## Résumé des leçons données a l’École Royale Polytechnique, sur le calcul infinitesimal… Tome premier [all published].

Paris: de l’Imprimerie Royale, chez Debure frères, 1823.

First edition of the rarest of Cauchy’s three great textbooks, which placed “the fundamental principles of the calculus on a satisfactory foundation” (Smith, History of Mathematics II, p. 700). The Résumé, often referred to as the Calcul infinitésimal, is divided into 40 lectures, the first twenty on differential calculus, the remainder on integral calculus, each lecture being precisely four pages in length (so that each lecture could be distributed to the students). It may be regarded as the second part of Cauchy’s famous Cours d’analyse (1821) (see below); it gives the first rigorous treatment of calculus proper (differentiation and integration), while the Cours had covered the necessary background concepts (limits, continuity, and convergence). “Within this single text [the Résumé], Cauchy succinctly lays out and rigorously develops all of the topics one encounters in an introductory study of the calculus, from his classic definition of the limit to his detailed analysis of the convergence properties of infinite series. In between, the reader will find a full treatment of differential and integral calculus, including the main theorems of calculus and detailed methods of differentiating and integrating a wide variety of functions. Real, single variable calculus is the main focus of the text, but Cauchy spends ample time exploring the extension of his rigorous development to include functions of multiple variables as well as complex functions” (Cates). “Cauchy’s work set a new standard of mathematical rigor in Europe for the remainder of the 19th century and would finally, after nearly 150 years of attempts, place the calculus on firmly defendable ground (ibid., p. viii). This copy is complete with the important ‘Addition’ (pp. 161-172), which treats Cauchy’s newly discovered mean value theorem and its applications (this is not mentioned in the Table de Matières and is lacking in many copies, including that in the BNF). “Although it is rather less famous than the Cours, the Résumé may have had a more direct influence on the establishment of mathematical analysis, since of course the calculus was the central part of the new discipline founded on limits” (Grattan-Guinness, Convolutions, p. 747). “Cauchy rivaled Euler in mathematical productivity, contributing some 800 books and articles on almost all branches of the subject. Among his greatest contributions are the rigorous methods which he introduced into the calculus in his three great treatises: the Cours d’analyse de l’École Polytechnique (1821), Résumé des leçons sur le calcul infinitesimal (1823), and Leçons sur le calcul differential (1829). Through these works Cauchy did more than anyone else to impress upon the subject the character which it bears at the present time” (Boyer, p. 271). Cauchy’s Résumé is a very rare book on the market – no copies are listed on ABPC/RBH.

The original development of the calculus by Leibniz and Newton had relied on intuitive geometric arguments. Although the majority of scientists and mathematicians accepted the truth of the calculus because of its impressive success in describing and predicting the workings of the natural world, especially in astronomy and mechanics, some, notably Bishop George Berkeley and Michel Rolle, were skeptical about the soundness of its foundations. Their criticisms were addressed by, among others, Colin Maclaurin and Jean le Rond d’Alembert. The next major development came in 1797, when Joseph-Louis Lagrange published his Théorie des fonctions analytiques, based on his lectures at the École Polytechnique. Lagrange used power series expansions to define derivatives, but his approach left open the question as to whether all functions could be expressed as power series. In 1815, just two years after Lagrange died, Cauchy joined the faculty at the École Polytechnique as professor of analysis and started to teach the same course that Lagrange had taught. He inherited Lagrange’s commitment to establish proper foundations for the calculus, but he followed Maclaurin and d’Alembert rather than Lagrange and sought those foundations in the formality of limits. A few years later he published his lecture notes as the Cours d’analyse (1821). As its subsidiary title ‘Première Partie: Analyse algébrique’ suggests, Cauchy intended to write a second part; this appeared two years later as the present work.

In the Foreword, Cauchy outlines the philosophy of this work. “This work, undertaken on the request of the Board of Instruction of the Royal Polytechnic School, offers a summary of the lectures that I gave to this school on the infinitesimal calculus. It will be composed of two volumes [sic, see below] corresponding to the two years which form the duration of the course. I publish the first volume today di- vided into forty lectures, the first twenty of which comprise the differential calculus, and the last twenty a part of the integral calculus. The methods that I follow differ in several respects from those which are found expressed in the works of similar type. My main goal has been to reconcile the rigor, which I have made a law in my Analysis Course [i.e., the Cours d’analyse], with the simplicity which results from the direct consideration of infinitely small quantities. For this reason, I thought obliged to reject the expansion of functions by infinite series, whenever the series obtained are not convergent; and I saw myself forced to return the formula of Taylor to the integral calculus, and this formula can only be admitted as general so long as the series that it contains is found reduced to a finite number of terms and supplemented by a definite integral. I am aware that the illustrious author of the Analytical Mechanics [i.e., Lagrange] has taken the formula in question for the basis of his theory of derived functions. But, despite all the respect that such a grand authority commands, the majority of mathematicians are now in accordance to recognize the uncertainty of the results which we can be led to by the employment of divergent series, and we add that in several cases, the Theorem of Taylor seems to provide the expansion of a function by convergent series, even though the sum of the series differs fundamentally from the proposed function (see the end of the thirty-eighth lecture). Moreover, those who will read my work, will be convinced, I hope, that the principles of the differential calculus and its most important applications, can be easily explained without the intervention of series.

“In the integral calculus, it seemed necessary to demonstrate generally the existence of integrals or primitive functions before making known their various properties. To achieve this, it was first necessary to establish the notion of integrals taken between given limits, or definite integrals. These latter objects can sometimes be infinite or indeterminate, it being essential to study in which cases they maintain a unique and finite value. The simplest means of resolving the question are the employment of singular definite integrals which are the subject of the twenty-fifth lecture. Moreover, among the infinite number of values that we can attribute to an indeterminate integral, there exists one which merits our particular attention and that we have named principal value. The consideration of singular definite integrals and those of the principal values of indeterminate integrals are very useful in the solution of a large group of problems. We deduce a great number of general formulas that work for the determination of definite integrals and are similar to those that I gave in a report presented to the Institute in 1814. We will find in lectures thirty-four and thirty-nine a formula of this type applied for the evaluation of several definite integrals, some of which were already known” (Cates, pp. xv-xvi).

Cauchy begins the 20 lectures devoted to the differential calculus by defining the derived function of a function f(x) as the limit of (f(x + h) – f(x))/h as h moves towards zero; he denotes it by f’(x). He goes on to develop the now standard material of a course of differential calculus: partial and total derivatives and differentials, the conditions for maxima and minima, including multipliers for constraints, and L’Hospital’s rule. Particularly noteworthy is his treatment of the mean value theorem in lecture 7 where we find the statement ‘we designate two very small (positive) numbers δ, ε’, the debut of these famous letters.

The twenty lectures on the integral calculus were equally blessed with novelties. He began by defining the definite integral of a continuous function over a finite interval using the now familiar method of partitioning the interval, and showing that the sum tends to a limit as the number of points in the partition grows ever larger. He extended this to complex integrals, but stopped short of the residue calculus, which he left for a later work (1825). He proved the fundamental theorem of calculus, that differentiation and integration are inverse processes. The other important result was the rigorous treatment of Taylor’s theorem, with the integral form of the remainder. Cauchy here answers in the negative the question, left open by Lagrange, of whether all functions are given by powers series: he gives the now familiar example of exp(-1/x2) as a function which has a vanishing Taylor expansion.

One of the most fascinating features of the work is the ‘Addition.’ It begins: ‘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 what is now called Cauchy’s mean value theorem; Cauchy had evidently just discovered this important theorem, realized its fundamental role in the differential calculus, and rushed to include it in the work by inserting the ‘Addition’ into copies remaining at the printer.

The reason the present work is titled ‘Tome premier’ was discovered only in the late 1970s, when Christian Gilain found the notes (actually printed proof sheets) for the second volume, which treated the theory of ordinary differential equations, including Cauchy’s famous existence theorem for solutions of such equations. Gilain explains why the second volume was not published. In 1820 the École Polytechnique prescribed that if a lecturer covered material not in the standard texts, he should write up notes on it for the students, who would be taking examinations the following September. This decree was the reason for the existence of both the Cours and the Résumé. But the printer’s dates on the differential equations notes show that with them Cauchy had fallen behind; printing was probably dropped once it was too late to help the students with their examinations.

Like the Cours, the Résumé was published by the publishing family of de Bure (‘Booksellers of the King’), into which Cauchy had conveniently married in 1818.

Cauchy (1789-1857) was one of the most prolific mathematicians in history (second only to Leonhard Euler), and contributed to almost every branch of mathematics. “Cauchy’s greatest contributions to mathematics, characterized by the clear and rigorous methods that he introduced, are embodied predominantly in his three great treatises: Cours d’analyse de l’École Royale Polytechnique (1821); Résumé des leçons sur le calcul infinitésimal (1823); and Leçons sur les applications du calcul infinitésimal à la géométrie (1826–28). The first phase of modern rigour in mathematics originated in his lectures and researches in analysis during the 1820s. He clarified the principles of calculus and put them on a satisfactory basis by developing them with the aid of limits and continuity, concepts now considered vital to analysis. To the same period belongs his development of the theory of functions of a complex variable (a variable involving a multiple of the square root of minus one), today indispensable in applied mathematics from physics to aeronautics … Cauchy made substantial contributions to the theory of numbers and wrote three important papers on error theory. His work in optics provided a mathematical basis for the workable but somewhat unsatisfactory theory of the properties of the ether, a hypothetical, omnipresent medium once thought to be the conductor of light. His collected works, Oeuvres complètes d’Augustin Cauchy (1882–1970), were published in 27 volumes” (Britannica).

Boyer, The History of the Calculus (1949), Ch. VII; Cates, Cauchy’s Calcul Infinitésimal. An Annotated English Translation, 2019. Gilain, Équations différentielles ordinaires by Augustin-Louis Cauchy (1981); Grattan-Guinness, Convolutions in French mathematics, 1800-1840 (1990); Grattan-Guinness (ed.), Landmark writings in western mathematics 1640-1940 (2005), Ch. 25.

Large 4to (259 x 205 mm), pp. xii, 172 (light browning and foxing). Contemporary half-calf and marbled boards, spine gilt with black lettering-piece.

Item #5815

Price: \$6,500.00

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