*The Art of the Infinite: the Pleasures of Mathematics*, Robert Kaplan and Ellen Kaplan, 2003, 348 pp, illustrations, $26.00, cloth, ISBN-0-19-514743-X, Oxford University Press, Inc., 198 Madison Avenue, New York, New York 10016,www.oup-usa.org/artoftheinfite

This book is a collection of mathematical ideas organized around the themes of infinity and the illumination of the nature of mathematical thought. The immediate questions that occur are: "What is the intended audience for this book?" and "Even though I don’t seem to be in it, can I use it anyway?" The answer to the second question is "yes." It can be used as a source of neat stuff to show students in informal settings, or as asides in courses. The authors present even the topics familiar to teachers in different and appealing ways, and many topics would be new. This book could be used as supplementary reading in a math for liberal arts course, but not as a text. The authors do not designate their intended audience, but it appears that at least one year of high school algebra and some plane geometry are required for a useful reading. Math teachers know much more. However, these very erudite authors use so many references and analogies drawn from such wide sources that it seems the audience must be well educated, possibly adult. This feature excludes many students.

The historical references are almost always interesting and accessible, but the others are riskier. There is a lot of serious mathematics in this book, and the descriptions of mathematical thought strike me as quite accurate, but the reader must have done, or do, some hard work to understand them. None the less, I would order it for my library and recommend it to the attention of my colleagues.

Albert W. Briggs, Jr. Professor of Mathematics, Emeritus, Washington College