186 THE PRINCIPLES OF SCIENCE. 



provided that the same roots are taken, that is that the 

 root of a shall really be related to a as the root of I is 

 to b. The ambiguity of meaning of an operation thus fails 

 in any way to shake the universality of the principle. 



We may go further and assert that, not only the above 

 common relations, but all other known or conceivable 

 mathematical relations obey the same principle. Let Pa 

 denote in the most general manner that we do something 

 with the quantity a ; then if a = b it foUows that 



Pa = PZ>. 

 Let us make Pa, for instance, mean 



a 3 ~3a 2 +2a + 5; 



then it necessarily follows that this quantity is exactly 

 equal to b s 3 b 2 + 2 I + 5. 



The reader will also remember that one of the most 

 frequent operations in mathematical reasoning is to sub- 

 stitute for a quantity its equal, as known either by 

 assumed, natural, or self-evident condition. Whenever a 

 quantity appears twice over in a problem, we may apply 

 what we learn of its relations in one place to its relations 

 in the other. All reasoning in mathematics, as in other 

 branches of science, thus involves the principle of treating 

 equals equally, or similars similarly. In whatever way we 

 employ quantitative reasoning in the remaining parts of 

 this work, we never can desert the simple principle on 

 which we first set out. 



Reasoning by Inequalities. 



I have stated that all the processes of mathematical 

 reasoning may be deduced from the principle of substitution. 

 Exceptions to this assertion may seem to exist in the use 

 of inequalities. The greater of a greater is undoubtedly a 

 greater, and what is less than a less is certainly less. 

 Snowdon is higher than the Wrekin, and Ben Nevis than 



