(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 Zürich, 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,
while the necessity for determining the convergence was generally lost sight of. Euler reached some very pretty results on infinite series, now well known, and also some very
absurd results, now quite forgotten. The faults of his time found their culmination in the Combinatorial School in Germany,
which has now passed into deserved oblivion. At the beginning of the period now under consideration, the doubtful, or plainly absurd, results obtained from infinite series stimulated profounder inquiries into the validity of operations with them. Their actual contents came to be the primary, form a secondary, consideration. The first important and strictly rigorous investigation of series was made by Gauss in connection