IV. 



ON MULTIPLE ALGEBRA. 



ADDRESS BEFORE THE SECTION OF MATHEMATICS AND ASTRONOMY OF THE AMERICAN 

 ASSOCIATION FOR THE ADVANCEMENT OF SCIENCE, BY THE VICE-PRESIDENT. 



[Proceedings of the American Association for the Advancement of Science, 

 vol. xxxv. pp. 37-66, 1886.] 



IT has been said that " the human mind has never invented a labor- 

 saving machine equal to algebra.'"* If this be true, it is but natural 

 and proper that an age like our own, characterized by the multi- 

 plication of labor-saving machinery, should be distinguished by an 

 unexampled development of this most refined and most beautiful of 

 machines. That such has been the case, none will question. The 

 improvement has been in every part. Even to enumerate the principal 

 lines of advance would be a task for any one ; for me an impossibility. 

 But if we should ask, in what direction the advance has been made 

 which is to characterize the development of algebra in our day, we 

 may, I think, point to that broadening of its field and methods which 

 gives us multiple algebra. 



Of the importance of this change in the conception of the office of 

 algebra, it is hardly necessary to speak : that it is really characteristic 

 of our time will be most evident if we go back some two or three- 

 score years, to the time when the seeds were sown which are now 

 yielding so abundant a harvest. The failure of Mobius, Hamilton, 

 Grassmann, Saint-Venant to make an immediate impression upon 

 the course of mathematical thought in any way commensurate with 

 the importance of their discoveries is the most conspicuous evidence 

 that the times were not ripe for the methods which they sought to 

 introduce. A satisfactory theory of the imaginary quantities of 

 ordinary algebra, which is essentially a simple case of multiple 

 algebra, with difficulty obtained recognition in the first third of this 

 century. We must observe that this double algebra, as it has been 

 called, was not sought for or invented; it forced itself, unbidden, 

 upon the attention of mathematicians, and with its rules already 

 formed. 



The Nation, vol. xxxiii, p. 237. 



