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 from elementary text-books would have been discarded over half a century ago. Cauchy was the first to publish a rigorous proof of Taylor's theorem. He greatly improved the exposition of fundamental principles of the differential calculus by his mode of considering limits and his new theory on the continuity of functions. The method of Cauchy and Duhamel was accepted with favour by Hoüel and others. In England special attention to the clear exposition of fundamental principles was given by De Morgan. Recent American treatises on the calculus introduce time as an independent variable, and the allied notions of velocity and acceleration—thus virtually returning to the method of fluxions.

Cauchy made some researches on the calculus of variations. This subject is now in its essential principles the same as when it came from the hands of Lagrange. Recent studies pertain to the variation of a double integral when the limits are also variable, and to variations of multiple integrals in general. Memoirs were published by Gauss in 1829, Poisson in 1831, and Ostrogradsky of St. Petersburg in 1834, without, however, determining in a general manner the number and form of the equations which must subsist at the limits in case of a double or triple integral. In 1837 Jacobi published a memoir, showing that the difficult integrations demanded by the discussion of the second variation, by which the existence of a maximum or minimum can be ascertained, are included in the integrations of the first variation, and thus are superfluous. This important theorem, presented with great brevity by Jacobi, was elucidated and extended by V. A. Lebesgue, C. E. Delaunay, Eisenlohr, S. Spitzer, Hesse, and Clebsch. An important memoir by Sarrus on the question of determining the limiting equations which must be combined with the indefinite equations in order to determine completely the maxima and minima of multiple integrals, was awarded a