166 THE PRINCIPLES OF SCIENCE. [CHAP. 



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 Ahbey from the assertions that they both 

 differ in height from St. Peter's at Eome. We need some- 

 thing more than 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 what other conditions are requisite, we find 

 ourselves lapsing into the use of equalities or identities. 



In the second place, every argument by inequalities 

 may be represented in the form of equalities. We 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 r - s. 



Every argument by inequalities may then be thrown 

 into the form of an equality ; but the converse is not true. 

 We cannot possibly prove that two quantities are equal 

 by merely asserting that they are both greater or both less 

 than another quantity. From e > f and g >/, or e <f 

 and g < /, we can infer no relation between e and y. And 

 if the reader take the equations x = y = 3 and attempt to 

 prove that therefore x = 3, by throwing them into in- 

 equalities, he will find it impossible to do so. 



From these considerations I gather that reasoning in 

 arithmetic or algebra by so-called inequalities, is only an 

 imperfectly expressed reasoning by equalities, and when 



