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 prize by the French Academy in 1845, honourable mention being made of a paper by Delaunay. Sarrus's method was simplified by Cauchy. In 1852 G. Mainardi attempted to exhibit a new method of discriminating maxima and minima, and extended Jacobi's theorem to double integrals. Mainardi and F. Brioschi showed the value of determinants in exhibiting the terms of the second variation. In 1861 Isaac Todhunter (1820-1884) of St. John's College, Cambridge, published his valuable work on the History of the Progress of the Calculus of Variations, which contains researches of his own. In 1866 he published a most important research, developing the theory of discontinuous solutions (discussed in particular cases by Legendre), and doing for this subject what Sarrus had done for multiple integrals.

The following are the more important authors of systematic treatises on the calculus of variations, and the dates of publication: Robert Woodhouse, Fellow of Caius College, Cambridge, 1810; Richard Abbatt in London, 1837; John Hewitt Jellett (1817-1888), once Provost of Trinity College, Dublin, 1850; G. W. Strauch in Zurich, 1849; Moigno and Lindelöf, 1861; Lewis Buffett Carll of Flushing in New York, 1881.

The lectures on definite integrals, delivered by Dirichlet in 1858, have been elaborated into a standard work by G. F. Meyer. The subject has been treated most exhaustively by D. Bierens de Haan of Leiden in his Exposé de la théorie des intégrals définies, Amsterdam, 1862.

The history of infinite series illustrates vividly the salient feature of the new era which analysis entered upon during the first quarter of this century. Newton and Leibniz felt the necessity of inquiring into the convergence of infinite series, but they had no proper criteria, excepting the test advanced by Leibniz for alternating series. By Euler and his contemporaries the formal treatment of series was greatly extended,