This book, first published in 1967, is something of a relic. It came, of course, well before the calculus reform movement and represents an approach to teaching calculus that — as far as I can tell — has mostly disappeared. One might be inclined to dismiss it out of hand, but I think there is some wisdom in the approach that would be a mistake to ignore.

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In the last few decades I have worked with a considerable variety of physicists, physical chemists, chemical engineers and computer scientists. When I have asked what elements of their mathematical training they have found the most valuable, nearly all of them point to a calculus book. Usually the book is something like an old version of George Thomas’s *Calculus with Analytic Geometry*. No doubt there is a selection effect here: these are people who did well in calculus and enjoyed it. There are no representatives from the same course who were overwhelmed, discouraged and dropped out. But in the current book Morris Kline concentrates much of his efforts on support for students who might be expected to struggle.

Kline’s plan for the book is clearly stated and is reflected in the subtitle. “Intuitive” means using plausible arguments to support the development of the basic concepts. A “physical approach” means relying extensively on applications to build intuition. Most of the applications are from physics, but there are a scattering of others from the biological and social sciences. None of the applications require a background in any of those fields. Kline believes that an approach with too much rigor is inappropriate for an introductory course. He also thinks that a compromised treatment with a bit of rigor (some theorems with proofs) mixed with heuristics and intuition is a bad idea because it confuses students about what proof really means. Fundamentally he feels that rigor should be postponed to a second course after an intuitive basis has been well established.

Kline’s approach is to proceed slowly and gently, and particularly so in the early development of the derivative. He recognizes that many beginning calculus students are not proficient in algebra and he accommodates this by working very slowly through the details starting with examples where algebraic manipulations are minimal.

The topics treated in the book are all standard ones. One notable variation is the introduction of the integral — really just the antiderivative — immediately following the first chapter on the derivative. Since Kline uses the concepts of distance and speed extensively to motivate the derivative, he wants to be able to go from speed back to distance. He doesn’t formally introduce the integral until several chapters later, but he does want the derivative and the integral firmly tied together in the student’s mind.

This is a very long book. When I first picked it up I assumed that it included all the usual topics of advanced calculus as well. But it concludes with chapters on partial differentiation, multiple integrals and an introductory bit of differential equations. The book is long because Kline simply goes more slowly and spends much more time on details than any other calculus book I have seen. Students who need that level of support and actually read the text would benefit. Others, especially stronger students, would find the exposition sometimes painfully drawn out.

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Bill Satzer (wjsatzer

mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.