The history of the physical sciences is filled with dead ends. No chemist today would try to create a new compound from the four ancient elements of earth, air, fire, and water. Deep space probes to the outermost reaches of the solar system have found no evidence of revolving crystalline spheres with the earth at the center. Mathematics, by contrast, is a steady progression. What was true for Euclid more than two millenniums ago is still true today. In this way, mathematics is its own history.

For evidence of this, one need only turn to a recent sample of books that attempt the unenviable task of explaining modern math to the popular reader. A good place to start is Keith Devlin's The Millennium Problems, an account of seven puzzles, the solution to which is worth a cool million each.

Before you leap to pick up a pencil and paper, though, it would be well to read this remarkable effort to explain exactly what is involved. Devlin is doubly qualified as the "Math Guy" from NPR's Weekend Edition and the author of numerous previous books on mathematics. Nonetheless, the job of describing these problems in understandable terms is far from easy. The scope of modern mathematics has become so complex and far-reaching that many professional mathematicians are not exactly clear about what each of these enigmas may mean. Devlin does a superb job trying, though, and his account is both fascinating and accessible to any reader who can remember some high school math.

For background it might be a good idea to look in Barry Mazur's Imagining Numbers, a poetic and profound meditation on the mathematical imagination. Mazur "does mathematics," as he says, at Harvard University and is an authority on (among other things) the branch of math called "number theory." This is the study of numbers and fractions, although considerably advanced from grade-school arithmetic. Mazur explores the history of imaginary numbers, that is, numbers that are some multiple of the square root of minus 1. He evocatively recreates the bewilderment of the Renaissance scholars who first grappled with this idea, and through the lens of their imagination, he leads the reader to a deeper comprehension of the realm of numbers.

The first of those seven $1 million millennium problems, by the way, is the oldest, devised by the German mathematician Bernhard Riemann in 1859. Riemann was interested in prime numbers (3, 7, 11, 13), those curious figures that cannot be evenly divided except by 1 and themselves. From the time of Euclid, the primes have presented a riddle for mathematicians trying to understand their puzzling pattern. This question has become extraordinarily significant for us because mathematical operations on prime numbers are the basis of most of the current systems that allow shoppers to send credit card numbers securely across the Internet.

In particular, the question of how many prime numbers are less than a given number is an important one for number theory. Remarkably, the answer seems to be somehow connected to the realm of complex numbers, that is, of numbers that have one real part and one imaginary. In a lecture Riemann gave on this topic, he offhandedly remarked that there was a curious feature of this relation, but since it was not really on the topic at hand, he was not going to go into it. His casual conjecture has since become known as the "Riemann Hypothesis," and it remains unproven to this day.

In The Music of the Primes, British mathematician Marcus du Sautoy attempts to explain some of the efforts that have been made on this "Everest of mathematics." Du Sautoy is a professor of mathematics at Oxford and a frequent guest on BBC television and radio. He notes that just about every important modern mathematician has tackled the problem, without much success. His account is fascinating, filled with odd twists such as the French mathematician André Weil, who managed to solve a related problem while in a World War II prison awaiting execution. Fortunately, he was pardoned, but somehow was never again able to achieve the level of concentration he had managed while in jail.

All of these works have their usual share of pixie dust, the popular science writer's practice of skipping over the hard parts. The most detailed, and consequently the most rewarding account of the Riemann Hypothesis is John Derbyshire's Prime Obsession. The author, a trained mathematician with a day job as an investment banker, moonlights as a novelist. This remarkable constellation of interests results in a math book that reads like a mystery novel. When, some 300 pages into the book, Derbyshire finally presents Riemann's conclusion, it is with literally breathtaking impact.

If the laws of nature are truly written in the language of mathematics, understanding Riemann's conjecture with the help of these passionate authors is like a glimpse into the innermost workings of the universe.

Around 300 BC, the Greek mathematician Euclid proved that there are an infinite number of prime numbers (figures that cannot be evenly divided except by 1 and themselves). But that was just the beginning. Since then, even with the most advanced computers, no one has been able to devise a formula for predicting the sequence or pattern of prime numbers. Indeed, their mysterious quality is the linchpin of modern-day banking security. Landon Clay, a Boston businessman, proposed a $1 million reward to anyone who could solve this conundrum, one of seven Millennium Problems announced in 2000 in Paris.

• *Frederick Pratter is freelance writer in Missoula, Mont., who enjoys his pi à la mode.*