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 Berlin, Klein of Göttingen, M. Nöther of Erlangen, C. Hermite of Paris, A. Capelli of Naples, L. Sylow of Friedrichshald, E. Netto of Giessen. Netto's book, the Substitutionstheorie, 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 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