If we may begin with a sweeping generalization, calculus is to mathematics what grammar is to literature: it is the necessary tool for the proper expression of ideas. But if students spent their time diagramming sentences and learning syntax, and occasionally applied their skills by writing practice letters to congress, they might be excused for thinking that literature has no beauty and little relevance to their lives. In Calculus Gems, Simmons provides the mathematical equivalent of Shakespeare, Browning, and Nash: important and enjoyable mathematics, accessible to those who have studied the basic tools of the subject.
The first part of the book consists of a series of biographies of prominent mathematicians, provided to give a human face to mathematics, as well as make connections to other disciplines. Simmons achieves these goals, though the biographies of the more recent mathematicians have an obituary-like quality about them: Fermat has ten pages, Euler eight, but Lagrange, Laplace, Fourier, and Cauchy have only two pages apiece. Unfortunately, despite Simmon's hope (and back cover claim) that the first part provides "a biographical history of mathematics from the earliest times to the late nineteenth century," this section omits entirely those whose work requires mathematics beyond calculus (such as Galois and Cantor); Islamic, Indian, and Chinese mathematicians are also omitted.
The second part of the book is the real treasure, showing how the basic tools of calculus, particularly infinite series and basic differential equations, can be used to derive some elegant results. The "gems" of the title include both pure and applied mathematics: for example, sections are devoted to the existence of transcendental numbers, the irrationality of π, an elementary derivation of E = mc2. There are even a few gems that do not require calculus: one chapter discusses several proofs of the Pythagorean Theorem, while another discusses the areas of lunes (regions bounded by intersecting circular arcs), and yet another various algebraic structures.
Many of the gems are drawn directly from the work of the masters. One finds, for example, that Leibniz found the series for π/4 by finding the area of a circular segment through integration, rather than through the use of the series for arctangent. By presenting both the original and modern derivations, Simmons draws notice to an important point: multiple proofs of the same result are commonplace, and often new mathematics allow us to simplify or improve existing proofs. The gem analogy is apt: jewelers are not satisfied with merely digging a gem out of the ground, but proceed to polish, facet, and set it, and later generations may take the stones and recut and reset them.
Overall, Calculus Gems is a worthwhile addition to any mathematics library. It is a useful source of interesting results to incorporate into many undergraduate mathematics courses, and could even work as a main text in a proofs-type course introducing students to higher mathematics.
Jeff Suzuki is an Associate Professor of Mathematics at Brooklyn College. His publications include A History of Mathematics (Prentice-Hall, 2002) and "The Lost Calculus (1637-1670): Tangency and Optimization Without Limits" (Mathematics Magazine, December 2005). His current interests include the mathematics of the eighteenth century, particularly the work of Lagrange, and geometric constructibility.