Cauchy's Cours d'Analyse: An Annotated Translation

As is the case with so many of us, I first discovered Augustin-Louis Cauchy (21 VIII 1789–23 V 1857) in the pages of E. T. Bell’s timeless Men of Mathematics. My local public library was in possession of a single copy of this idiosyncratic book, in its old (original) hard-cover form, replete with marvelous photographs, daguerrotypes, and renderings of old painted portraits. I was very happy to be able to obtain a few years ago a cherry isomorphic copy of this book via the magic of online booksellers, and it is indeed like visiting an old friend to leaf through these familiar pages from time to time. It is a pleasure to do so now in connection with the book under review.

On pp 270–293 of Men of Mathematics we find the article, “Mathematics and Windmills,” a biography of Cauchy and a compact representation of Bell’s unquestionably dubious psychoanalytic skills and historical sensibilities, as well as, more importantly, his marvelous ability to make his subjects come to life on the printed page. So long as Bell’s now well-established leanings in the direction of embellishment are taken into account, I can still heartily recommend this sketch of Cauchy and his place in 19th century mathematics to anyone with an interest in the personalities and characters of mathematical figures that are now household names. Again, however, be forewarned: Bell is, for better or for worse, a raconteur.

In any event, we learn from Bell (p. 273) that, “Cauchy’s childhood fell in the bloodiest period of the Revolution… [T]o escape the obvious danger, Cauchy[’s father] moved his family to his country place in the village of Arcueil. There he sat out the Terror, half-starved himself and feeding his wife and infant son largely from what scanty fruits and vegetables he could raise.” Bell proceeds to note that “Cauchy senior undertook the education of his children [himself, writing] his own textbooks, several of them in the fluent verse of which he was a master.” Thus, from infancy and throughout his youth, Cauchy experienced, albeit amidst considerable hardship, the exceptional privilege of a serious if eccentric private education. Perhaps this qualifies as suitable foreshadowing for Cauchy’s subsequent unique career as a scholar. Says Bell along these lines (p. 271): “He had an extraordinary fertility in mathematical inventiveness and a fecundity that has been surpassed only twice — by Euler and Cayley. His work, like his times, was revolutionary.”

In broad outlines then, the events of Cauchy’s life are as follows, again in Bell’s words. “Having outlived the Terror, he graduated from the [École] Polytechnique into the service of Napoleon.” (p. 271) “Whatever else may be said of Napoleon, he took ability wherever he found it.” (p. 275) “The Moscow fiasco of 1812, war against Prussia and Austria, and … the battle of Leipzig in October, 1813, … distracted Napoleon’s dream of invading England, and the works at Cherbourg [where Cauchy was stationed] languished. Cauchy returned to Paris in 1813, worn out by overwork. He was then only 24, but he had already attracted the attention of the leading mathematicians of France by his brilliant researches, particularly the memoir on polyhedra and that on symmetric functions.” (p. 277) (The former dealt with the Euler-Poincaré formula, sans Poincaré, of course; the latter is nothing less than a treatise on the theory of substitutions and transformation groups, evidently containing what we now call Cauchy’s Theorem in group theory.)

“[Accordingly, b]y the age of 27 (in 1816) Cauchy had raised himself to the front rank of living mathematicians.” (p. 284) “[In 1815] Cauchy created a sensation by proving one of the great theorems Fermat had bequeathed to a baffled posterity: every integer is a sum of three [triangular numbers], four squares, five [pentagonal numbers], six [hexagonal numbers], and so on…” (p. 284) “Honorable and important positions now came thick and fast to the greatest mathematician in France — still well under thirty. His mathematical activity was incredible; sometimes two full-length papers would be laid before the Academy [of Sciences] in the same week.” (p. 286) “In the midst of all this work, Cauchy found time to do his courting … Aloise de Bure, whom he married in 1818 and with whom he lived for nearly forty years, was the daughter of a cultured old family and, like himself, an ardent Catholic. They had two daughters, who were brought up as Cauchy had been.” (loc. cit.)

Then Bell goes on to say, “encouraged by Laplace and others, Cauchy in 1821 wrote up for publication the course of lectures on analysis he had been giving at the Polytechnique. This is the work which for long set the standard in rigor,” and, indeed, it is the book now under review, Cauchy’s Cours d’analyse, An Annotated Translation, by Robert E. Bradley and C. Edward Sandifer.

And it is a marvel indeed to encounter the thoughts and words of one of the Olympians of analysis (and of mathematics itself) in such an accessible and well-crafted form. It is unquestionably the case that Bradley and Sandifer are to be credited with first-rate scholarship, both as regards the quality of the translation and the critical labor of supplying the indicated footnotes, which add a great deal to the finished product. These footnotes serve well to tie Cauchy’s École Polytechnique lecture notes (so to speak) to modern conceptions and phrasings of what are now household notions; consider e.g., footnote 20 in Chapter Six: “This is another implicit application of the Comparison Test and Cauchy’s notion of absolute convergence.” This speaks volumes: we are indeed present at the birth of modern analytic rigor, or at least its early infancy, so we should not expect to come across more historically mature formulations of the according notions. Additionally, even given Cauchy’s status as a pioneer of rigor, we must factor in the reality that the style of communication Cauchy embraced differs dramatically from what is now standard in the field. There is no question that the almost exactly four hundred pages of Cauchy’s Cours d’analyse, An Annotated Translation read like what they are: an early nineteenth century textbook of mathematics written for students of a polytechnic university.

But it is still Cauchy, of course, and it cannot be stressed strongly enough that in the pages of this wonderful book we are presented with first-generation evidence of a mathematical revolution that left the subject forever changed (indeed, corrected) presented by one of the two or three players solely responsible for this sea-change. (Carl Friedrich Gauss is often properly credited with being mathematically rigorous well ahead of his era, but it was Karl Weierstrass (1815–1897) who gave us the arithmetization of analysis; this said, Cauchy might be classed as something of a precursor to Weierstrass, and then some: the notion of a Cauchy sequence and the concomitant completion of Q into R surely qualify as a prime example of true rigor entering into analysis and analysis situs.)

The book under review is split into nine chapters, the mere titles of which tell the reader what he’s in for. I want to address only a few points in this connection: Chapter 2 is titled “On infinitely small and infinitely large quantities,” which illustrates the fact that Leibniz’ ghosts of departed quantities had not yet departed at the time of Cauchy’s writing; I guess they were not properly exorcised till Weierstrass entered the game. (And then there’s the relatively recent work of Abraham Robinson (1918–1974) letting them in again … ). In any case, no explicit epsilon-delta definitions are to be found in Cauchy’s Cours d’analyse. But not to worry: consider Chapter 6, for example, where the focus falls on, among other things, “rules for the convergence of series.” Bell tells of Laplace’s great worry that his classic Méchanique Céleste, published in five volumes from 1799 to 1825, would be undone by some of the series occurring there failing to pass Cauchy’s convergence tests: happily this fear proved unfounded (but just barely, if Bell is to be believed). Thus, the mere fact that Cauchy presented these tests in his text suffices to underscore its great novelty.

Complex analysis starts in Chapter 7 and then is present throughout the rest of the book. Interestingly, this early presentation of complex variables by the master himself is void of any discussion of what we now know as the Cauchy integral theorem and formula; indeed, integration itself is not even part of the text (although the derivative occurs on p.44: again, a sign of the times, of course (note that Riemann (1826–1866) had not even been born at the time of the first publication of Cauchy’s Cours d’analyse). Additionally, it is the case that Cauchy apparently did not prove his theorem until 1825 (although there is some evidence of an earlier (1814) related submission: see Stillwell, Mathematics and Its History), but, in any event, it appears that this material, now at the heart of “Cauchy theory” in complex analysis was not yet ready for inclusion in lectures at the École at this point in time.

But there is so much else there. Consider Chapter 10 for example: “On real or imaginary roots of algebraic equations for which the left-hand side is a rational and integer function of one variable. The solution of equations of this kind by algebra or trigonometry.” (How’s that for a beefy chapter-heading?) This chapter entails a dramatic illustration of the preeminent role played by the theory of equations in France at this point in mathematical history and of Cauchy’s leading part in the play. Recall in this connection that Évariste Galois (1811–1832) submitted two papers to the Académie des Sciences, refereed by Cauchy, who rejected them “for reasons that still remain unclear. In spite of many claims to the contrary, it appears that Cauchy had recognized the importance of Galois’ work, and that he merely suggested combining the two papers into one in order to enter it in the competition for the Academy’s Grand Prize in Mathematics. Cauchy … considered Galois’ work to be a likely winner.” (http://en.wikipedia.org/wiki/%C3%89variste_Galois).

It should also be noted that the book under review comes equipped with a well-written Translator’s Preface, full of interesting and relevant historical data, placing Cauchy’s work in the present connection in the proper historical context. This preface is evocative also because of the introduction of a wealth of familiar French scholars of Cauchy’s era, all of whom are now part of the analysis courses we teach today. And here is a telling phrase: “The Cours d’analyse begins with a short Introduction, in which Cauchy acknowledges the inspiration of his teachers, particularly Pierre Simon Laplace (1749–1827) and Siméon Denis Poisson (1781–1840), but most especially his colleague and former tutor André Marie Ampère (1775–1836).” It is here that he gives his oft-cited intent in writing the volume, “As for methods, I have sought to give them all the rigor which one demands from geometry, so that one need never rely on arguments drawn from the generality of algebra.’’ This appears to fit with our earlier informal estimate of Cauchy’s rigor vis à vis, say, that of Weierstrass coming a generation later: it was, and is, an evolution.

Cauchy’s Cours d’analyse, An Annotated Translation is a major contribution to mathematical historical scholarship, and it is most welcome indeed to have occasion to examine the infancy of a part of modern analysis, to recognize familiar things in archaic and even arcane phrasings (caveat: the notation is Cauchy’s, too), and, through it all, to witness a grandmaster in action.

Finally, we turn to Bell again (p. 293) to complete our biographical remarks regarding the author of the Cours d’analyse: “Cauchy died rather unexpectedly in his 68th year on May 23, 1857. Hoping to benefit a bronchial trouble, he retired to the country to recuperate, only to be smitten with a fever that proved to be fatal.” Happily we have his (voluminous) writings in his Oeuvres Complètes; and now we also have a marvelous English translation of one of his textbooks, a true historical landmark.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.