164 THE PRINCIPLES OF SCIENCE. [CHAP. 



The general conclusion then must be that where there 

 is equality there may be inference, but where there is not 

 equality there cannot be inference. A plain induction 

 will lead us to believe that equality is the condition of 

 inference concerning quantity. All the three axioms may 

 in fact be summed up in one, to the effect, that "in 

 whatever relation one quantity stands to another, it stands 

 in the same relation to tlic equal of that other" 



The active power is always the substitution of equals, 

 and it is' an accident that in a pair of equalities we can 

 make the substitution in two ways. From a = b = c we 

 can infer a = c, either by substituting in a = b the value 

 of b as given in b = c, or else by substituting in b = c the 

 value of b as given in a = b. In a = b ~ d we can make 

 but the one substitution of a for b. In e ~ f ~ g we can 

 make no substitution and get no inference. 



In mathematics the relations in which terms may stand 

 to each other are far more varied than in pure logic, yet 

 our principle of substitution always holds true. We may 

 say in the most general manner that In ^vhatever relation 

 one quantity stands to another, it stands in tlic same relation 

 to the equal of that other. In this axiom we sum up a 

 number of axioms which have been stated in more or less 

 detail by algebraists. Thus, " If equal quantities be added 

 to equal quantities, the sums will be equal." To explain 

 this, let 



a = b, c d. 



Now a + c, whatever it means, must be identical with 

 itself, so that 



a + c = a + c. 



[11 one side of this equation substitute for the quantities 

 their equivalents, and we have the axiom proved 



a + c = b + d. 



The similar axiom concerning subtraction is equally evi- 

 dent, for whatever a c may mean it is equal to a c, 

 and therefore by substitution to b d. Again, " if equal 

 quantities be multiplied by the same or equal quantities, 

 the products will be equal." For evidently 



ac = ac, 

 and if for c in one side we substitute its equal d, we have 



ac = ad, 

 and a second similar substitution gives us 



