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 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 (1886).

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