Algebra

are his researches on Linear Associative Algebra. The first of

several papers thereon was read at the first meeting of the American Association for the Advancement of Science in 1864. Lithographed copies of a memoir were distributed among friends in 1870, but so small seemed to be the interest taken in this subject that the memoir was not printed until 1881 (Am. Jour. Math., Vol. IV., No. 2). Peirce works out the multiplication tables, first of single algebras, then of double algebras, and so on up to sextuple, making in all 162 algebras, which he shows to be possible on the consideration of symbols $A$ , $B$ , etc., which are linear functions of a determinate number of letters or units $i$ , $j$ , $k$ , $l$ , etc., with coefficients which are ordinary analytical magnitudes, real or imaginary,–-the letters $i$ , $j$ , etc., being such that every binary combination $i^{2}$ , $i j$ , $j i$ , etc., is equal to a linear function of the letters, but under the restriction of satisfying the associative law.56 Charles S. Peirce, a son of Benjamin Peirce, and one of the foremost writers on mathematical logic, showed that these algebras were all defective

forms of quadrate algebras which he had previously discovered by logical analysis, and for which he had devised a simple notation. Of these quadrate algebras quaternions is a simple

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

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