ers 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 + cx^2 + dx^3 + ... 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 - x^2z) (1 - x^3z) ...
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 x^a, x^b, viz. x^a + x^b = x^(a+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