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 example; nonions is another. C. S. Peirce showed that of all linear associative algebras there are only three in which division is unambiguous. These are ordinary single algebra, ordinary double algebra, and quaternions, from which the imaginary scalar is excluded. He showed that his father's algebras are operational and matricular. Lectures on multiple algebra were delivered by J. J. Sylvester at the Johns Hopkins University, and published in various journals. They treat largely of the algebra of matrices. The theory of matrices was developed as early as 1858 by Cayley in an important memoir which, in the opinion of Sylvester, ushered in the reign of Algebra the Second. Clifford, Sylvester, H. Taber, C. H. Chapman, carried the investigations much further. The originator of matrices is really Hamilton, but his theory, published in his Lectures on Quaternions, is less general than that of Cayley. The latter makes no reference to Hamilton.

The theory of determinants[73] was studied by Hoëné Wronski in Italy and J. Binet in France; but they were forestalled by the great master of this subject, Cauchy. In a paper (Jour. de l'ecole Polyt., IX., 16) Cauchy developed several general theorems. He introduced the name determinant, a term previously used by Gauss in the functions considered by him. In 1826 Jacobi began using this calculus, and he gave brilliant proof of its power. In 1841 he wrote extended memoirs on determinants in Crelle's Journal, which rendered the theory easily accessible. In England the study of linear transformations of quantics gave a powerful 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,"