vii i.] PRINCIPLES OF NUMBER. 165 



ac = bd. 



We might prove a like axiom concerning division in an 

 exactly similar manner. I might even extend the list of 

 axioms and say that " Equal powers of equal numbers are 

 equal." For certainly, whatever a x a x a may mean, it 

 is equal to a x a x a ; hence by our usual substitution it 

 is equal to b x b x b. The same will be true of roots of 

 numbers and l]a = ?Jb provided that the roots are so 

 taken that the root of a shall really be related to. a as 

 the root of b is to b. The ambiguity of meaning of an 

 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 prin- 

 ciple. Let Qa denote in the most general manner that we 

 do something with the quantity a ; then if a = b it follows 

 that 



Qa = Q&. 



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 conditions. 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 substi- 

 tution. 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 Snowdon ; therefore Ben Nevis is higher 

 than the Wrekin. But a little consideration discloses 

 sufficient reason for believing that even in such cases, 



