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The Moment Of Proof: Mathematical Epiphanies

Donald C. Benson
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Ed Sandifer
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Why are you reading this book review? Although there is a chance you are reading it because you are related to the author, more likely you are a mathematics teacher, a mathematics student, or a mathematician. Now, reflect a moment on how you came to be so mathematical. You probably had some aptitude. Encouraging that aptitude, you probably had good teachers, and nourishing it, there were probably books.

Not textbooks. Interesting books. Books that captured the excitement of mathematics. Such books are absolutely essential. They gave us the promise that, beyond the tedium of calculation and routine algebraic manipulation, there were interesting and exciting things to learn and to do.

My favorites were by Irving Adler and George Gamow. The generation before me might have read Kastner and Newman's "Mathematics and the Imagination", and many mathematical people younger than me discovered mathematics in the works of Douglas Hofstadter and Martin Gardner. Even Andrew Wiles reports that his interest in mathematics in general, Fermat's Last Theorem in particular, began with a popular book on mathematics from his local public library.

Such books age quickly, though, as mathematics and culture both grow and change. Today's popular mathematics books seldom mention "rubber sheet geometry", for example, and a chapter on Rubik's cube today is not nearly as interesting as it would have been fifteen years ago (is it already that long?) Thus, there is a need, and always will be a need, for new popular mathematics books.

The Moment of Proof: Mathematical Epiphanies by Donald C. Benson is a worthy addition to the family of popular mathematics books. Its twenty chapters, each about 15 pages long, each try to climax with a discovery or an insight that will carry the reader to that wonderful feeling of "I understand!" The author calls these events "Mathematical Epiphanies" and says that they are found at the "moment of proof", though they might better be described as arriving at the moment of understanding.

The topics the author has chosen resemble those in a typical liberal arts mathematics textbook, like For All Practical Purposes by COMAP or Excursions in Modern Mathematics, by Tannenbaum and Arnold.

In Chapter 9, "Divide and Conquer", we learn about the complexity of sorting algorithms, starting with the insertion sort that most people use to sort a hand of playing cards. The mathematical epiphany arrives when we learn how the quicksort works, why it works, and why it is faster than more familiar techniques, if the lists are very long.

Chapter 14, "Two Pearls" describes Archimedes' beautiful demonstration that the volume of a cone plus the volume of a sphere equals the volume of a cylinder.

Chapter 8, "The Power of Two" is a slightly longer chapter that tours through binary arithmetic, card tricks, the games of Nim and Kayles, base three arithmetic and the Cantor set before reaching the use of chaos in drawing a Sierpinski triangle.

Chapter 12, "Cutouts", is devoted to clever geometry proofs, especially the ones that use dissection. There is President Garfield's proof of the Pythagorean theorem (though our author gets Garfield's first name wrong. It's James, not John.) There's a neat proof relating double angle triangles, those where the measure of one angle is twice that of another, to Pythagorean triangles, and a refreshingly brief mention of Fermat's last theorem. This is not a textbook. Its personality is more that of a tour guide than of a teacher, and there are no exercises. It is hard to know what books will entice today's and tomorrow's budding mathematicians to pursue their interests and develop their talents, but The Moment of Proof has a chance of being one of those books.

Finally, if you are related to the author, you should read this book, and maybe you'll understand why he thinks mathematics is such a beautiful subject. Maybe you'll agree.

Ed Sandifer ( is a professor of mathematics at Western Connecticut State University and has run the Boston Marathon 26 times.

The table of contents is not available.