Descripción
First edition, first issue (see below), containing Lagrange's formulation of calculus in terms of infinite series, which provided the basis for Cauchy's development of complex function theory. "The full title of the Théorie explains its purpose: 'Theory of analytical functions containing the principles of the differential calculus disengaged from all consideration of infinitesimals, vanishing limits or fluxions and reduced to the algebraic analysis of finite quantities'. Lagrange's goal was to develop an algebraic basis for the calculus that made no reference to infinitely small magnitudes or intuitive geometrical and mechanical notions . . . Part One of the Théorie begins with some historical matters and examines the basic expansion of a function as a Taylor power series. There is considerable discussion of values where the expansion may fail, and a derivation of such well-known results as l'Hôpital's rule. Lagrange then turned to methods of approximation and an estimation of the remainder in the Taylor series, followed by a study of differential equations, singular solutions and series methods, as well as multi-variable calculus and partial differential equations . . . Part Two on geometry opens with an investigation of the geometry of curves. Here Lagrange examined in detail the properties that must hold at a point where two curves come into contact the relationships between their tangents and osculating circles. Corresponding questions concerning surfaces are also investigated, and Lagrange referred to Gaspard Monge's memoirs on this subject in the Académie des Sciences. He derived some standard results on the quadrature and rectification of curves. The theory of maxima and minima in the ordinary calculus, a topic Lagrange suggested could be understood independently of geometry as part of analysis, is taken up. Also covered are basic results in the calculus of variations, including an important theorem of Adrien-Marie Legendre in the theory of sufficiency . . . The third part on dynamics is somewhat anticlimactic, given the publication nine years earlier of his major work Méchanique analitique. In this part Lagrange presented a rather kinematically-oriented investigation of particle dynamics, including a detailed discussion of the Newtonian problem of motion in a resisting medium. He also derived the standard conservation laws of momentum, angular momentum and live forces. The book closes with an examination of the equation of live forces as it applies to problems of elastic impact and machine performance . . . some of the major original contributions of this work: the formulation of a coherent foundation for analysis; Lagrange's conception of theorem-proving in analysis; his derivation of what is today called the Lagrange remainder in the Taylor expansion of a function; his formulation of the multiplier rule in the calculus and calculus of variations; and his account of sufficiency questions in the calculus of variations" (Fraser, Joseph Louis Lagrange, Théorie des fonctions analytiques (1797), Chapter 19 in Landmark Writings. Grattan-Guinness, Convolutions in French Mathematics 1800-184, pp. 129-133). There are two issues of this work: Version A, with 276 pages, and Version B, with 277, as well as the journal issue. Version B is the true first edition. This is revealed by the 8-page index that appears at the beginning of both A and B, which refers to the "Conclusion" on pp. 276-277. Since Version A does not have a page 277, it is clear that the index must have been prepared for Version B, making B the earlier issue. Version B's priority is confirmed by the fact that, in Version A the errata listed on p. 277 of Version B have been corrected. The journal issue is also version B. Riccardi I (2), 3; Norman 1258. 4to, pp. [iv], viii, 277, [1], unopened (stamp of Ecole Polytechnique on title). Original marbled wrappers, manuscript paper label on spine (rubbed). N° de ref. del artículo ABE-1587403220478
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