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COMBINATION—COMBINATORIAL ANALYSIS

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 and exhibit all the possible ways in which a given number of Historical Introduction. 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.