184 THE PRINCIPLES OF SCIENCE. 



be that ' Two things of which one is equal and the other 

 unequal to a third common thing, are unequal to each 

 other/ An equality and inequality, in short, may give an 

 inequality, and this is equally true with the first axiom of 

 all kinds of quantity. If Venus, for instance, agrees with 

 Mars in density, but Mars differs from Jupiter, then Venus 

 differs from Jupiter. A third axiom must exist to the 

 effect that 'Things unequal to the same thing may or 

 may not be equal to each other/ Two inequalities give 

 no ground of inference whatever. If we only know, for 

 instance, that Mercury and Jupiter differ in density from 

 Mars, we cannot say whether or not they agree between 

 themselves. As a fact they do not agree ; but Venus and 

 Mars on the other hand both differ from Jupiter and yet 

 closely agree with each other. The force of the axioms 

 can be most clearly illustrated by drawing lines d . 



The general conclusion must be then 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 the 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 

 d 'Elementary Lessons in Logic' (Macmillan), p. 123. 



