When I first approached the idea of reviewing Toeplitz’s book, I felt a mixture of curiosity and apprehension about giving my opinion on a book written some 80 years ago. What type of book would it be, how did the author and the public think then, and how would I — and the modern audience — receive it now?
First of all, as both Alfred Putnam, the writer of the preface of the first American edition, and David Bressoud, the author of the current edition’s foreword, mention, this is not a textbook. I would like to argue the point even further and say that it is not even a Calculus book, in some sense. This I will explain in a moment, but until then let me just say that it is not a history book either, even though it does incorporate significant moments of the history of calculus.
The book is comprised of four chapters, which deal respectively with the development of the concept of limit, the definite integral, differential and integral calculus and applications to physics.
I personally found the first chapter the most interesting. I have to confess to having a soft spot for limits and I found the detailed history fascinating. This is not a book in which the author inserted a few historical snippets, just to take us in a few pages to the modern definition of the concept. On the contrary, this first chapter takes almost a quarter of the book and presents the development of the idea of convergence as it arose from specific problems. Two examples are the development of trigonometry by Archimedes (section 5) and Bernoulli’s question about defining continuously compound interest (section 7).
The reason I said that this is not really a calculus book is that to me this whole chapter, which sets the foundation for all that will follow, is a chapter on real analysis. And to support this thesis even further, after giving us a historical and at the same time rigorous foundation on limits, the author goes on with a treatment of sequences, series, special limits (such as ), and limit theorems.
The second chapter deals with areas and definite integrals. Toplitz leads us carefully from the classical works of the Greeks (again, presented with full proofs), to more modern results of Cavalieri and Fermat, which together with Napier’s work on logarithms give us a preliminary glimpse into what will be the Fundamental Theorem of Calculus. And then he points out that to define an area concept in a rigorous manner it is in fact equivalent to defining a definite integral. He defines the latter, not as a Riemann integral, but by means of left and right sums, which in the case of monotonic functions that he chooses to deal with is the same as using Darboux sums. One delightful bit of this chapter are the area paradoxes presented in section 14, which arise from non-rigorous applications of infinitesimal methods.
The centerpiece of the third chapter is the development of logarithms by Napier and Bürgi, and the discovery of the Fundamental Theorem by Barrow, Newton and Leibniz. It is after this presentation that the book really turns to “calculus” and talks about differentiation rules and techniques of integration.
The author makes a very deep comment about the development of the function concept in two distinct forms. One concept he calls “geometric”, and it is the one that views a function as a rule, and allows us to think of such abnormalities as the Dirichlet function. The “computational” functions, on the other hand, are our daily polynomials, radicals, exponential and trig functions. These are much better behaved and don’t necessarily require all the analysis care in handling them. At the same time, they help us understand the fundamental ideas without having to deal with pathological distractions. These two concepts are, according to Toeplitz, what made the difference between the progress made by the works of Barrow (a purist), and Newton (a computationalist).
The final chapter feels closer to physics than mathematics, and it does a historical presentation of the laws of planetary motion in parallel with Newton’s laws, as expressed in the Principia.
Finally, a very short section of exercises follows, which is yet another reason for which this is not a textbook. It does not contain either a complete collection of techniques, or enough problems to practice them.
One remarkable feat accomplished by this book is that even though it is not a history book, as its focus is obviously the mathematics, it is made out of history. It does not fall into the pattern of all other “historical approach to…” books that I have seen, which start with a bit the history and then get down to business as it is done now (or 80 years ago, for that matter). This difference makes Toeplitz’s book both interesting and challenging at the same time.
So who is this book for? (I must address this question now!) I would say it is for people who already have some notion of calculus (or even real analysis). This was agreed upon before me, by the authors of both forewords. It is definitely a wonderful resource, but most of all, a source of enjoyment for the individual reader.
Ioana Mihaila (email@example.com) is Assistant Professor of Mathematics at Cal Poly Pomona. Her research area is analysis, and she is also interested in mathematics competitions.