Paris: Plassan, 1822.
First edition, extremely rare separately-paginated offprint (journal pagination 161-174), inscribed by Cauchy (1789-1857) to his senior colleague Sylvestre François Lacroix (1765-1843). In this paper Cauchy published for the first time many of the most important results of his great work Mémoire sur les intégrales définies, submitted to the Académie des Sciences on 22 August 1814 but not published until 1827, which laid the foundations of complex function theory (the delay was possibly due to its length of almost 200 pages). “This remarkable paper [i.e., the 1814 paper] is a major seed for 19th century mathematics, for in it Cauchy inaugurated the theory of complex variable functions and their integration” (Grattan-Guinness, p. 656). “The discoveries with which Cauchy’s name is most firmly associated in the minds of both pure and applied mathematicians are without doubt his fundamental theorems on complex functions” (DSB). In this 1822 work we find, in particular, the first published statement of ‘Cauchy’s integral formula’ (p. 10) and the ‘residue theorem’ (on p. 9) – these are the central results of complex function theory. On p. 11 of the present paper, Cauchy also argues for the first time that the definite integral should be defined in terms of ‘partial sums,’ an approach he developed in detail in his Résumé des Leçons (1823). This was an essential step towards the theory of complex integration he presented in Mémoire sur les intégrales définies, prises entre des limites imaginaires (1825), and was the basis for all rigorous approaches to integration until the twentieth century. We can find no record of this offprint in auction records, nor of any offprint inscribed by Cauchy. Not on OCLC.
Provenance: Sylvestre François Lacroix, inscription on title page “à M. Lacroix de la part de l’auteur”. Although Lacroix’s mathematical work contained little that was absolutely new and original, he was a great teacher and writer of textbooks. His monumental three-volume treatise on calculus, Traité du Calcul Différentiel et du Calcul Intégrale, incorporated the various advances made since the middle of the eighteenth century, and synthesised the works of Euler, Lagrange, Laplace, Monge, Legendre, Poisson, Gauss, and Cauchy, whose writings are followed up to the year 1819. As a student at the École Polytechnique (from which he graduated in 1807), Cauchy studied analysis under Lacroix. Several years later, Lacroix was one of the examiners of Cauchy’s 1814 memoir (the other was Legendre), and its publication was recommended by their report to the Institut de France.
One of Cauchy’s motivations in writing the 1814 paper was to justify the use of complex variables to evaluate real definite integrals. “The practice had begun with Euler and continued with Laplace, Poisson, and Legendre, in his Exercises de Calcul Intégrale (1811). In the Mémoire on this topic that he presented in 1814 Cauchy commented that many of the integrals had been evaluated for the first time “by means of a kind of induction” based on “the passage from the real to the imaginary”, and that no less a figure than Laplace had remarked that the method “however carefully employed, leaves something to be desired in the proofs of the results”. Cauchy accordingly set himself the task of finding a “direct and rigorous analysis” of this dubious passage” (Gray, pp. 59-60). Cauchy’s defence of this technique led, among other things, to the Cauchy-Riemann equations. However, his chief interest in the 1814 paper was in double integrals.
“Mathematicians of the period, with their somewhat formal attitude to the processes of the calculus, not only tended to regard a double integral as a nested pair of single integrals but to be indifferent to the order of integration. This formal approach precluded interpreting double integrals in terms of line integrals along the boundaries of a rectangle in the plane of complex numbers; they were strictly about integrals on a rectangle in the real plane. However, it was known that in some cases this attitude was too naive and led to conflicting results, and it was this problem that Cauchy addressed in the second part of the Mémoire …
“Cauchy showed varying the order of integration of a double integral can yield two distinct definite values when the function becomes infinite or indeterminate for certain values of the variables within the domain of integration. To investigate the matter further, he introduced what he called singular integrals (‘intégrales singulières’), which he defined as limits as the end points of the integral tend to the same fixed value “without the integrals being zero.” Singular integrals are a profound idea. They were to crop up repeatedly in real analysis and, in an intriguing different way, in Cauchy's later work on complex integrals” (Gray, p. 61).
In the 1822 paper, Cauchy began by recalling his 1814 definition of a singular integral and then drew upon results from the same paper on the evaluation of functions of a complex variable over real-valued limits. This led to the formula at the top of p. 169, the first published version of the residue theorem. “This pretty form of a residue theorem was put through some of its paces in the rest of Cauchy’s paper [i.e., the offered paper]. For example, he found an integral formula for [the nth derivative], and two pages later he obtained a couple of evaluations of which Poisson had given special cases … in a postscript he reproved some of those results using Parseval’s theorem” (Grattan-Guinness, p. 740).
“At this point [p. 171, second paragraph] Cauchy made an interesting remark that helps to explain his motivation for introducing the concept of definite integral the way he did in his Résumé. He said that he considered each definite integral, taken between real limits, ‘as nothing other than the sum of the values of the differentials corresponding to the various values of the variable between the given limits. This way of thinking of a definite integral, it seems to me, must be preferred because it leads equally to all cases, even those in which one docs not know at all how to pass generally from the function placed under the integral sign to the primitive function [i.e., the function whose derivative is the function to be integrated].’ Thus, his research work on definite integrals, including in particular his theory of singular integrals, apparently persuaded him to abandon the prevailing view at the time that the definite integral should be defined via the primitive function and, instead, to define the definite integral as limits of partial sums. In addition, Cauchy stated, this view ‘has the advantage’ of always providing real values for integrals of real functions, as well as allowing one to separate ‘easily’ any imaginary equation into its real and imaginary parts. This will not be the case when considering a definite integral as the difference of the values taken by a (discontinuous) primitive function at the limits of integration, or when allowing the variable to take imaginary values” (Gray & Bottazzini, pp. 121-122).
“… although there is hardly any doubt that by the early 1820s Cauchy was in possession of a wealth of results that, in an incoherent way, prefigured his later achievements, a satisfactory theory of complex functions was still lacking. An essential step towards such a theory was his decision to get rid of the ‘old’ definition of definite integrals based on primitive functions, and instead to define them as limits of partial sums” (ibid., p. 128).
“Cauchy showed understandable signs of impatience in [the present] paper at the non-appearance of his 1814 paper on complex integration. Not only did he announce – falsely, it turned out – that it ‘will soon be published’ [p. 1, footnote], but he also mentioned other texts which in the end were never published and seem now to be lost … [The present paper] appeared from the Société Philomatique in their November Bulletin. At 14 pages it was rather long for them … so they accepted his offer to pay for the printing of eight of the pages” (Grattan-Guinness, p. 738).
Grattan-Guinness, Convolutions in French Mathematics 1800-1840, Section 11.4.4; Gray, The Real and the Complex: A History of Analysis in the 19th Century, 2015; Gray & Bottazzini, Hidden Harmony – Geometric Fantasies. The Rise of Complex Function Theory, 2013; Smithies, Cauchy and the Creation of Complex Function Theory, Sections 3.5-3.7.
4to (265 x 210 mm), pp.  2-14. Seven uncut loose leaves, at some point extracted from a binding. Light spotting, edges slightly frayed.