impulse. Cayley developed skew-determinants and Pfaffians, and introduced the use of determinant brackets, or the familiar pair of upright lines. More recent researches on determinants appertain to special forms. "Continuants" are due to Sylvester; "alternants," originated by Cauchy, have been developed by Jacobi, N. Trudi,
H. Nägelbach, and G. Garbieri; "axisymmetric determinants,"
first used by Jacobi, have been studied by V. A. Lebesgue,
Sylvester, and Hesse; "circulants" are due to E. Catalan of
Liège, W. Spottiswoode (1825–1883), J. W. L. Glaisher, and
R. F. Scott; for "centro-symmetric determinants" we are
indebted to G. Zehfuss. E. B. Christoffel of Strassburg and
G. Frobenius discovered the properties of "Wronskians," first
used by Wronski. V. Nachreiner and S. Günther, both of
Munich, pointed out relations between determinants and continued fractions; Scott uses Hankel's alternate numbers in his
treatise. Text-books on determinants were written by Spottiswoode (1851), Brioschi (1854), Baltzer (1857), Günther
(1875), Dostor (1877), Scott (1880), Muir (1882), Hanus
Modern higher algebra is especially occupied with the theory of linear transformations. Its development is mainly the work of Cayley and Sylvester.
Arthur Cayley, born at Richmond, in Surrey, in 1821, was educated at Trinity College, Cambridge.74 He came out Senior Wrangler in 1842. He then devoted some years to the study and practice of law. On the foundation of the Sadlerian professorship at Cambridge, he accepted the offer of that chair, thus giving up a profession promising wealth for a very modest provision, but which would enable him to give all his time to mathematics. Cayley began his mathematical publications in the Cambridge Mathematical Journal while he was still an undergraduate. Some of his most brilliant discoveries were made during the time of his legal practice. There is hardly any subject in pure mathematics which the genius of Cayley has not enriched, but most important is his creation of a new branch of analysis by his theory of invariants. Germs of the principle of invariants are found in the writings of Lagrange, Gauss, and particularly of Boole, who
showed, in 1841, that invariance is a property of discriminants
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