then he studied medicine, taking his degree in 1867, and setting up in practice at Pons in Charente-Inférieure. In 1881 he presented himself as a political candidate for Saintes, but was defeated. In 1885 he was elected to the senate by the department of Charente-Inférieure. He sat in the Democratic left, and was elected vice-president in 1893 and 1894. The reports which he drew up upon educational questions drew attention to him, and on the 3rd of November 1895 he entered the Bourgeois cabinet as minister of public instruction, resigning with his colleagues on the 21st of April following. He actively supported the Waldeck-Rousseau ministry, and upon its retirement in 1903 he was himself charged with the formation of a cabinet. In this he took the portfolio of the Interior, and the main energy of the government was devoted to the struggle with clericalism. The parties of the Left in the chamber, united upon this question in the Bloc republicain, supported Combes in his application of the law of 1901 on the religious associations, and voted the new bill on the congregations (1904), and under his guidance France took the first definite steps toward the separation of church and state. He was opposed with extreme violence by all the Conservative parties, who regarded the secularization of the schools as a persecution of religion. But his stubborn enforcement of the law won him the applause of the people, who called him familiarly le petit père. Finally the defection of the Radical and Socialist groups induced him to resign on the 17th of January 1905, although he had not met an adverse vote in the Chamber. His policy was still carried on; and when the law of the separation of church and state was passed, all the leaders of the Radical parties entertained him at a noteworthy banquet in which they openly recognized him as the real originator of the movement.
COMBINATION (Lat. combinare, to combine), a term meaning
an association or union of persons for the furtherance of a common
object, historically associated with agreements amongst workmen
for the purpose of raising their wages. Such a combination was
for a long time expressly prohibited by statute. See Trade Unions; also Conspiracy and Strikes and Lock Outs.
COMBINATORIAL ANALYSIS. The Combinatorial Analysis, as it was understood up to the end of the 18th century, was of limited scope and restricted application. P. Nicholson, in his Essays on the Combinatorial Analysis, published
in 1818, states that “the Combinatorial Analysis is a
branch of mathematics which teaches us to ascertain
Historical Introduction.
and exhibit all the possible ways in which a given number of
things may be associated and mixed together; so that we may be
certain that we have not missed any collection or arrangement of
these things that has not been enumerated.” Writers on the
subject seemed to recognize fully that it was in need of cultivation,
that it was of much service in facilitating algebraical
operations of all kinds, and that it was the fundamental method
of investigation in the theory of Probabilities. Some idea of its
scope may be gathered from a statement of the parts of algebra
to which it was commonly applied, viz., the expansion of a
multinomial, the product of two or more multinomials, the
quotient of one multinomial by another, the reversion and
conversion of series, the theory of indeterminate equations, &c.
Some of the elementary theorems and various particular problems
appear in the works of the earliest algebraists, but the true
pioneer of modern researches seems to have been Abraham
Demoivre, who first published in Phil. Trans. (1697) the law
of the general coefficient in the expansion of the series
a + bx + cx² + dx³ + . . . raised to any power. (See also Miscellanea Analytica,
bk. iv. chap. ii. prob. iv.) His work on Probabilities
would naturally lead him to consider questions of
this nature. An important work at the time it was published
was the De Partitione Numerorum of Leonhard
Euler, in which the consideration of the reciprocal of the
product (1 − xz) (1 − x²z) (1 − x³z) . . . establishes a fundamental
connexion between arithmetic and algebra, arithmetical addition
being made to depend upon algebraical multiplication, and a close
bond is secured between the theories of discontinuous and
continuous quantities. (Cf. Numbers, Partition of.) The
multiplication of the two powers xa, xb, viz. xa + xb = xa+b,
showed Euler that he could convert arithmetical addition into
algebraical multiplication, and in the paper referred to he gives
the complete formal solution of the main problems of the partition
of numbers. He did not obtain general expressions for the coefficients
which arose in the expansion of his generating functions,
but he gave the actual values to a high order of the coefficients
which arise from the generating functions corresponding to various
conditions of partitionment. Other writers who have contributed
to the solution of special problems are James Bernoulli, Ruggiero
Guiseppe Boscovich, Karl Friedrich Hindenburg (1741–1808),
William Emerson (1701–1782), Robert Woodhouse (1773–1827),
Thomas Simpson and Peter Barlow. Problems of combination
were generally undertaken as they became necessary for the
advancement of some particular part 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 problem of distribution, of which the partition of a number is a particular case. He introduced the method of symmetric functions and the method of differential operators, applying both methods to the two important subdivisions, the theory of composition and the theory of partition. He introduced the notion of the separation of a partition, and extended all the results so as to include multipartite as well as unipartite numbers. He showed how to introduce zero and negative numbers, unipartite and multipartite, into the general theory; he extended Sylvester’s graphical method to three dimensions; and finally, 1898, he invented the “Partition Analysis” and applied it to the solution of novel questions in arithmetic and algebra. An important paper by G. B. Mathews, which reduces the problem of compound partition to that of simple partition, should also be noticed. This is the problem which was known to Euler and his contemporaries as “The Problem of the Virgins,” or “the Rule of Ceres”; it is only now, nearly 200 years later, that it has been solved.
