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The Calculus Story: A Mathematical Adventure

David Acheson
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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Based on the title of this book, I had thought that it would focus largely on the history of calculus, but this turned out to be a somewhat incorrect assumption. The approach to calculus in this book does follow a more-or-less historical path, and some famous events in the history of calculus (such as the Newton-Leibniz controversy and Bishop Berkeley’s The Analyst) are talked about, but not in great depth. Primarily, however, this book appears to be an attempt to explain the basic ideas of calculus to a layperson. (Usually the author of a book spells out his or her goals explicitly in a preface, but for some reason there is no preface here.) As a math-for-layperson book, this one is partially successful. As will be explained below, it likely also has other uses as well.

The book certainly has the physical trappings of a text that is designed to be inviting and unintimidating to a general population. It is quite short (less than 200 pages of actual text), very small, with large font, lots of pictures, wide margins, and paragraphs that are almost all one sentence long. It actually does seem to assume at least some prior mathematical knowledge on the part of a reader, however, for example some familiarity with the concept of a function and its graph, as well as some rudimentary analytic geometry. In short, the book seems to assume roughly the kind of background that a typical American first-year undergraduate calculus student possesses, and the exposition, though considerably less formal than is found in a typical mammoth-sized calculus book, follows familiar lines: after some preliminary material on the nature of mathematics and the Pythagorean theorem, the actual study of calculus begins with a discussion of Newton and how the attempt to generalize the notion of “slope” from straight lines to curves leads to the familiar limit definition of the derivative.

In one respect, though, this book differs considerably from the approach of a standard first-semester calculus textbook. Although the area under a curve is talked about before page 45, the integral sign does not appear in the text until page 91, and, if I counted correctly, subsequently shows up on only 3 more pages of the text.

A surprising amount of material is covered in a short amount of time. In the first 35 pages, for example, the author gives the definition of a derivative, uses it compute the derivative of a couple of functions and explains how the derivative can be used to solve a standard optimization problem (maximizing the area of a rectangular field with a fixed amount of fence). By the time we reach the end of the book, we have also seen discussions of (among other things) the derivatives of trigonometric, exponential and logarithmic functions, infinite series (including some famous examples, such as Leibniz’s formula for \(\pi/4\)), Newton’s laws of planetary motion, ordinary differential equations, partial differential equations and the wave equation, Maxwell’s equations and chaos. These are “big picture” discussions, with most or all of the technical details often put to one side. For example, when discussing the definition of the derivative, the author avoids technicalities by simply using the word “limit” without any kind of definition or discussion; much later in the book, he circles back to the subject and engages in a more systematic discussion. (This, of course, again follows history — the ideas of calculus were used long before they were precisely defined.)

Based on the succinctness of the exposition and the lack of lots of worked out examples, I’m not at all confident that the prototypical “person on the street” is likely to get a great deal out of the book, especially the later portions of it. I do think, however, that a student in a beginning calculus course might read some of these chapters with profit as a useful supplement to more detailed lectures (which will undoubtedly have many more worked out examples). Likewise, a professor might find some good fodder for lectures in these pages.

Mark Hunacek ( teaches mathematics at Iowa State University. 

1. Introduction
2. The spirit of mathematics
3. Infinity
4. How steep is a curve?
5. Differentiation
6. Greatest and least
7. Playing with infinity
8. Area and volume
9. Infinite series
10. 'Too much delight'
11. Dynamics
12. Newton and planetary motion
13. Leibniz's paper of 1684
14. 'An enigma'
15. Who invented calculus?
16. Round in circles
17. Pi and the odd numbers
18. Calculus under attack
19. Differential equations
20. Calculus and the electric guitar
21. The best of all possible worlds?
22. The mysterious number e
23. How to make a series
24. Calculus with imaginary numbers
25. Infinity bites back
26. What is a limit, exactly?
27. The equations of nature
28. From calculus to chaos
Further Reading