This is a survey of several areas of mathematics, chosen because they of representative of current research and because they can be understood without a lot of mathematical background. The authors say in the Preface (p. vii), "We want the reader to see mathematics as a living subject in which new results are constantly being obtained." The title is a quote from the Major-General's Song in Gilbert & Sullivan's *The Pirates of Penzance* and indicates the book's broad and varied scope.

The book often recurs to earlier topics and is not as miscellaneous as it looks on first glance. For example, it starts with elementary set theory but comes back at the end of the book to look at transfinite cardinals. (And it's refreshing to see a discussion elementary set theory that does not begin and end with Venn diagrams.) Similarly there is a chapter on permutations that is followed by a chapter on group theory.

The book is now 35 years old but has aged well, and the subjects covered are still lively research topics even though the statuses given here are not completely up to date. The big weakness of the book, in my view, is that it concentrates on mathematical concepts rather than mathematical problems. Giving more attention to problems would have made it more interesting and would have reinforced the message that math is a living subject.

This is an unusual book and it doesn't fit nearly into a well-known category or have an obvious audience. Although its prerequisites are low, it is a rigorous math book, with definitions, proofs, abstraction, some intricate reasoning, and challenging exercises. The authors suggest (p. vi) that "it could be used in courses designed for students who intend to teach mathematics". I think it would be most useful for pre-service or in-service high-school math teachers, especially those who want a good grounding in mathematical processes. It's definitely not a "popular math" book, and is probably too hard for a college math appreciation class.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning.