Exercices d'Analyse et de Physique mathématique

CAUCHY, Augustin-Louis

Verlag: Bachelier, 1840

First edition, very rare complete set, of the second Paris series of Cauchy's 'Exercices', containing most of his research in the period 1840-1853 (see below), in a wide variety of fields including differential equations, the wave theory of light, complex function theory, and group theory. Early in his career Cauchy (1789-1857) found that the volume of his research output could not be accommodated in the Memoirs of the Academy, so he started to publish his work in what was essentially his own mathematical periodical, the 'Exercices de Mathématiques'. This appeared yearly from 1826 until 1830, when Cauchy exiled himself from France following the July Revolution of 1830, moving to the University of Turin. In 1833 he was called to Prague, where Charles X had settled, to assist in the education of the crown prince. There he continued the publication of his 'journal', under the title 'Nouveaux Exercises de Mathématiques'. He returned to Paris in 1838, where the 'Exercices' were once more continued, this time with the title 'Exercices d'Analyse et de Physique mathématique'. Tome I contains a number of important papers that continue research on the wave theory of light that Cauchy had begun in Prague. The problem he studied was how to design an aether with properties that would lead to the observed features of the propagation of light. The interaction of the 'molecules' of the aether had to be such that the aether would only propagate transverse waves, and not longitudinal waves, for example. The investigation of systems of interacting molecules naturally led to the solution of differential equations, a subject that had long been pursued by Cauchy and one that is the subject of several papers scattered throughout these volumes. "A fundamentally new development of the theory of ordinary differential equations in the nineteenth century was defined by the research of Cauchy. Instead of the problem of finding the general solution of a differential equation, the task of solving an initial value problem (the Cauchy problem) came to the fore . . . in solving the initial value problem, Cauchy posed the existence problem, which was to be so fundamental for analysis in the nineteenth century" (Kolmogorov & Yushkevich, Mathematics of the 19th century, p. 193). The highlight of Tome III is undoubtedly 'Mémoire sur les arrangements que t'on peut former avec des lettres données, et sur les permutations ou substitutions a l'aide desquelles on passé d'un arrangement a un autre' (pp. 151-252). This paper, which Peter Neumann has shown was written in 1845, constitutes, with the work of Galois, the birth of group theory. Starting in Tome III, but particularly in Tome IV, Cauchy attempts to establish a firm foundation for the notion of complex numbers. In 'Mémoire sur les quantités géométriques' (IV, pp. 157-80) he proposed the replacement of the theory of imaginary expressions by a theory of geometric quantities - in which he had been anticipated by Argand and others - but he went further, and in 'Sur les fonctions des quantités géométriques' (IV, pp. 308-13) he finally established a visual representation of a function of a complex variable (which he had actually been using implicitly for many years). He took this study one stage further in 'Sur les différentielles de quantités algebriques ou géométriques, et sur les dérivés des fonctions de ces quantités' (pp. 336-47), which is particularly notable for containing his first clear statement of the Cauchy-Riemann equations. ABPC/RBH lists only the Stanitz copy, sold Sotheby's 1984, $770 (lacking half-titles). Four vols., 4to, pp. viii, 424; [4], 412; 395, [1]; 404. Later 19th or early 20th century calf-backed boards, vellum tips, red lettering-pieces on spines (sporadic foxing, heavy on a few leaves, old library stamps on endpapers, titles and first text leaf of each volume, two small holes in margin of 5 leaves of Tome II not affecting text, one other small hole not affecting text). Bestandsnummer des Verkäufers ABE-18795747949

Bibliografische Details

Titel: Exercices d'Analyse et de Physique ...
Verlag: Bachelier
Erscheinungsdatum: 1840
Einband: Hardcover
Zustand: Good
Auflage: 1st Edition