MULTIPLE ALGEBRA. 97 



In the language of multiple algebra 9 is called the product of 

 and <3>. It is of course interesting to see how it is derived from 

 the latter, and it is little more than a schoolboy's exercise to determine 

 this. Now this matrix has the property that its determinant 

 is equal to the products of the determinants of *Sf and $. And this 

 property is all that is generally stated in the books, and the funda- 

 mental property, which is all that gives the subject its interest, that 

 is itself the product of and 3? in the language of multiple algebra, 

 i.e., that operating by 9 is equivalent to operating successively by 

 $ and , is generally omitted. The chapter on this subject, in most 

 treatises which I have seen, reads very like the play of Hamlet with 

 Hamlet's part left out. 



And what is the cause of this omission ? Certainly not ignorance 

 of the property in question. The fact that it is occasionally 

 given would be a sufficient bar to this answer. It is because 

 the author fails to see that his real subject is matrices and not 

 determinants. Of course, in a certain sense, the author has a right 

 to choose his subject. But this does not mean that the choice 

 is unimportant, or that it should be determined by chance or by 

 caprice. The problem well put is half solved, as we all know. 

 If one chooses the subject ill, it will develop itself in a cramped 

 manner. 



But the case is really much worse than I have stated it. Not only 

 is the true significance of the formation of 9 from ^ and f> not given, 

 but the student is often not taught to form the matrix which is the 

 product of "SF and <, but one which is the product of one of these 

 matrices and the conjugate of the other. Thus the proposition which 

 is proved loses all its simplicity and significance, and must be recast 

 before the instructor can explain ' its true bearings to the student. 

 This fault has been denounced by Sylvester, and if anyone thinks 

 I make too much of the standpoint from which the subject is viewed, 

 I will refer him to the opening paragraphs of the "Lectures on 

 Universal Algebra " in the sixth volume of the American Journal of 

 Mathematics, where, with a wealth of illustration and an energy 

 of diction which I cannot emulate, the most eloquent of mathe- 

 maticians expresses his sense of the importance of the substitution 

 of the idea of the matrix for that of the determinant. If then so 

 important, why was the idea of the matrix let slip ? Of course the 

 writers on this subject had it to commence with. One cannot even 

 define a determinant without the idea of a matrix. The simple fact is 

 that in general the writers on this subject have especially developed 

 those ideas which are naturally expressed in simple algebra, and have 

 postponed or slurred over or omitted altogether those ideas which 

 find their natural expression in multiple algebra. But in this subject 



G. II. G 



