194: Josiah Willard Gibhs. 



The theory of dyadics * as developed in the vector analysis 

 of 1884 must be regarded as the most important published 

 contribution of Professor Gibbs to pure mathematics. For 

 the vector analysis as an algebra does not fulfill the definition 

 of the linear associative algebras of Benjamin Peirce, since 

 the scalar product of vectors lies outside the vector domain ; 

 nor is it a geometrical analysis in the sense of 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 nevertheless 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 ^i = 3, a linear associative algebra of nine 

 units, namely nonions, the general nonion satisfying an identi- 

 cal 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 Association 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, 

 " Quarternions 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 equa- 

 tions subsist between the factors) and, after calling attention 

 to the brief development of this product in Grassmann's work 

 of 181:4, atiirms that Sylvester's assignment of the date 1858 

 to the " second birth of Algebra " (this being the year of Cay- 

 ley's Memoir on Matrices) must be changed to 1844. Grass- 

 mann, however, ascribes very little importance to the open 

 product, regarding it as oifering no useful applications. On 

 the contrary. Professor Gibbs assigns to it the first place in the 

 three kinds of multiplication considered in the Ausdehnungs- 

 lehre, 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 7^" 



* For the following account of the mathematical relations of this theory the 

 writer is indebted to Professor Percey F. Smith. 



