began his papers in the Cambridge and Dublin Mathematical Journal on the Calculus of Forms. After this, discoveries followed in rapid succession. At that time Cayley and Sylvester were both residents of London, and they stimulated each other by frequent oral communications. It has often been difficult to determine how much really belongs to each.
James Joseph Sylvester was born in London in 1814, and educated at St. Johns College, Cambridge. He came out Second Wrangler in 1837. His Jewish origin incapacitated him from taking a degree. In 1846 he became a student at the Inner Temple, and was called to the bar in 1850. He became professor of natural philosophy at University College, London; then, successively, professor of mathematics at the University of Virginia, at the Royal Military Academy in Woolwich, at the Johns Hopkins University in Baltimore, and is, since 1883, professor of geometry at Oxford. His first printed paper was on Fresnel's optic theory, 1837. Then followed his researches on invariants, the theory of equations, theory of partitions, multiple algebra, the theory of numbers, and other subjects mentioned elsewhere. About 1874 he took part in the development of the geometrical theory of link-work movements, originated by the beautiful discovery of A. Peaucellier, Capitaine du Génie à Nice (published in
Nouvelles Annales, 1864 and 1873), and made the subject of close study by A. B. Kempe. To Sylvester is ascribed the
general statement of the theory of contravariants, the discovery
of the partial differential equations satisfied by the invariants and covariants of binary quantics, and the subject
of mixed concomitants. In the American Journal of Mathematics are memoirs on binary and ternary quantics, elaborated partly with aid of F. Franklin, now professor at the Johns
Hopkins University. At Oxford, Sylvester has opened up a new subject, the theory of reciprocants, treating of the functions
of a dependent variable and the functions of its differential coefficients in regard to , which remain unaltered by the interchange of and . This theory is more general than one on differential invariants by Halphen (1878), and has
been developed further by J. Hammond of Oxford, McMahon
of Woolwich, A. R. Forsyth of Cambridge, and others. Sylvester
playfully lays claim to the appellation of the Mathematical Adam, for the many names he has introduced into mathematics. Thus the terms invariant, discriminant, Hessian,
Jacobian, are his.
The great theory of invariants, developed in England mainly by Cayley and Sylvester, came to be studied earnestly in Germany, France, and Italy. One of the earliest in the field was Siegfried Heinrich Aronhold (1819–1884), who demonstrated
the existence of invariants, and , of the ternary cubic. Hermite discovered evectants and the theorem of reciprocity named after him. Paul Gordan showed, with the aid of