of mathematical science: it was not
recognized that the theory of combinations is in reality a science by
itself, well worth studying for its own sake irrespective of
applications to other parts of analysis. There was a total absence of
orderly development, and until the first third of the 19th century had
passed, Euler's classical paper remained alike the chief result and the
only scientific method of combinatorial analysis.

In 1846 Karl G. J. Jacobi studied the partitions of numbers by means of
certain identities involving infinite series that are met with in the
theory of elliptic functions. The method employed is essentially that of
Euler. Interest in England was aroused, in the first instance, by
Augustus De Morgan in 1846, who, in a letter to Henry Warburton,
suggested that combinatorial analysis stood in great need of
development, and alluded to the theory of partitions. Warburton, to some
extent under the guidance of De Morgan, prosecuted researches by the aid
of a new instrument, viz. the theory of finite differences. This was a
distinct advance, and he was able to obtain expressions for the
coefficients in partition series in some of the simplest cases (_Trans.
Camb. Phil. Soc._, 1849). This paper inspired a valuable paper by Sir
John Herschel (_Phil. Trans._ 1850), who, by introducing the idea and
notation of the circulating function, was able to present results in
advance of those of Warburton. The new idea involved a calculus of the
imaginary roots of unity. Shortly afterwards, in 1855, the subject was
attacked simultaneously by Arthur Cayley and James Joseph Sylvester, and
their combined efforts resulted in the practical solution of the problem
that we have to-day. The former added the idea of the prime circulator,
and the latter applied Cauchy's theory of residues to the subject, and
invented the arithmetical entity termed a denumerant. The next distinct
advance was made by Sylvester, Fabian Franklin, William Pitt Durfee and
others, about the year 1882 (_Amer. Journ. Math._ vol. v.) by the
employment of a graphical method. The results obtained were not only
valuable in themselves, but also threw considerable light upon the
theory of algebraic series. So far it will be seen that researches had
for their object the discussion of the partition of numbers. Other
branches of combinatorial analysis were, from any general point of view,
absolutely neglected. In 1888 P. A. MacMahon investigated the general
prob