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, Wilhelm Fiedler, P. A. McMahon, J. W. L. Glaisher of Cambridge, Emory McClintock of New York. McMahon discovered that the theory of semi-invariants is a part of that of symmetric functions. The modern higher algebra has reached out and indissolubly connected itself with several other branches of mathematics—geometry, calculus of variations, mechanics. Clebsch extended the theory of binary forms to ternary, and applied the results to geometry. Clebsch, Klein, Weierstrass, Burckhardt, and Bianchi have used the theory of invariants in hyperelliptic and Abelian functions.

In the theory of equations Lagrange, Argand, and Gauss furnished proof to the important theorem that every algebraic equation has a real or a complex root. Abel proved rigorously that the general algebraic equation of the fifth or of higher degrees cannot be solved by radicals (Crelle, I., 1826). A modification of Abel's proof was given by Wantzel. Before Abel, an Italian physician, Paolo Ruffini (1765–1822), had printed proofs of the insolvability, which were criticised by his countryman Malfatti. Though inconclusive, Ruffini's papers are remarkable as containing anticipations of Cauchy's theory of groups.[76] A transcendental solution of the quintic involving elliptic integrals was given by Hermite (Compt. Rend., 1858, 1865, 1866). After Hermite's first publication, Kronecker, in 1858, in a letter to Hermite, gave a second solution in which was obtained a simple resolvent of the sixth degree. Jerrard, in his Mathematical Researches (1832–1835), reduced the quintic to the trinomial form by an extension of the method of Tschirnhausen. This important reduction had been effected as early as 1786 by E. S. Bring, a Swede, and brought out in a publication of the University of Lund. Jerrard, like Tschirnhausen, believed that his method furnished a general algebraic solution of equations of any degree. In 1836 William R. Hamilton made a report on the validity of Jerrard's