In his introduction to Théorie des fonctions analytiques, Lagrange analyses the metaphysics of infinitesimal calculus and gives John Landen the credit for having worked out a purely analytical method, in which the finite differences of the variables substitute the infinitesimal differences, although he adds: "on doit convenir que cette manière de rendre le Calcul différentiel plus rigoureux dans ses principes lui fait perdre ses principaux avantages, la simplicité de la méthode et la facilité des opérations". Preceded in 1755 by Mathematical Locubrations, in which the method of fluxions is systematically applied to the solution of algebraic equations and the inverse calculus of fluents, in 1758 John Landen (1719-1790) published A Discourse concerning the Residual Analysis, an announcement of the treatise which was to appear in 1764: The Residual Analysis, a New Branch of the Algebraic Art. In this John Landen intended to release the method of fluxions from the principles derived from the doctrine of motion, and from a basis of pure algebra he would develop methods for immediate application to the main problems of analysis: maxima and minima of functions, curvature, quadrature and rectification of curves. Only finite increments are considered, which are equalized to zero only after having simplified the factor that in their ratio makes them null. An algebraic identity on the differences of rational powers plays a central role. While teaching at the military academy in Turin in the years 1756-59, Lagrange wrote a treatise of Cartesian geometry and differential calculus in which he accomplished a compromise between the Newton method of the “prime and ultimate ratios” and the Leibnitzian notation. Differential calculus and integral calculus were preceded by the algebraic calculus of finite differences from whose formulas those of differential calculus are obtained. The initial concordance then becomes an explicit influence in the basic inspiration of the theory of analytic functions, and even Lacroix was to give credit to Landen for having solved the drawbacks of the theory of fluxions, using a "très-élégant" algebraic identity as his basis.

Residual Analysis versus Analytic Functions

BORGATO, Maria Teresa
2010

Abstract

In his introduction to Théorie des fonctions analytiques, Lagrange analyses the metaphysics of infinitesimal calculus and gives John Landen the credit for having worked out a purely analytical method, in which the finite differences of the variables substitute the infinitesimal differences, although he adds: "on doit convenir que cette manière de rendre le Calcul différentiel plus rigoureux dans ses principes lui fait perdre ses principaux avantages, la simplicité de la méthode et la facilité des opérations". Preceded in 1755 by Mathematical Locubrations, in which the method of fluxions is systematically applied to the solution of algebraic equations and the inverse calculus of fluents, in 1758 John Landen (1719-1790) published A Discourse concerning the Residual Analysis, an announcement of the treatise which was to appear in 1764: The Residual Analysis, a New Branch of the Algebraic Art. In this John Landen intended to release the method of fluxions from the principles derived from the doctrine of motion, and from a basis of pure algebra he would develop methods for immediate application to the main problems of analysis: maxima and minima of functions, curvature, quadrature and rectification of curves. Only finite increments are considered, which are equalized to zero only after having simplified the factor that in their ratio makes them null. An algebraic identity on the differences of rational powers plays a central role. While teaching at the military academy in Turin in the years 1756-59, Lagrange wrote a treatise of Cartesian geometry and differential calculus in which he accomplished a compromise between the Newton method of the “prime and ultimate ratios” and the Leibnitzian notation. Differential calculus and integral calculus were preceded by the algebraic calculus of finite differences from whose formulas those of differential calculus are obtained. The initial concordance then becomes an explicit influence in the basic inspiration of the theory of analytic functions, and even Lacroix was to give credit to Landen for having solved the drawbacks of the theory of fluxions, using a "très-élégant" algebraic identity as his basis.
2010
Lagrange; Landen; Residual analysis; Analytic functions
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11392/1401364
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