*The Calculus Gallery: Masterpieces from Newton to Lebesgue*, William Dunham, 2005, 288 pages, 19 halftones 44 line illustrations, $29.95, ISBN (Cloth) 0-691-09565-5. Princeton University Press, 41 William Street, Princeton, NJ 08540. (800) 777-4726, http://www.pup.princeton.edu/

As the title and Dunham’s introduction suggest, this book is meant to be a look at the works of the masters, akin to a gallery exhibition. What an exhibition it is! Newton and Leibniz are here, as are Euler, Cauchy, and Riemann. This is no leisurely stroll through a museum, however; the reader needs to do some work. As the author says, “I would not recommend this for the mathematically faint-hearted.”

Who should read this? Teachers and students of calculus, certainly. The first four chapters (Newton, Leibniz, the Bernoulli’s, Euler) contain a tour de force of what can be done with calculus. The Newton and Leibniz material rounds up some of the usual suspects: Newton’s agility with infinite series and Leibniz’ transmutation theorem. Other amazing accomplishments with infinite series can be found in the chapters on the Bernoullis and Euler. Regular readers of history of mathematics books have seen this material before, but this is all written in the usual Dunham style, so it is fresh and interesting. Students and teachers alike will be impressed with what can be done with a little calculus knowledge.

That being said, the people who should rush right out and get this book are students and teachers of what is usually called “advanced calculus” or “real analysis.” The last two-thirds of the book reads like a trip through an undergraduate real analysis course: convergence tests, the intermediate value and mean value theorems from Cauchy; Riemann’s definition of the integral and integrability condition, along with his rearrangement of series theorem; Weierstrassian rigor and pointwise versus uniform convergence of functions. Classic functions abound: Dirichlet’s function, Riemann’s infinitely discontinuous but integrable function, and Weierstrass’s everywhere continuous but nowhere differentiable function. Dunham walks us through these results, and in so doing gives instructors of real analysis access to the original source material, and provides students of real analysis all the motivation they need.

Dunham takes us to the periphery of real analysis as well. Volterra’s result that two functions cannot have complementary sets of continuity and discontinuity deserves to be more well known, and using the Baire Category Theorem in the context of sets of continuity and discontinuity of functions makes it more palatable and less abstract. Finally, Lebesgue’s measure of sets and his integral are made accessible to undergraduates, and Dunham puts Lebesgue’s theorem on a necessary and sufficient condition for Riemann integrability within reach of undergraduate real analysis students.

Before you teach real analysis again, read this book. If you are about to take a real analysis course, take this book along with you. I anxiously await *The Real Analysis* *Gallery* for graduate students of analysis. Professor Dunham…?

Gary S. Stoudt, Professor of Mathematics, Indiana University of Pennsylvania