symbolic methods, that the number of distinct forms for a binary quantic is finite. Clebsch proved this to be true for
quantics with any number of variables. A very much simpler proof of this was given in 1891, by David Hilbert of Königsberg.
In Italy, F. Brioschi of Milan and Faà de Bruno
(1825–1888) contributed to the theory of invariants, the latter writing a text-book on binary forms, which ranks by the side of Salmon's treatise and those of Clebsch and Gordan. Among other writers on invariants are E. B. Christoffel,
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