*The Honors Class: Hilbert’s Problems and Their Solvers*, Ben H. Yandell, 2002. 486 pages, illustrations, bibliography, $24.95, paperbound. ISBN 1-56881-216-7. AK Peters Ltd., 63 South Avenue, Natick, MA 01760. www.akpeters.com

At the beginning of the twentieth century, David Hilbert (1862-1943) was an elder statesman and leader of the mathematics community. A creative and versatile mathematician, he headed the mathematics program at Göttingen University making the German institution the world center of mathematical activity. In his address at the Second International Congress of Mathematicians held in Paris in 1900, Hilbert presented a list of twenty-three unsolved problems worthy of attention and, in a sense, set a research agenda for the new century. Sixteen of his problems have since been solved; four others really involve broad research programs which are well underway and three problems still remain unsolved. Meeting Hilbert’s challenge has brought together some of the greatest mathematical minds of our time. Working alone and in concert, they have greatly advanced mathematical understanding and opened new horizons for further research. Those skilled and talented enough to resolve one of the problems are considered members of the mathematical “Honors Class,” thus the title of Yardell’s book.

This is an excellent, interesting book; carefully researched and well written. The author discusses each of Hilbert’s problems, supplies mathematical and historical background and carefully traces the work of the problem solvers. He provides insights into the lives and personalities of the mathematicians he discusses. From Henri Poincaré to Paul Cohn, to Julia Robinson to Andrei Kolmogorov and many others, the reader is taken through a “Who’s Who” of twentieth century mathematicians. Photographs and illustrations visually enrich the presentation.

Many standard texts on the history of mathematics usually limit their examination of mathematical developments to the beginning of the twentieth century. It is felt that beyond this point the level and scope of mathematical activity exceeds the capacity of the usual reader. In such situations, *The Honors Class* can serve as a ready reading supplement. The book can also supply the basis for a seminar on twentieth century mathematics. I highly recommend its acquisition for personal reading and library reference.

Frank J. Swetz, Professor Emeritus, The Pennsylvania State University