*Unknown Quantity, A Real and Imaginary History of Algebra*, John Derbyshire, 2006, 352 pp., illustrations and photographs, $27.95 cloth, ISBN 0-309-09657-x, Joseph Henry Press, 500 5th Street NW, Washington DC 20001 (800) 624-6242, www.jhpress.org

This book is a delightful tour through the ages, beginning the reader with the days when “algebra” meant the use of letter symbols to “relieve the imagination” (Leibniz), and taking him or her through the classic stories of the solutions to polynomial equations. The latter half of the book leads the reader into an abrupt new definition of the word “algebra”, rather “an algebra”, leaving numbers behind, and concentrating not at all on the objects being manipulated but totally on the overall structures and operations.

While the book's title is *Unknown Quantity*, the author, John Derbyshire, is a very known quantity, having written the remarkable *Prime Obsession* three years before. His dual disciplines, linguistics and mathematics, are engagingly entwined in his story telling. This book captures the personalities involved in more than the typical snapshot silhouette, and humanizes them suitably for classroom use. He also is able to bring the reader to the very edge of deep mathematical chasms, enough for us to see as if we were “standing on the shoulders of giants”, and then back off to approach those thoughts from different, equally accessible vantage points. You will want to buy this, read this, loan this, re-read this, and, no doubt, purchase the next book that the author writes, as well.

Don Crossfield, Mathematics Teacher, Roseburg High School