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*Cogwheels of the Mind: The Story of Venn Diagrams*, A.W.F.Edwards, foreward by Ian Stewart, John Hopkins University Press, 2004, 110pp + xvi, hardcover, ISBN 080187434, $25 http://www.press.jhu.edu .

*Cogwheels* is a very short readable book on attempts to physically represent the intersection of any number of sets. As a successor of John Venn at Gonville and Caius College, Cambridge, Edwards became interested in this problem. *Cogwheels* is the story of how he and others have extended Venn’s original ideas.

The first two chapters of the book give a brief early history of the problem. The contributions of Venn, Euler, Boole, Jevons, H.S. Smith, and Lewis Carroll are discussed. In Chapter 3 the author outlines his own method of using spherical surfaces to extend these diagrams to five or more sets. Ian Stewart first referred to these diagrams in 1989 as “les dentelures d’Edwards-Venn” in an article in the magazine Pour la Science, the French version of Scientific American. The English translation of this article in 1992 was called “Cogwheels of the Mind”, hence, the title of the book.

The remaining four chapters of the book discuss applications to the gray code, binomial coefficients, the revolving-door algorithm, trig metric representations, mapping the hypercube, and rotational symmetries. There are two appendices containing previously written articles, a bibliography and an index. It should be mentioned that nearly one third of the book consists of diagrams not all of which are well placed to aid understanding.

While reading this book, I initially felt that its main objective was to establish the author’s place in the development of these diagrams. However, further on, when I reached the point where the author expresses his joy at solving the problem of representing a symmetrical seven set diagram, I began to appreciate the author’s position. He clearly states: “Most deep understanding of other people‘s work comes from having unwittingly repeated it oneself, for only *then* does one see what the others meant all along.”

I have read few better explanations on the advantages of a constructivist education. It is certainly a true endorsement for using the history of mathematics in allowing students to develop their own understanding of mathematics.

Jim Kiernan, Adjunct Professor, Brooklyn College, CUNY

Jim Kiernan, reviewer, "Cogwheels of the Mind: The Story of Venn Diagrams," *Convergence* (July 2007)