nants generally, and who applied it to the theory of orthogonal substitution. Cayley set himself the problem to determine a priori what functions of the coefficients of a given equation possess this property of invariance, and found, to begin with, in 1845, that the so-called "hyper-determinants " possessed it. Boole made a number of additional discoveries. Then Sylvester 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 linkwork 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 dis

covery 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 y and the functions of its differential coefficients in regard to x, which remain unaltered by the interchange of x and y. 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, S and T, of the ternary cubic. Hermite discovered evectants and the theorem of reciprocity named after him. Paul Gordan showed, with the aid of 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. Chris

toffel, 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.

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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. 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

method, and showed that by his process the quintic could be transformed to any one of the four trinomial forms. Hamilton defined the limits of its applicability to higher equations. Sylvester investigated this question, What is the lowest degree an equation can have in order that it may admit of being deprived of i consecutive terms by aid of equations not higher than ith degree. He carried the investigation as far as i = 8, and was led to a series of numbers which he named "Hamilton's numbers." A transformation of equal importance to Jerrard's is that of Sylvester, who expressed the quintic as the sum of three fifth-powers. The covariants and invariants of higher equations have been studied much in recent years.

Abel's proof that higher equations cannot always be solved algebraically led to the inquiry as to what equations of a given degree can be solved by radicals. Such equations are the ones discussed by Gauss in considering the division of the circle. Abel advanced one step further by proving that an irreducible equation can always be solved in radicals, if, of two of its roots, the one can be expressed rationally in terms of the other, provided that the degree of the equation is prime; if it is not prime, then the solution depends upon that of equations of lower degree. Through geometrical considerations, Hesse came upon algebraically solvable equations of the ninth degree, not included in the previous groups. The subject was powerfully advanced in Paris by the youthful Evariste Galois (born, 1811; killed in a duel, 1832), who introduced the notion of a group of substitutions. To him are due also some valuable results in relation to another set of equations, presenting themselves in the theory of elliptic functions, viz. the modular equations. Galois's labours gave birth to the important theory of substitutions, which has been greatly advanced by C. Jordan of Paris, J. A. Serret (18191885) of the Sorbonne in Paris, L. Kronecker (1823-1891) of

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 E. H. Moor 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." 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