xx JOSIAH WILLARD GIBBS. 



Grassmann, the vector product satisfying the combinatorial law, but 

 yielding a vector instead of a magnitude of the second order. While 

 these departures from the systems mentioned testify to the great 

 ingenuity and originality of the author, and do not impair the utility 

 of the system as a tool for the use of students of physics, they never- 

 theless expose the discipline to the criticism of the pure algebraist. 

 Such objection falls to the ground, however, in the case of the theory 

 mentioned, for dyadics yield, for n = 3, a linear associative algebra of 

 nine units, namely nonions, the general nonion satisfying an identical 

 equation of the third degree, the Hamilton-Cayley equation. 



It is easy to make clear the precise point of view adopted by 

 Professor Gibbs in this matter. This is well expounded in his vice- 

 presidential address on multiple algebra, before the American Asso- 

 ciation for the Advancement of Science, in 1886, and also in his warm 

 defense of Grassmann's priority rights, as against Hamilton's, in his 

 article in Nature "Quaternions and the Ausdehnungslehre." He 

 points out that the key to matricular algebras is to be found in the 

 open (or indeterminate) product (i.e., a product in which no equations 

 subsist between the factors), and, after calling attention to the brief 

 development of this product in Grassmann's work of 1844, affirms 

 that Sylvester's assignment of the date 1858 to the " second birth of 

 Algebra" (this being the year of Cayley's Memoir on Matrices) must be 

 changed to 1844. Grassmann, however, ascribes very little importance 

 to the open product, regarding it as offering no useful applications. 

 On the contrary, Professor Gibbs assigns to it the first place in the 

 three kinds of multiplication considered in the Ausdehnungsfahre, 

 since from it may be derived the algebraic and the combinatorial 

 products, and shows in fact that both of them may be expressed in 

 terms of indeterminate products. Thus the multiplication rejected 

 by Grassmann becomes, from the standpoint of Professor Gibbs, the 

 key to all others. The originality of the latter's treatment of the 

 algebra of dyadics, as contrasted with the methods of other authors in 

 the allied theory of matrices, consists exactly in this, that Professor 

 Gibbs regards a matrix of order n as a multiple quantity in n 2 units, 

 each of which is an indeterminate product of two factors. On the 

 other hand, C. S. Peirce, who was the first to recognize (1870) the 

 quadrate linear associative algebras identical with matrices, uses for 

 the units a letter pair, but does not regard this combination as a 

 product. In addition, Professor Gibbs, following the spirit of 

 Grassmann's system, does not confine himself to one kind of multi- 

 plication of dyadics, as do Hamilton and Peirce, but considers two 

 sorts, both originating with Grassmann. Thus it may be said that 

 quadrate, or matricular algebras, are brought entirely within the 

 wonderful system expounded by Grassmann in 1844. 



