MULTIPLE ALGEBKA. 95 



combinatorial multiplication. His principal application was to the 

 theory of elimination. In this application, as in the law of multi- 

 plication, he had been anticipated by Grassmann. 



We come next to Cayley's celebrated Memoir on the Theory of 

 Matrices* in 1858, of which Sylvester has said that it seems to him 

 to have ushered in the reign of Algebra the Second, t I quote this 

 dictum of a master as showing his opinion of the importance of the 

 subject and of the memoir. But the foundations of the theory of 

 matrices, regarded as multiple quantities, seem to me to have been 

 already laid in the Ausdehnungslehre of 1844. To Grassmann's 

 treatment of this subject we shall recur later. 



After the Ausdehmingslehre of 1862, already mentioned, we come 

 to Hankel's Vorlesungen iiber die complexen Zahlen, 1867. Under 

 this title the author treats of the imaginary quantities of ordinary 

 algebra, of what he calls alternirende Zahlen, and of quaternions. 

 These alternate numbers, like Cauchy's clefs, are quantities subject 

 to Grassmann's law of combinatorial multiplication. This treatise, 

 published twenty- three years after the first Ausdehnungslehre, marks 

 the first impression which we can discover of Grassmann's ideas upon 

 the course of mathematical thought. The transcendent importance 

 of these ideas was fully appreciated by the author, whose very able 

 work seems to have had considerable influence in calling the attention 

 of mathematicians to the subject. 



In 1870, Professor Benjamin Peirce published his Linear Associative 

 Algebra, subsequently developed and enriched by his son, Professor 

 C. S. Peirce. The fact that the edition was lithographed seems to 

 indicate that even at this late date a work of this kind could only 

 be regarded as addressed to a limited number of readers. But the 

 increasing interest in such subjects is shown by the republication of 

 this memoir in 1881,! as by that of the first Ausdehnungslehre in 

 1878. 



The article on quaternions which has just appeared in the Encyclo- 

 pcedia Britannica mentions twelve treatises, including second editions 

 and translations, besides the original treatises of Hamilton. That all 

 the twelve are later than 1861 and all but two later than 1872 shows 

 the rapid increase of interest in this subject in the last years. 



Finally, we arrive at the Lectures on the Principles of Universal 

 Algebra by the distinguished foreigner whose sojourn among us has 

 given such an impulse to mathematical study in this country. The 

 publication of these lectures, commenced in 1884 in the American 

 Journal of Mathematics, has not as yet been completed, a want but 

 imperfectly supplied by the author's somewhat desultory publication 



* Phil. Trans., vol. cxlviii. turner. Journ. Math., vol. vi, p. 271. 



$Amer. Journ. Math., vol. iv. 



