has been translated into English by F. N. Cole of
the University of Michigan, who contributed to the theory. A simple group of 504 substitutions of nine letters, discovered by Cole, has been shown by E. H. Moore of the University of
Chicago to belong to a doubly-infinite system of simple groups.
The theory of substitutions has important applications in the theory of differential equations. Kronecker published, in 1882,
his Grundzüge einer Arithmetischen Theorie der Algebraischen Grössen.
Since Fourier and Budan, the solution of numerical equations has been advanced by W. G. Horner of Bath, who gave
an improved method of approximation (Philosophical Transactions, 1819). Jacques Charles François Sturm (1803–1855), a
native of Geneva, Switzerland, and the successor of Poisson
in the chair of mechanics at the Sorbonne, published in 1829 his celebrated theorem determining the number and situation of roots of an equation comprised between given limits. Sturm tells us that his theorem stared him in the face in the midst of some mechanical investigations connected with the motion of a compound pendulum.77 This theorem, and Horner's method, offer together sure and ready means of finding the real roots of a numerical equation.
The symmetric functions of the sums of powers of the roots
of an equation, studied by Newton and Waring, was considered
more recently by Gauss, Cayley, Sylvester, Brioschi. Cayley
gives rules for the "weight" and "order" of symmetric functions.
The theory of elimination was greatly advanced by Sylvester,
Cayley, Salmon, Jacobi, Hesse, Cauchy, Brioschi, and
Gordan. Sylvester gave the dialytic method (Philosophical
Magazine, 1840), and in 1852 established a theorem relating to the expression of an eliminant as a determinant. Cayley made a new statement of Bézout's method of elimination and
established a general theory of elimination (1852).