A Tour of the Calculus

By David Berlinski

Vintage Books

331 pp., $14 (paper)

Mathematics: From the Birth of Numbers

By Jan Gullberg

W.W. Norton

1,093 pp., $50

It is a necessary but unfortunate condition that in most schools, math classes beyond simple algebra and geometry generally are taught by mathematicians.

Most students complete their courses in Cartesian geometry or the calculus with a shudder of relief but little appreciation of the magnificent human achievement that is mathematics.

The skills and intuition required for higher math appear to be incompatible with the ability to describe the wonder and beauty of the mathematical universe, except in the dry symbols of algebraic notation.

David Berlinski is an alienated academic. His slender volume, A Tour of the Calculus, eloquent and cogent, almost succeeds in making the calculus accessible. Aimed at bridging the gap between the "two cultures," science and the humanities, it has just been reprinted in paperback.

Trained as a mathematician, the author confesses that a student once charged him with being immoderately fond of the sound of his own voice. He writes in a superabundant, racy style recalling the 18th-century novels of Sterne and Fielding. This book might aptly be retitled "Mathematics for Poets."

The author's strategy is to teach by metaphor and indirection. He begins with a little history, bringing life to Gottfried Wilhelm Leibnitz and Isaac Newton as well as lesser figures like Augustin-Louis Cauchy and Leonhard Euler.

Most of the book's first part is written in a discursive style that is diverting yet surprisingly informative. The profound contribution of 17th-century mathematics to human understanding was the notion that the world could be understood in terms of the set of real numbers.

Berlinski reviews in a clear and engaging style the discovery that motion and extent were measurable quantities, related by simple mathematical functions. He is less successful in trying to explain the details of the calculus. The literary manner is not well suited to presenting these formal concepts, and he is forced by the nature of the material to use more algebraic notation and less anecdote.

Finally, if the reader is not already somewhat familiar with the concepts of functions and limits, derivatives and integrals, this presentation is not going to help much.

One of the most delightful aspects of the history of mathematics and science in general is that until relatively recently much of the truly significant work of discovery was accomplished by amateurs. Jan Gullberg is an amateur par excellence. A surgeon by profession, he has a deep and profound understanding of most of the important areas of higher math, and a genuine gift for presenting them.

Some years ago, when his son was studying engineering in college, Dr. Gullberg was prompted to write a little introductory handbook of mathematical formulas. He writes "I mainly wrote for myself as a means of exploring mathematics and its history more completely than I ever had before."

The result is Mathematics: From the Birth of Numbers, more than 1,000 pages of history, derivations, problems, and lucid, clear explication.

This volume is essential for anyone who has to deal with numbers, which nowadays is just about all of us. It starts, as promised, with the invention of number systems, with a multicultural perspective that is breathtaking. The first chapter, for example, compares Hebrew, Greek, and Roman numbers with those of the Greenland Inuit and the Maori of New Zealand. Gullberg seems to have read, and understood, everything.

The presentation is formal but even very difficult concepts are presented clearly and fairly completely. Where material presented earlier is required for an understanding, the relevant chapters and page numbers are cited. This makes the book an excellent reference source.

It is also a treasure of fascinating mathematical gems. For example, Richard Feynman once commented that the most beautiful fact in mathematics was Swiss mathematician Euler's formula about trigonometric identities linking the three great natural numbers: ei = -1. Gullberg explains why it is necessary that this should be so.

Mathematical innumeracy is not just an inconvenient handicap. The proliferation of unscientific studies is a social problem of genuine consequence. Bombarded with this or that claim presented as scientific truth, the responsible citizen has an obligation to delve through to the underlying argument.

Even more important, it is the duty of all of us, especially parents, to provide for children a secure understanding of the rules of proof. These are unavailable to anyone without a good grounding in mathematics.

Both of these introductory volumes, then, represent a hopeful and a positive effort. An educated person needs to understand enough math to be able to function in our complex and often confusing world.

At the same time, mathematics is a creative tool for description and prediction that can enlighten and delight. The study of mathematics is, therefore, both essential and rewarding for its own sake.

* Frederick Pratter is a freelance writer living in Hull, Mass.