American Scientist

The Calculus Gallery: Masterpieces from Newton to Lebesgue. William Dunham. xvi + 236 pp. Princeton University Press, 2005. $29.95.

Musings of the Masters: An Anthology of Mathematical Reflections. Edited by Raymond G. Ayoub. xvi + 277 pp. The Mathematical Association of America, 2004. $47.95 ($37.95 for members).

The history of calculus is fairly well known and is competently described in many places. The prehistory of calculus begins with Archimedes in the third century B.C. and continues through the works of some Islamic mathematicians in the medieval period. The first two-thirds of the 17th century saw the development of integration by such mathematicians as Johannes Kepler, Bonaventura Cavalieri, Evangelista Torricelli and Pierre de Fermat; during the same period, Fermat, René Descartes and others devised approaches to finding maxima, minima and tangents. Toward the end of the century, Isaac Newton and Gottfried Leibniz wove together many of these earlier ideas into "the calculus," although they were not able to put their creation on a rigorous footing. The 18th century saw great development of the subject and its applications to more and more areas of science. But it was only in the 19th century that Augustin-Louis Cauchy, Karl Weierstrass and others gave calculus a foundation as secure as the paradigmatic mathematical topic of geometry.

Although this outline is familiar, the mathematical details are less well known—and quite fascinating. Yet they are often skipped over in general histories. William Dunham, however, has remedied this situation in his brilliant book The Calculus Gallery, at least for the period beginning with Newton and Leibniz. Dunham picks out important results in calculus from the works of 13 mathematicians, sets them in the context of their times and explores the original proofs. Although some of the ideas he discusses are rather difficult, he manages to make them accessible to anyone with a background equivalent to that of a senior in college majoring in mathematics. As Dunham points out, these ideas are some of the masterworks of analysis, and anyone studying mathematics should be as familiar with these as someone studying art is with the paintings of Michelangelo and Renoir. I predict that Dunham's book will itself come to be considered a masterpiece in its field.

Each chapter of The Calculus Gallery is an outstanding piece of exposition, often based on a talk Dunham has given. I will discuss just four chapters. In the one on Leonhard Euler (1707-1783), Dunham not only analyzes Euler's differential approach to the calculation of the derivative of the sine function but also shows how Euler found integrals of "bizarre" functions as well as sums of several interesting infinite series. In particular, Dunham demonstrates why the sum of the series of the reciprocals of the integral squares is π²/6, showing that Euler found this sum, along with numerous others, as a corollary to a general result on the sums of reciprocals of the roots of polynomial functions or functions represented by power series.

In the chapter on Bernhard Riemann (1826-1866), we see one of the earliest examples of a "pathological" function, a function with properties that surprised many mathematicians of the day and led, along with other such functions, to the necessity for a reevaluation of intuition in analysis. Riemann's example was of a function that had infinitely many discontinuities on a finite interval and yet was Riemann-integrable.

The chapter on Karl Weierstrass (1815-1897), in addition to discussing his influence on the general development of rigor in analysis, presents one of his own creations, a pathological function that is continuous everywhere but differentiable nowhere. Although Weierstrass's proof that his function satisfies those conditions is long and tricky, Dunham succeeds in making the explanation understandable to those with the patience to go through each step.

Finally, in the chapter on René Baire (1874-1932), we get a marvelous discussion of the famous Baire category theorem. This result always appears in graduate courses on analysis, but for many of us, there was little besides the strange name by which to remember it. Dunham changes all that by presenting in great detail the context of the theorem. He then carefully states and proves it, using Baire's original proof. Finally Dunham demonstrates that the theorem enables one to prove easily many of the results that he had presented in the chapters on earlier mathematicians. For example, one corollary of the theorem is Georg Cantor's result that, given any sequence of distinct real numbers and any interval, there is a point in the interval not a member of the sequence. This immediately shows, as Cantor demonstrated, that the set of real numbers has cardinality larger than that of the natural numbers.

Although The Calculus Gallery requires some background in advanced calculus for a complete understanding, the other book under review, Musings of the Masters, has few explicit mathematical prerequisites. The editor of this anthology, Raymond Ayoub, has collected 17 essays, originally written between 1869 and 1978, by prominent mathematicians, each reflecting the author's views on what the editor calls the "humanistic" side of mathematics. Most of the essays are, in fact, the transcripts of lectures given by these mathematicians to general audiences at such occasions as the International Congress of Mathematicians.

Given that the book's title includes the word "musings," it is not surprising that these essays form a varied lot, dealing with topics as diverse as the existence of God and Goethe's opinions about mathematics. To help us put things in context, each piece is prefaced by both a biographical sketch of the author and a short summary by Ayoub. The book contains too many essays to discuss them all here individually, but let me offer my own brief musings about some of them.

The oldest essay in the collection is James Joseph Sylvester's presidential address to the British Association in 1869. This noted English mathematician criticizes remarks of the biologist T. H. Huxley to the effect that mathematicians spend their time making "subtle" deductions from a few simple, "self-evident" propositions, and that mathematics "knows nothing" of observation, experiment, induction or causation. Sylvester then brings to bear his immense erudition to show that, in fact, Huxley's ideas are totally opposite to actual practice: Mathematicians have always observed and experimented. From this Sylvester concludes that the study of Euclid in English schools should be "honorably shelved" and replaced by more "living" topics that would better stimulate the minds of students. Interestingly enough, his wishes with regard to Euclid were achieved. But more than a century later it seems doubtful that, in England or elsewhere, the replacements do a better job of providing stimulation.

G. H. Hardy's 1929 lecture on "Mathematical Proof" deals with the philosophies of mathematics that were then current. Hardy explains why he is not convinced that any of those philosophies could be acceptable to the vast majority of working mathematicians, including himself. As he points out, mathematicians believe that when they have proved a theorem, they really "know" something; the philosophies he discusses apparently are too restrictive in confirming such beliefs.

The only contribution by a woman in this volume is a lecture by Mary Cartwright on "Mathematics and Thinking Mathematically," given at Goucher College in 1969. In her lecture, Cartwright attempts to distinguish abstract mathematical thinking from the applications of mathematics in the real world. But, as she points out, some of the best "abstract" mathematicians formulate ideas in terms of that real world. For example, her frequent collaborator J. E. Littlewood worked on antiaircraft gunfire in the First World War and thereafter often translated more abstract problems into the language of "trajectories."

Several of the essays deal, directly or indirectly, with the history of mathematics. André Weil, in a talk given at the International Congress of Mathematicians (ICM) in 1978, presents his thoughts on the reasons for studying the history of mathematics. My own opinion is that too often he looks at ideas from a past era through a modern lens, neglecting the context in which the ideas arose. Raymond Wilder, in an ICM talk from 1950, introduces us to his views on the cultural bases of mathematics, ideas that culminated in a book published in 1974. Finally, André Lichnerowicz, in a lecture from 1955, discusses the meaning of being a scientist, both historically and in modern times.

Although many of the essays in this collection will stimulate thought and discussion, it would have been no great loss if some of these "musings" had remained unpublished. Nevertheless, the collection as a whole will prove valuable in bringing its readers the thoughts of well-known research mathematicians on topics outside their areas of specialty.