170 THE PRINCIPLES OF SCIENCE. [CHAP. 



It should be understood that this solution applies only 

 to the terms of the example quoted above, and not to the 

 general problem for which De Morgan intended it to serve 

 as an illustration. 



As a second instance, let us take the following ques- 

 tion : The whole number of voters in a borough is a ; 

 the number against whom objections have been lodged by 

 liberals is &; and the number against whom objections 

 have been lodged by conservatives is c; required the 

 number, if any, who have been objected to on both sides. 

 Taking 



A = voter, 



B = objected to by liberals, 

 C = objected to by conservatives, 



then we require the value of (ABC). Now the following 

 equation is identically true 



(ABC) = (AB) + (AC) + (Abe) - (A). (i) 



For if we develop all the terms on the second side we 

 obtain 



(ABC) = (ABC) + (ABc) + (ABC) + (A6C) + (A6c) 



- (ABC) - (ABc) - (A&C) - (A6c) ; 



and striking out the corresponding positive and negative 

 terms, we have left only (ABC) = (ABC). Since then 

 (i) is necessarily true, we have only to insert the known 

 values, and we have 



(ABC) = & + c - a + (Abe). 



Hence the number who have received objections from both 

 sides is equal to the excess, if any, of the whole number 

 of objections over the number of voters together with the 

 number of voters who have received no objection (Abe). 



The following problem illustrates the expression for 

 the common part of any three classes : The number of 

 paupers who are blind males, is equal to the excess, if 

 any, of the sum of the whole number of blind persons, 

 added to the whole number of male persons, added to the 

 number of those who being paupers are neither blind nor 

 males, above the sum of the whole number of paupers 

 added to the number of those who, not being paupers, 

 are blind, and to the number of those who, not being 

 paupers, are male. 



The reader is requested to prove the truth of the above 

 statement, (i) by his own unaided common sense; (2) by 



