Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/182

166 reservation that the addition both of vectors and of points had been given by earlier writers. In both systems as finally developed we have the linear vector function, the theory of which is identical with that of strains and rotations. In BAmilton's system we have also the linear quaternion function, and in Grassmann's the linear function appHed to the quantities of his algebra of points. This application gives those transformations in which projective properties are preserved, the doctrine of reciprocal figures or principle of duality, etc. (Grassmann's theory of the linear function is, indeed, broader than this, being coextensive with the theory of matrices; but we are here considering only the geometrical side of the theory.)

In his earliest writings on quaternions, Hamilton does not discuss the linear function. In his Lectures on Quaternions (1853), he treats of the inversion of the linear vector function, as also of the linear quaternion function, and shows how to find the latent roots of the vector function, with the corresponding axes for the case of real and unequal roots. He also gives a remarkable equation, the symbolic cubic, which the functional symbol must satisfy. This equation is a particular case of that which is given in Prof. Cayley's classical Memoir on the Theory of Matrices (1858), and which is called by Prof. Sylvester the Hamilton- Cayley equation. In his Elements of Quaternions (1866), Hamilton extends the symbolic equation to the quaternion function.

In Grassmann, although the linear function is mentioned in the first Ausdehnungslehre, we do not find so full a discussion of the subject until the second Ausdehnungslehre (1862), where he discusses the latent roots and axes, or what corresponds to axes in the general theory, the whole discussion relating to matrices of any order. The more difficult cases are included, as that of a strain in which all the roots are real, but there is only one axis or unchanged direction. On the formal side he shows how a linear function may be represented by a quotient or sum of quotients, and by a sum of products, Lückenausdruck.

More important, perhaps, than the question when this or that theorem was first published is the question where we first find those notions and notations which give the key to the algebra of linear functions, or the algebra of matrices, as it is now generally called. In vol. xxxi, p. 35, of Nature, Prof. Sylvester speaks of Cayley's "ever-memorable" Memoir on Matrices as constituting "a second birth of Algebra, its avatar in a new and glorified form," and refers to a passage in his Lectures on Universal Algebra, from which, I think, we are justified in inferring that this characterization of the memoir is largely due to the fact that it is there shown how matrices