166 QUATERNIONS AND THE AUSDEHNUNGSLEHRE. 



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 Hamilton's system we have also the linear quaternion 

 function, and in Grassmann's the linear function applied 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. (Grass- 

 mann'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 repre- 

 sented by a quotient or sum of quotients, and by a sum of products, 

 Luckenausdruck. 



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 



