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Math Equations + Science = Great Discoveries


By Michael Guillen

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277 pp., $22.95


By John L. Casti

John Wiley & Sons

235 pp., $24.95

Michael Guillen seems to have led an exciting life in contrast to the droll existence with which we stereotype mathematicians. As science editor for ABC's ''Good Morning America,'' he has traveled to the Monteverde Cloud Forest in Costa Rica, the caves of Qumran (where the Dead Sea Scrolls were discovered). He has covered the eruption of volcanic Mt. Pinatubo in the Philippines and the burning oil fields of Kuwait. He is also an instructor of physics and mathematics at Harvard. He is by training a physicist whose job is explaining physical science theories and concepts to a television audience.

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His previous popular book, ''Bridges to Infinity,'' was intended to give ''readers a sense of how mathematicians think and what they think about ... all without subjecting the reader to a single equation.'' In this new volume he examines five great equations in the history of science, each of which he equates with a great discovery: Newton's universal law of gravity; Bernoulli's law of hydrodynamic pressure; the law of electromagnetic induction discovered by Faraday; Clausius' second law of thermodynamics; and Einstein's theory of special relativity, the famous E=MC2.

Each section provides a brief biographical sketch of the scientists, an account of how they came to their conclusion, and some of the subsequent effects of their discovery. Several of these are obvious, such as the path from relativity to the atom bomb, or from Newton to the space shuttle. Somewhat less well known, the Bernoulli principle explains why airplanes fly, while the second law of thermodynamics predicts the eventual running down of the universe to a cold, steady state.

Reading this book is a lot like eating a hearty bowl of rich soup that the chef has topped with a dollop of whipped cream. There is plenty of nourishment here, but it is accompanied by a large helping of filler. In his effort to make these great discoveries understandable to a wide audience, the author has watered down their impact. Also, for a scientific book the number of goofs is surprising. Guillen refers several times to King Charles I of England, beheaded at Whitehall in 1649, as ''King William.'' The author also has the British amateur George Cayley experimenting with man-carrying gliders in 1753, a century off.

If Guillen's book is soup, ''Five Golden Rules'' is caviar for the inquiring reader. John Casti is at the Santa Fe Institute and a professor at the Technical University of Vienna. He describes this work as ''mathematics for poets.'' Anyone who enjoyed solving math problems in high school will be able to follow the author's explanations, even if high school was a long time ago.

There are plenty of equations but nothing so formidable that it can't be followed with a little patient application. There is joy here too, in watching the unfolding of these intricate and beautiful techniques. Casti's gift is to be able to let the non-mathematical reader share in his understanding of the beauty of a good theory. The five techniques described are all widely used in a variety of disciplines. The Minimax Theorem from Game Theory models complex real-world decisionmaking in politics and the military. The Fixed-Point Theorem investigates the properties of space. Morse's Singularity Theorem explains why living things must take the shapes they do. Turing's Halting Theorem is the basis for all modern computers. The Simplex Method of linear programming can be used to make investment decisions or decide the optimum height for an apartment building.

All the practical consequences of these techniques are finally less interesting than the understanding of how the universe is shaped, what Faraday knew as ''the intricate contrivances of Nature.''

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