326 MEMOIRS OF THE AMERICAN ACADEMY. 



The Signs for Multiplication. 



I shall adopt for the conception of multiplication the application of a relation, in 

 such a way that, for example, Zw shall denote whatever is lover of a woman. This 

 notation is the same as that used by Mr. De Morgan, although he appears not to 

 have had multiplication in his mind. s(m -|j- w) will, then, denote whatever is ser- 

 vant of anything of the class composed of men and women taken together. So 



that 



s(m -\j w) = sm -(5- sw 



(Z -fr «)w will denote whatever is lover or servant to a woman, and 



(I -\y s)w = Zw -jj sw . 



(s/)w will denote whatever stands to a woman in the relation of servant of a lover, 



and 



(sl)w = s(7w) 



Thus all the absolute conditions of multiplication are satisfied. 



The term "identical with — " is a unity for this multiplication. That is to say, 

 if we denote "identical with — " by / we have 



xi = x , 



whatever relative term x may be. For what is a lover of something identical with 

 anything, is the same as a lover of that thing. 



A conjugative term like giver naturally requires two correlates, one denoting the 

 thing given, the other the recipient of the gift. We must be able to distinguish, in 

 our notation, the giver of A to B from the giver to A of B, and, therefore, I suppose 



the signification of the letter equivalent to such 



distinguish th 



lates as first, second, third, etc., so that " giver of — to — " and " giver to — of — " 

 will be expressed by different letters. Let 3. denote the latter of these conjugative 

 terms. Then, the correlates or multiplicands of this multiplier cannot all stand di- 

 rectly after it, as is usual in multiplication, but may be ranged after it in regular 

 order, so that 



S X V 



will denote a giver to x of y. But according to the notation, x here multiplies y, so 

 that if we put for x owner (0), and for y horse (h), 



5 oh 



