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*Yearning for the Impossible: The Surprising Truths of Mathematics*, John Stillwell; 2006, 320pp. hardcover, $29.95, ISBN 156881-254X A.K. Peters, Ltd., 63 South Avenue, Natick, MA 01760 http://www.akpeters.com

*Yearning for the Impossible* offers a fascinating, historical look at some popular mathematical concepts used in music, art and philosophy.. The book begins with the Pythagoreans and their search for a frequency ratio to unite mathematics and music. It explores the mathematics of the different musical scales developed in Eastern and Western cultures.

The concept of irrational numbers is traced throughout the early cultures. The different terms used to designate rational numbers in the various civilizations (logos, Greek; audible, Arabic) and irrationals (alogos, Greek; inaudible, Arabic) were quite interesting. Some of the more common proofs concerning irrational numbers are included.

Geometry was the basis of the proofs for algebraic laws. Imaginary numbers were the result of Cardano’s solution of a cubic equation, another topic familiar to mathematicians. The imaginary number was an unrecognized entity; although it did not formally appear until 1572 in the literature, its concept had been used by Diophantus as he split numbers into sums of squares. Operations with the complex numbers were also explained through geometry.

The introduction of perspective to art was extremely fascinating reading, and the visuals used added a great deal to the topic. The principles employed were the following: straight lines remain straight under projection, intersections remain intersections, and parallel lines stay parallel or meet at the horizon. The use of tilings in art was also examined and discussed. Impossibilities in art later focused on the work of Escher and Magritte.

The final third of the book explored the development and proofs of some of the basic area and volume formulas as well as the development of Calculus through infinitesimals, topics again familiar to math historians. The final chapters involved higher-level math as they discussed curved space (both spherical and cylindrical), quaternions and ideals.

This book is an interesting find and provides a readable approach to some higher-level mathematics. The chapters can be read independently, and the reader can dig deeper into textbooks and history books for additional problems and details. I give a high recommendation for this book!

Lynn Godshall, Susquehanna Township High School, Harrisburg, PA (retired)

Lynn Godshall reviewer, "Yearning for the Impossible: The Surprising Truths of Mathematics," *Loci* (July 2007)