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Cauchy's Calcul Infinitésimal: A Complete English Translation

Augustin-Louis Cauchy, translated by Dennis M. Cates
Fairview Academic Press
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[Reviewed by
Fernando Q. Gouvêa
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Augustin-Louis Cauchy is a towering figure, both in the history of mathematics and in the history of how it has been taught. Having been appointed professor of analysis at the École Polytechnique, he proceeded to completely recast the subject, mostly with the goal of making it more rigorous. Many of the topics we teach in today’s calculus courses were first taught to undergraduates in Cauchy’s courses, and even the sequence of materials owes something to his ideas. On the other hand, his approach was intensely intellectual and “proof-oriented,” as we now say. So much so, in fact, that students at the Polytechnique held protests against the young professor.

Cauchy was famously prolific, writing memoirs and papers very quickly and fluidly. Focused on research, he seems to have been much less interested in writing for students. The most famous of his textbooks was the Cours d’Analyse, published in 1821, which was lovingly translated into English and annotated by Ed Sandifer and Rob Bradley. This was an introductory volume dealing with what Cauchy called “algebraic analysis”: the theory of continuity and convergence, including sequences and series and the elementary functions of analysis. The book was supposed to be only the first volume, but Cauchy never did write the second.

For Cauchy’s calculus course, we have only a “Résumé des Leçons,” what we would describe as lecture notes. The first words of the “Avertissement” (translated here as “Foreword”) are

This work, undertaken on the request of the Board of Instruction of the Royal School Polytechnic, offers a summary of the lectures that I gave to this school on the infinitesimal calculus. It will be composed of two volumes corresponding to the two years which form the duration of the education. I publish the first volume today divided into forty lectures, the first twenty of which comprise the differential calculus, and the last twenty a part of the integral calculus. (p. xii)

Once again, the promised second volume seems to have never appeared.

Cauchy’s calculus course is easily available online (for example, through Gallica), scanned either from the original edition or from Cauchy’s Collected Works (series II, volume 4). I own a printed facsimile, part of a series called “Les Cours Historiques de l’École Polytechnique.” That’s great for those who read French, but in our monolingual times many readers will want to see a translation, and so will welcome Dennis M. Cates’ effort.

In contrast to the Bradley-Sandifer edition of the Cours d’Analyse, Cates does not attempt analysis or annotation, except for occasional footnotes, for example to point out typos in the original publication. He translates carefully, if sometimes a little woodenly, as is already apparent in the lines we quoted from the preface. “École Polytechnique” should either be untranslated or given as “Polytechnic School” and “duration of the course” would be a more idiomatic version of “durée de l’enseignement.” (Engineering students were expected to move to a more specialized school after their two years at the Polytechnique.)

Reading Cauchy is fun. Today’s readers will enjoy both the moments of recognition (“see, he did it the way we do it!”) and the moments where something surprising happens. Cauchy tries to have it both ways: he wants to be rigorous, but he also wants to use infinitesimal arguments. “My main goal has been to reconcile the rigor, which I have made a law in my Analysis Course, with the simplicity which results from the direct consideration of infinitely small quantities,” he says. Thus, for example, he defines the differential df rather than the derivative function f'(x): for him the derivative is indeed derived from the differential! “To differentiate a function is to find its differential.”

This combination of familiarity and difference makes reading these notes very interesting. Indeed, students in a real analysis course would learn a lot by trying to figure out which of Cauchy’s arguments would pass muster in their course.

One of the most fascinating features of these lecture notes is the addendum that appears at the end (the title in the translation is “Addition”, which will confuse English readers, I’m afraid). Here’s how it starts:

Since the printing of this work, I recognized that with the help of a very simple formula we could bring back to the differential calculus the solution of several problems that I returned to the integral calculus. I will, in the first place, give the formula; then, I will indicate its main applications.

The formula in question is the (Cauchy) mean value theorem, and the applications include L’Hospital’s rule and Taylor’s theorem. In the main text, L’Hospital’s rule had actually been discussed in the section on differential calculus, but with an informal argument based on infinitesimals. Taylor and Maclaurin series had indeed been treated in the context of the integral calculus. The “addition” appears already in the 1823 first edition, so what Cauchy may mean is that he wrote it up and added it after seeing the first printed pages. In any case, we see history happening: Cauchy discovers the mean value theorem, realizes its fundamental role in the differential calculus, and rushes to add an explanation of this at the last minute.

If you can’t read the French and you’d like to see how Cauchy invented the calculus course, here’s your chance!

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College. He loves to read old books.

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