BOOK REVIEW: SCIENCE : Reaching a New Understanding of the World We Live In : POETRY OF THE UNIVERSE: A Mathematical Exploration of the Cosmos <i> by Robert Osserman</i> , Anchor Books $18.95, 210 pages

Wow. This short book manages to pack a lot in. It is the story of how civilization discovered--long before Columbus--that the Earth is round (even though it looks flat) and, by analogy, how scientists have discovered that the universe is curved (even though it, too, looks flat).

More important, it repeatedly brushes up against and finally asks the deep philosophical question first put by Eugene Wigner, the Nobel Prize-winning physicist: Why does mathematics, a construct of our minds, so precisely describe the physical world?

This is no idle question, though no one has a clue what the answer is. Robert Osserman shows that many branches of mathematics that were developed purely intellectually without any conceivable practical use subsequently wound up being used by physicists and cosmologists to make sense of the universe.

The most famous example--but hardly the only one--is non-Euclidean geometry, which seems to violate our common-sense experience of the world. It was developed in the 19th Century as an alternative to Euclid, and in the 20th Century, it became the geometry of Einstein’s space/time reality.

“The lesson of the past is that we cannot tell in advance how long we may have to wait to find a use for each new mathematical creation or where in the real world it might arise,” Osserman writes.

But I am getting ahead of myself (probably because I am fascinated by these unanswerable philosophical conundrums).

Osserman, a mathematician at Stanford and at the Mathematical Sciences Research Institute in Berkeley, starts at the beginning and does a masterful job of describing how the ancients discovered that the Earth is round and then went about measuring its circumference--with surprising accuracy.

He then clearly lays out the centuries-long efforts to map a spherical surface on a two-dimensional flat map without distorting something along the way. This, we now know, cannot be done. Think of peeling an orange and then trying to flatten the peel in one piece. It’s impossible.

As a result, Osserman writes, “Every map, in fact, is a compromise.” As in everything, you give up something to get something, and you have to decide what you want and what you’re prepared to give up to get it.

He then shows how the very same issues that hobbled humanity’s understanding of a spherical Earth have similarly hamstrung its understanding of a curved universe.

“Our experience confirms that Euclidean geometry offers a good description of space on a small scale,” he says. “But we have absolutely no reason to assume that it also holds on a larger intergalactic scale. It is that extrapolation, from small to large scale, that makes us think in terms of a flat universe exactly as an earlier age believed in a flat Earth.”

Furthermore, he observes, “While we have learned to overcome our flat-Earth mentality in measurements on a global scale, we have not yet overcome our tendency to think in terms of a flat universe.”

Unfortunately, the mathematics behind the curved universe is more complicated than most of us can easily grasp. To fully understand modern physics and cosmology, you need a fair amount of advanced math, which leaves most of us out.

Nonetheless, under the circumstances, Osserman does a solid job of explaining things, making good use of analogies and diagrams. The problem of mapping a curved universe on a flat plane is the same as the problem of mapping the Earth, and similar techniques and tricks have been applied. Having followed his discussion of mapping the Earth, it is easier to understand the complexities of the problem on the intergalactic scale.

Through these pages wander some of the great names in the history of science and thought: From Euclid, Pythagoras and Eratosthenes (who got the circumference of the Earth almost exactly right in the Third Century B.C.) to Gauss and Riemann in the 19th Century and Einstein and Hubble in our own.

The capsule profiles of these people help to humanize the story of their immense intellectual accomplishments.

Throughout, Osserman keeps in mind the sheer beauty of these discoveries, how elegant mathematics combined with technological advances (the optical telescope and the radio telescope, for openers) have produced a deep understanding of the universe, where it came from and what it looks like. Where it is going is a question we’re still not so sure about.

For as far back in history as we know and can imagine, people have looked at the night sky and wondered, “What is out there?”

In this century, Osserman writes, “Mathematical imagination and imagery, closely linked, provide the vision that allows us to see the hidden but exquisite structure below the surface.”