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A Guide to Cauchy's Calculus: A Translation and Analysis of Calcul Infinitésimal

Dennis M. Cates
Fairview Academic Press
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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About one year ago, I reviewed Dennis Cates’s translation of Cauchy’s Calcul Infinitésimal. In that review, I noted that the book did not include annotations or commentary. Now here is A Guide to Cauchy’s Calculus, whose aim is to provide an interpretive guide to those parts of the original book that remain part of the standard first year calculus sequence.

That leads at once to an important caveat for historians: this book does not include the complete text of Cates’s translation. We get Lectures One through Eight in sequence, but then skip to Twelve, then Nineteen, and so on. The others deal with topics not included in the modern calculus course, and so are left out. That left me very curious, so I checked. Lectures One to Twenty are on differential calculus. From that section, we don’t get

  • Lecture Nine: multivariable chain rule, implicit function theorem.
  • Lecture Ten: homogeneous functions and maxima and minima of functions in several variables.
  • Lecture Eleven: Lagrange multipliers (not with that name).
  • Lecture Thirteen: higher order derivatives of functions of several variables.
  • Lecture Fourteen: total differentials and techniques related to them.
  • Lecture Fifteen: higher order derivatives in one variable and their use to find maxima and minima.
  • Lecture Sixteen: same as fifteen, but in several variables.
  • Lecture Seventeen: conditions for the total differential to have constant sign.
  • Lecture Eighteen: differentials of general functions of several variables which are themselves linear functions of other variables, factoring polynomials with real coefficients.
  • Lecture Twenty: decomposition of rational functions.

That seems like a pity: the fact that these topics are included is one of the things that makes Cauchy’s approach so different from what we do today. It’s also a fairly curious selection, since Lecture Eight (on partial derivatives) is included, while Lecture Fifteen, which is single-variable calculus, is not.

For each Lecture, we get a short introduction, Cauchy’s actual notes (in translation, of course), and finally a long “analysis” of the material in the notes. Each chapter concludes with a set of exercises on the material covered in that Lecture.

All of this reflects Cates’s notion that today’s students could learn calculus by reading Cauchy. I’m afraid I am skeptical. Certainly a course like this would, if it worked, teach students a lot more than they learn from the procedural approach favored in most calculus courses. But I think it would be very difficult to make it work with any but the best students.

As a result, I find myself feeling that the book hangs awkwardly in space somewhere between a historical study and a calculus textbook. Too bad.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.