A history of mathematics

Bibliographic data

Monograph

Persistent identifier:: 889759502

Author:: Cajori, Florian

Title:: A history of mathematics

Scope:: 1 Online-Ressource (VIII, 516 Seiten)

Edition title:: Second edition, revised and enlarged

Year of publication:: 1922

Place of publication:: New York, NY

Identifier (digital):: 889759502

Language:: English

Publisher of the original:: Macmillan

Usage licence:: Public Domain Mark 1.0

Publisher of the digital copy:: Technische Informationsbibliothek Hannover

Place of publication of the digital copy:: Hannover

Year of publication of the original:: 2017

Document type:: Monograph

Collection:: Mathematics

Chapter

Title:: EUROPE DURING THE SIXTEENTH, SEVENTEENTH AND EIGHTEENTH CENTURIES

Document type:: Monograph

Structure type:: Chapter

Chapter

Title:: Euler, Lagrange, and Laplace

Document type:: Monograph

Structure type:: Chapter

EULER, LAGRANGE AND LAPLACE 257 Foncenex in the Miscellanea Taurinensia for 1759, believed to have been written for Foncenex by Lagrange himself. 1 In 1810 he began a thorough revision of his Mécanique analytique, but he died before its completion. The Théorie des fonctions, the germ of which is found in a memoir of his of 1772, aimed to place the principles of the calculus upon a sound foundation by relieving the mind of the difficult conception of a limit. John Landen’s residual calculus, professing a similar object, was unknown to him. In a letter to L. Euler of Nov. 24, 1759, La grange says that he believed he had developed the true metaphysics of the calculus; at that time he seems to have been convinced that the use of infinitesimals was rigorous. He “used both the infinitesimal method and the method of derived functions side by side during his whole life” (Jourdain). Lagrange attempted to prove Taylor’s theorem (the power of which he was the first to point out) by simple algebra, and then to develop the entire calculus from that theorem. The principles of the calculus were in his day involved in philosophic difficulties of a serious nature. The infinitesimals of G. W. Leibniz had no satisfactory metaphysical basis. In the differential calculus of L. Euler they were treated as absolute zeros. In I. Newton’s limit ing ratio, the magnitudes of which it is the ratio cannot be found, for at the moment when they should be caught and equated, there is neither arc nor chord. The chord and arc were not taken by Newton as equal before vanishing, nor after vanishing, but when they vanish. “That method,” said Lagrange, “has the great inconvenience of con sidering quantities in the state in which they cease, so to speak, to be quantities; for though we can always well conceive the ratios of two quantities, as long as they remain finite, that ratio offers to the mind no clear and precise idea, as soon as its terms become both nothing at the same time.” D’Alembert’s method of limits was much the same as the method of prime and ultimate ratios. When Lagrange endeavored to free the calculus of its metaphysical difficulties, by resorting to common algebra, he avoided the whirlpool of Charybdis only to suffer wreck against the rocks of Scylla. The algebra of his day, as handed down to him by L. Euler, was founded on a false view of infinity. No rigorous theory of infinite series had then been established. Lagrange proposed to define the differential coefficient of f(x) with respect to x as the coefficient of h in the expansion of f(x+h) by Taylor’s theorem, and thus to avoid all reference to limits. But he used infinite series without ascertaining carefully that they were convergent, and his proof that f{x+h) can always be expanded in a series of ascending powers of h, labors under serious defects. Though Lagrange’s method of developing the calculus was at first greatly applauded, its defects were fatal, and to-day his “method of 1 Philip E. B. Jourdain in Proceed. 5th Intern. Congress, Cambridge, 1912, Cam bridge, 1913, Vol. II, p. 540.

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