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Writing about this area of physics, as Greene does, without assuming that the reader has any mathematical background is the hardest challenge of popular science writing. Michio Kaku, a physicist at City College in New York, provided a very nice introduction to superstrings in ''Beyond Einstein: The Cosmic Quest for the Theory of the Universe.'' But Greene goes beyond Kaku's book, exploring the ideas and recent developments with a depth and clarity I wouldn't have thought possible. Like Simon Singh in ''Fermat's Enigma,'' he has a rare ability to explain even the most evanescent ideas in a way that gives at least the illusion of understanding, enough of a mental toehold to get on with the climb.

For many physicists the main objection to string theory has not been the extra dimensions but the fact that until recently there appeared to be five different string theories competing to describe the same world. Greene shows how, in the past few years, physicists have made a strong case that these are simply different views of a single underlying theory -- five blind men looking at the same elephant. This more than anything is what has impressed many of the skeptics.

But as enthusiastic as he is about the theory, which he himself has contributed to, Greene doesn't gloss over the potential pitfalls. In the past, theoretical breakthroughs -- like the ''eightfold way,'' a scheme for arranging the bits and pieces of the subatomic world -- have become accepted only after they predicted new particles, or other phenomena, that were later found by experimenters. Not only has string theory not been verified, it has not even been developed to the point where it can make precise, testable predictions.

WHEN these forecasts do come, as Greene is optimistic that they will, verifying them will not necessarily require an accelerator the size of the universe. He suggests many practical experiments that might indirectly indicate whether or not the theory is on the right track. Not everyone will find these convincing. But even quarks, now considered gospel by particle physicists, have been detected by only the most oblique means. Superstrings aspire to supplant quarks as crude approximations of a deeper physics.

Like all popular science writers, Greene is burdened with the task of having to start from scratch, explaining ideas about relativity and quantum mechanics that were current in the 30's. (Imagine having to spend a hundred pages of every book about baseball reviewing the terminology of the game.) But with Greene this familiar ground is well worth retreading. Rather than recycling the tired old set pieces science writers too often fall back on, he develops one fresh new insight after another. Some of his examples, involving George Burns and Gracie Allen floating around in space suits or having a drink at the H-Bar (a physicist's joke), are a little silly. And toward the end of the book, the tower of abstractions gets harder and harder to ascend. But that comes with the territory. In the great tradition of physicists writing for the masses, ''The Elegant Universe'' sets a standard that will be hard to beat.

Illustration (Christoph Niemann)