XVI. On the Foundation of Algebra, No. II. By Augustus De Morgan, 

 F.R.A.S., F.C.P.S.; of Trinity College; Professor of Mathematics 

 in University College, London. 



[Read November 29, 1841.] 



In presenting to the Society a continuation of the Paper on the 

 Foundation of Algebra, printed in Vol. vn, p. 173, I wish to make 

 the principal point of the new communication, which is the filling up 

 of an unfinished difficulty of the old one, subservient to such a view 

 of the transition from semi-logical to logical algebra as may perhaps 

 be useful to any one who may hereafter have to deal with an unexplained 

 result. By the semi-logical algebra I mean the ordinary science, in 

 which the explanations are insufficient to include J- 1 ; and in which 

 therefore the results, though always intelligible when J- 1 disappears, 

 can only be considered as demonstrated upon the assumption that the 

 symbolical laws of algebra must in some, though an unknown, manner, 

 admit of a wider explanation. 



The first step to logical algebra is the separation of the rules of the 

 ordinary science from its principles, or rather of its laws of operation 

 from the explanation of the symbols operated upon or with. As far 

 as I can see (and I believe no writer has professed to throw together 

 in one place every thing that is essential to algebraical process) the laws 

 of operation are as follow : 



1. The literal symbols, a, b, c, &c. have no necessary relation except 

 this, that whatever any one of them may mean in any one part of a 

 process, it means the same in every other part of the same process. 



HH2 



