# The Calculus Gallery

This is a slightly corrected reprint of the book originally published in 2005. The fact that it is now made available in the *Princeton Science Library* series as a cheaper version is a confirmation of its quality.

Dunham has chosen to tell the history of calculus from its origin, as conceived by Leibniz and Newton, till the moment that Lebesgue redefined Riemann's concept of an integral. Of course there exist several books on the history of mathematics, but Dunham has chosen to tell the story as if he is the intendant of a mathematical art exhibition. He chose a number of key results that he discusses in some detail, that means including the ideas of the original proofs (although translated in a for us readable form). These are the stepping stones that tell us about the evolution taking place. So Dunham walks with the reader through the historical museum and tells us why a particular result is important in the chain of ideas that brought us to our current understanding of the subject, and eventually how the current abstraction became a necessity. The museum where his exhibition is displayed has twelve period rooms corresponding to as many chapters in the book, named after the artist-mathematicians who, if not produced, at least published the result(s). So each chapter has a short introductory biographical sketch but the emphasis lies on the discussion of the mathematics and why these are important in an historical perspective. The museum has also two lounge rooms, two interludes, where there is time to summarize the history so far, looking at remaining problems and at what is ahead, and where a somewhat broader bird's-eye view is given because the twelve mathematicians selected are of course not the only ones that have shaped the history of mathematics.

The names of the twelve chapters chosen to support the evolution are Newton, Leibniz, Jakob and Johann Bernoulli, Euler, Cauchy, Riemann, Liouville, Weierstrass, Cantor, Volterra, Baire, and Lebesgue. This includes obviously some of the usual suspects but a somewhat surprising name in the list is Baire and one may wonder why Liouville and Volterra are featuring while for example Gauss is not. So Dunham justifies his choice in the introduction. To answer the question which functions were continuous, differentiable, or integrable, one needs to know something about the continuum of the real numbers. Here Liouville was important for the discussion about irrational (algebraic, transcendental) numbers and how close these could be approximated by rationals, somewhat similar to what Weierstrass did for the approximation of continuous functions by polynomials. Volterra was instrumental in helping to answer the question of how irregular a function can be and still be (Riemann-)integrable. He was able to construct some pathological example that had everywhere a bounded derivative and yet was not integrable. Baire fits in this story because with his category theory, functions were finally classified with respect to their irregularity, which settled the discussion.

Because Dunham digs into primary sources, we learn how also these brilliant pioneers who paved the way, had their struggles with concepts and approaches that for us seem clumsy. But we should realize that our calculus courses are the results of many years of filtration, polishing and reshaping of these original ideas. For example we know how to deal with infinitesimals as quantities that go to zero in the limit, but in the early days, without limits, serious resistance against the new ideas of calculus was raised because the infinitesimals were non-zero at some points and were replaced by zero at others. Manipulations that were considered by opponents to be all but sound mathematics. This issue was only solved with the introduction of the limit by d'Alembert.

We also see that although Newton's fluxion stands for the derivative, both Newton's and Leibniz' approach was via integration, heavily relying on series expansions for small perturbations. The role of the integral for the origin of calculus can be seen in an historical context where geometry was dominant in solving mathematical problems and computing a surface area is a geometric problem. But calculus gradually moves away from geometry as we read on. Series however remained important issues in the early days. The Bernoulli's as well as Euler have analysed their convergence or divergence, but Cauchy was the one to formulate sound convergence criteria, while Riemann later showed the importance of differentiating between absolute and conditional convergence.

With Riemann we are back to integration. Integrability was however related to the construction of pathological functions which were often of "ruler type" like being equal to 1 for *x* rational and 0 for *x* irrational. Weierstrass could construct a function continuous everywhere and yet nowhere differentiable. So this goes hand in hand with a discussion about algebraic and transcendental irrational numbers (hence the Liouville chapter). With this fundamental discussion of the number system, set theory enters the scene with Cantor's fundamental contributions and Dedekind's cuts. Topological aspects such as density of a subset of an interval has eventually triggered Lebesgue to redefine the concept of the integral to circumvent the problems raised when using Riemann's concept. With this evolution, for the finer details of calculus one has to leave not only geometry but also algebra to take off in a more abstract topological realm.

Many generations of students are currently instructed in calculus courses, more or less advanced. Some may feel annoyed with the abstraction and may not see why it is needed. This book will reveal how and why their modern calculus course was shaped into its current form. This book is unique in its content because it is not a full history book, and it is not a calculus course. There are however many proofs that require some knowledge of (modern) calculus, and some of them are quite involved. But by restricting the discussion to functions of one real variable, the mathematics stay within the reach of students familiar with a basic calculus course at the level of a first year at the university. The nice thing about these proofs is that they follow the original ideas. Also Dunham's style is pleasant and much more entertaining than a formal course text. Princeton University Press has made a proper choice by promoting this book to their *Science Library* series and making it in this cheaper form available to a broader readership. My warm recommendation is only appropriate.

**Submitted by Adhemar Bultheel |

**13 / Dec / 2018