combinatorial multiplication. His principal application was to the theory of elimination. In this application, as in the law of multiplication, he had been anticipated by Grassmann.
We come next to Cayley's celebrated Memoir on the Theory of Matrices[1] in 1858, of which Sylvester has said that it seems to him to have ushered in the reign of Algebra the Second.[2] 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 Ausdehnungslehre of 1862, already mentioned, we come to Hankel's Vorlesungen über 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,[3] as by that of the first Ausdehnungslehre in 1878.
The article on quaternions which has just appeared in the Encyclopædia 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 countiy. The publication of these lectures, commenced in 1884 in the American Journal of Mathemathics, has not as yet been completed,—a want but imperfectly supplied by the author's somewhat desultory publication
