PRINCIPLES OF NUMBER. 187 



Snowdon ; therefore Ben Nevis is higher than the Wrekin. 

 But a little consideration discloses much reason for be 

 lieving that even in such cases, where equality does not 

 apparently enter, the force of the reasoning entirely 

 depends upon underlying and implied equalities. 



In the first place, two statements of mere difference do 

 not give any ground of inference. We learn nothing 

 concerning the comparative heights of St. Paul s and 

 Westminster Abbey from the assertions that they both 

 differ in height from St. Peter s at Rome. Thus we need 

 something more than mere inequality ; we require one 

 identity in addition, namely the identity in direction of 

 the two differences. Thus we cannot employ inequalities 

 in the simple way in which we do equalities, and, when 

 we try to express exactly what other conditions are 

 requisite, we shall find ourselves lapsing into the use of 

 equalities or identities. 



In the second place, every argument by inequalities may 

 be represented with at least equal clearness and force in 

 the form of equalities. Thus we clearly express that a 

 is greater than b by the equation 



a = b + p, (i) 



where p is an intrinsically positive quantity, denoting the 

 difference of a and b. Similarly we express that b is 

 greater than c by the equation 



b = c + q, (2) 



and substituting for b in (i) its value in (2) we have 



a = c + q + p. (3) 



Now as p and q are both positive, it follows that a is 

 greater than c, and we have the exact amount of excess 

 specified. It will be easily seen that the reasoning con 

 cerning that which is less than a less will result in an 



equation of the form 



c a q p. 

 Every argument by inequalities may then be thrown 



