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*Archimedes, What Did He Do Besides Cry Eureka?*, Sherman Stein, 1999. 168 pp., $27.95 paperbound. ISBN 0-88385-718-9. The Mathematical Association of America, P.O. Box 91112, Washington, DC 20090-1112. (800-331-1622) www.maa.org .

Many people know the story of Archimedes, who, when asked to determine whether a crown was made of pure gold or a fraudulent silver alloy, was suddenly inspired by the displacement of water in his bath tub to think of a method for solving the problem and jumped from the bath shouting, “Eureka!” They may also know of his theory of the lever, invention of the screw and his method for estimating the value of pi. But they will not appreciate his true genius until they have seen how he achieved his remarkable mathematical discoveries. Sherman Stein’s book enables anyone with high school mathematics skills to understand how Archimedes did it.

The keystone of the book is “The Method” of Archimedes, a description of which he sent to his friend Eratosthenes but which was lost to history until 1906 when a tenth century copy was discovered in Constantinople (Istanbul). Archimedes used his “method of mechanics” to discover, and in many cases, prove mathematical theorems. The author shows how Archimedes used his method to determine the volumes of paraboloids and spheres and the centers of gravity of paraboloids and hemispheres. In later chapters he addresses Archimedes’ work on parabolic sections, floating bodies, spirals, spheres and estimation of pi. He also includes a chapter on Archimedes’ evaluation of geometric and other series and, in an appendix, shows how many of Archimedes’ results can now be achieved by affine transformations.

The book’s author has certainly made Archimedes’ discoveries accessible and will inspire mathematics teachers to introduce Archimedes to their students.

Tim Keenan, Mathematics Teacher, Osbourn Park High School, Prince William County, Virginia

Tim Keenan, reviewer, "Archimedes: What Did He Do Besides Cry Eureka?," *Loci* (July 2007)