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The Calculus Gallery: Masterpieces from Newton to Lebesgue

William Dunham
Princeton University Press
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Ed Sandifer
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How many excellent books can William Dunham write? He has already written two of my favorites, Journey Through Genius and Euler: The Master of Us All. Now the pair becomes a triad, with The Calculus Gallery: Masterpieces from Newton to Lebesgue. Like his other books, The Calculus Gallery is a collection of characters, concepts and stories from the history of mathematics, ruthlessly edited until only the very best remain.

The dozen who survive the cut fall into three eras, or, in the metaphor of the book, three wings of the gallery. The Early Era features the birth and early days of calculus. We see the familiar concepts of derivatives and integrals, mostly treated with the archaic tools of infinite series, at the hands of Newton, Leibniz, Euler and the Bernoulli Brothers, Johann and Jakob.

Dunham makes the transition from the Early Era to what follows with an “Interlude,” a kind of half-length chapter that gives the “big picture.” He describes the status of calculus after Euler and before Cauchy. He points out now-familiar elements like piecewise-defined functions and the very concept of inequalities that were not yet part of the calculus repertoire. He also makes passing mention of the contributions of lesser luminaries like D’Alembert and Berkeley, who were important, but not Olympian. This idea of such an interlude works very well, and I have shamelessly stolen it in some of my own writing.

The second wing of The Calculus Gallery is dubbed the Classical Era, featuring the works of Cauchy, Riemann, Liouville and Weierstrass. This era begins in 1821 with Cauchy’s Cours d’Analyse and ends in 1872 with the pathological Weierstrass function. This was the time that rigor came to calculus, and we learned why some apparently obvious theorems about convergence that seemed so simple were so hard to prove; what seems “obvious” in calculus sometimes just isn’t true, as in Cauchy’s “proof” that a pointwise limit of continuous functions is continuous. Other times, theorems seemed obvious, but their proofs were difficult because certain properties of the real numbers had not yet been isolated, as in the case of the Intermediate Value Theorem. I found Dunham’s account of Cauchy’s role in this episode particularly interesting.

A second interlude is devoted mostly to pathological functions designed by Dirichlet, Thomae, and Darboux, and the questions they exposed; “Can we construct a function continuous at each rational and discontinuous at each irrational?”, “How discontinuous can a derivative be?”, and others.

The third wing, starring Cantor, Volterra, Baire and Lebesgue, brings us into the mid-20th century and tells how the mathematical community strived to answer the questions that Dunham described in the second interlude. It contains my favorite chapter of the book, the tragic and brief story of René Baire. Dunham makes the idea of “nowhere dense sets” and the Baire Category Theorem seem far more interesting than they seemed when I learned of them in my real analysis course. Perhaps I am not the only one who failed to appreciate the ideas, for, as Dunham tells us, “The completely non-descriptive ‘first category’ is about as colorless a term as there is, and conjures up no image in the mind’s eye.”

Dunham is one of the world’s best mathematical lecturers. He is witty, meticulously prepared and informative. Those of us who have been lucky enough to hear him speak can hear his voice in the words of The Calculus Gallery even more clearly than in his other books. For example, early in the book he is writing about some of Isaac Newton’s work on series. Newton had written a function z as a power series in x, and he was sought to invert the series and find a series for x in terms of z. Dunham prepares us for the difficulties of the details, writing, “The resulting technique involves a bit of heavy algebraic lifting, but it warrants our attention…” Then, after a page of intricate details, he celebrates our perseverance by telling us that Newton “had a remarkable tolerance for algebraic monotony [and] seemed able to continue such calculations ad infinitum (almost).” I like the image that genius is not easy, and that it sometimes involves heavy lifting and a tolerance for monotony.

The Calculus Gallery is full of mathematical details without being cluttered with them. Dunham has an artist’s touch in choosing what is necessary to include, and a surgeon’s skill in leaving out what is unnecessary. Since the topics in The Calculus Gallery are more advanced than those he has treated in his earlier books, I expect that this book will not find so wide an audience as the others have. It requires an understanding of the foundations of rigorous calculus to fully appreciate the book, but anyone who has felt the thrill of a good counterexample in calculus and who appreciates masterful mathematical exposition should enjoy this book. I haven’t asked him about it, but I suspect that The Calculus Gallery is Bill Dunham’s favorite of his own books, and I highly recommend it.

Ed Sandifer ( writes the column “How Euler Did It ” for MAA Online. He is professor of mathematics at Western Connecticut State University and has run the Boston Marathon 35 times.

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