Beschreibung
The true first edition of the most advanced statement of Lagrange's programme of algebraic analysis, which forms a supplement to, and commentary on, his 'Théorie des fonctions analytiques'. (Please note that the 'Nouvelle edition' statement on the title refers to the fact that the Seances were first published as a series of 6 vols.; in the 'Nouvelle edition', the first 6 vols. were reset and four further vols. were added, but the offered work is the first printing of vol. 10.) This first edition of the 'Leçons', which forms the complete Tome 10 of Séances des Écoles Normales, is rare; the 'Leçons' were reprinted, in quarto format, in the Journal de l'Ecole Polytechnique, Tome 5, Cahier 12 (An XII = 1803/4), and then in a revised version in 1806 as a separate work, the form in which the 'Leçons' are almost always encountered. (DSB states that the 'Leçons' first appeared in the Journal de l'Ecole Polytechnique in 1801, and this error is often reproduced in dealer descriptions). "The most detailed attempt to provide a systematic foundation of the calculus was contained in two treatises by Lagrange published at the end of the [eighteenth] century: the 'Théorie des fonctions analytiques' (1797) and 'Leçons sur le calcul des fonctions' (1801; rev. ed. 1806) . . . Lagrange's goal was to develop an algebraic basis for the calculus that made no reference to infinitely small magnitudes or intuitive geometrical notions . . . Lagrange used the term 'algebraic analysis' to designate the part of mathematics that results when algebra is enlarged to include calculus-related methods and functions. The central object here was the concept of an analytical function. Such a function y = f(x) is given by a single analytical expression that is constructed from variables and constants using the operations of analysis . . . The idea behind Lagrange's theory was to take any function f(x) and expand it in a Taylor power series . . . The derived function or derivative f'(x) is defined to be the coefficient of the linear term in this expansion; f'(x) is a new function of x with a well-defined algebraic form . . . Lagrange's 'Théorie' and 'Leçons', written when he was in his sixties, were notable for their success in developing the entire differential and integral calculus on the basis of the concept of an analytic function. They contained several important technical advances. Lagrange introduced inequality methods to obtain numerical estimates of the values of functions, thereby providing a source of techniques that Augustin Cauchy was later able to use in his arithmetical development of the calculus. Another significant contribution was contained in Lagrange's exposition of the calculus of variations. To obtain the variational equations he modelled the derivation after an earlier argument in the theory of integrability. Although his derivation never quite achieved acceptance among later researchers, it remains historically noteworthy as an example of advanced reasoning in algebraic analysis. Lagrange also introduced the multiplier rule in both the calculus and the calculus of variations, a powerful method that allows one to solve a range of problems in the theory of constrained optimization" (The Cambridge History of Science: Volume 4, Eighteenth-Century Science, Roy Porter, ed., pp. 322-4). For a detailed analysis of the 'Leçons', see C. G. Fraser, Joseph Louis Lagrange's algebraic vision of the calculus, Historia Mathematica 14 (1987), 38-53. 8vo, pp. [6], 7-528, [4]. Contemporary quarter-calf and marbled boards, spine decorated in gilt and with red lettering-piece (pp. 131-4 misbound after p. 180, ink stamp on title, small piece missing from lower inner margin of final leaf, probably a paper flaw, nowhere near text). Bestandsnummer des Verkäufers ABE-18719155539
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