost. Stein describes the main results of nine of these twelve surviving mathematical works. Generally, he sticks as close to Archimedes' exposition as possible, but occasionally he sums a geometric series or moves the proof of a lemma to the appendix to clarify things.

Chapters 4 and 5 are devoted to Archimedes' Method of Treating Mechanical Problems. The first chapter details its dramatic discovery as a palimpsest in the first years of the 20th Century, and its equally dramatic re-emergence and auction in 1998. The second of the two chapters describes some the sensational contents of the palimpsest. It gives a sense of how much Archimedes' methods resembled modern calculus.

Chapters 6 and 7 describe Archimedes' quadrature of the parabola, another example of his calculus-like tools.

Chapter 11 gives details of Archimedes' approximation of pi, how he showed that it is larger than 3 1/7 yet smaller than 3 10/71.

Stein wisely concentrates on the surviving mathematical works, but, strangely, he doesn't even mention the works attributed to Archimedes. There is no mention of his famous "Cattle Problem", or of the Stomachion, the fragment that describes polygon puzzles and games. He understandably ignores On Conoids and Spheroids, a seldom-mentioned treatise, but, mysteriously, he overlooks The Sand Reckoner, in which Archimedes estimates the number of grains of sand that it would take to fill the universe. It is a particularly interesting work because it entirely debunks the myth that the Ancients did not realize that the Earth was round, and it also develops a way to give names to very large numbers. The Sand Reckoner is written in a style that is a bit awkward for a modern reader, or at least the Heath translation makes it seem so. That makes it all the stranger that Stein did not use his expository skills to clarifying the article.

The book is most timely, appearing, as it does, so soon after the October, 1998 auction, for $2 million, of the Archimedes Palimpsest, and, in fact, while the Palimpsest itself was on exhibition at The Walters Gallery in Baltimore. By the end of the 155 pages, the reader is convinced that Archimedes deserves his position on Gauss' pedestal. It also follows closely Bill Dunham's book on Euler, that did for Euler what Stein has done for Archimedes.

More books like Stein's book on Archimedes would be welcome.

Ed Sandifer ( [email protected]) is a professor of mathematics at Western Connecticut State University and has run the Boston Marathon 27 times.