Poetry of the Universe, by Robert Osserman. Published by Doubleday, 210 pages, 1995, $18.95.

Reviewed by Gene Mirabelli

If you've ever wondered what shape the universe is in you should read Robert Osserman's dazzling little book on the subject. It's easy to overlook this gem, since it's hidden under the sickly title Poetry of the Universe, and that's a shame. Osserman, a Stanford mathematician, leads the reader on a fascinating trip through the cosmos or, to be precise, he provides the reader with a series of maps.

The author begins with the story of Eratosthenes, the librarian at Alexandria in the third century B.C. who realized the earth was round and calculated its circumference. Eratosthenes measured the angle of a noonday shadow on the summer solstice and performed a very simple geometric calculation to arrive at his answer. He came up with an amazingly accurate figure, the Alexandrian equivalent of 25,000 miles. With Osserman as the teacher you can do as well as Eratosthenes, which is part of the charm of this book.

Now ancient map makers were faced with a frustrating problem. How do you make an accurate flat map of a spherical world? The inaccuracies are relatively small on the maps we use to get around town, but they are there. They have to be there. You knew that didn't you? Even if you didn't, the author makes you feel as if you knew it all along. And he explains how those tiny errors add up when the map maker tries to encompass the whole globe -- as on a Mercator projection where a step around the North Pole is as long as a hike along the equator. There are, of course, many ways of mapping the spherical onto the flat, but each one produces its own grotesque distortions. Furthermore, these inaccuracies are even worse than they look at first, because the earth is not perfectly spherical -- it's somewhat squashed at the poles and bumpy with hills and valleys.

Osserman points out that Eratosthenes had been able to derive the shape and size of the earth by taking a few measurements on a very small part of it -- a part which looked flat. In the nineteenth century Carl Friedrich Gauss did somewhat the same thing on a more profound level. He composed a set of formulas which allowed a surveyor to determine the nature of the local curvature of any area of the earth by making simple measurements and calculations. Thanks to the author, you can grasp the fundamentals of Gauss's work even if you have only basic geometric skills. It was about this time that Nikolai Ivanovich Lobachevsky, just for the mathematical fun of it, proposed a non-euclidean geometry -- it would be the same as Euclid's geometry but with one of the crucial rules, the parallel axiom, changed. It worked fine; it didn't touch the world of the street where people were walking in euclidean space, but like an old game with a new set of rules, it worked.

A short time later Georg Friedrich Riemann came up with an even bolder idea; we should try to free ourselves from all our preconceptions about space -- maybe space isn't the familiar euclidean volume we think it is. Now, Riemann surfaces are not for the mathematically naive, but our author avoids that fascinating detour and stays with his narrative. We should explore the geometry of space, Riemann suggested, by taking measurements and recording what we find. Albert Einstein did so and came to an astonishing conclusion. Rather like Eratosthenes who while living in flat Alexandria figured out that the real estate was actually on a sphere, Einstein concluded that space itself was curved, a concept almost as hard to grasp as the idea that the earth is round.

Poetry of the Universe is remarkable for its clarity, its brevity, its easy-going presentation of subtle concepts. Osserman pretends to tell a guileless tale about the difficulty of making maps, but along the way he mentions a little bit of this and a little bit of that. By the end of the story the reader has effortlessly taken the steps necessary to understand that if he traveled far enough outward, say to the edge of the cosmos, he might at last come to that place where he could look further out into the heart of the place he left behind. That's Einstein and it's the same map that Dante drew back in the fourteenth century.