*Abel’s Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability*, Peter Pesic, 2004, 221 pp. $14.95 paper. ISBN 0-262-16216-4. The MIT Press, 5 Cambridge Center, Cambridge, MA 02142. www.mitpress-orders@mit.edu

Although the title of the book sounds quite intimidating, the content is quite readable and interesting. It presents not only high-powered mathematics but also the human side of mathematicians.

The book begins back in ancient Greek times, way before the birth of the Norwegian mathematician Niels Henrik Abel (1802-1829). Pesic explains how mathematics developed out of specific needs (shapes of objects, counting things, accounting needs of merchants) in the various cultures. He traces the development of mathematics from its rudimentary stages of arithmetic through algebra and geometry. Numerous milestones – the discovery of 0, the use of symbolism, irrational numbers, for example – are described as are the events leading up to Abel’s work and his discovery that the quintic equation cannot be solved by algebraic means. The difficulties, successes and failures of the development of mathematics are described both from mathematical and human viewpoints. In the process of solving problems many ideas were originally rejected because they seemed so unusual, and as a result, progress in mathematics often suffered.

The connection between algebra and geometry is explained in great detail, and the diagrams connecting the two disciplines are quite helpful. Students often do not see the dependence of one discipline on the other, so this section of the book could be quite useful in the classroom. Students often think that discoveries happen over night, and after reading certain sections of this book, they may better realize that people in the past struggled to solve mathematical problems just as they do today. Our students often feel that mathematics has no relevance, and this, too, is far from the truth. Most upper level students and their teachers should be able to appreciate many of the ideas presented in *Abel’s Proof* and gain a better understanding of the difficulties many of the early mathematicians experienced.

This book can be a valuable resource in every high school classroom, one that is used by teachers and students alike. Its price also makes it quite affordable, and its length makes it quite readable!

Lynn Godshall, Susquehanna Township High School, Harrisburg, Pa.

See also the MAA Review by David Graves.