PRINCIPLES OF NUlfBER. 185 



our principle of substitution always holds true. We may 

 say in the most general manner that In whatever relation 

 one quantity stands to another, it stands in the 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 e . 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. 



In 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, 

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



ac = ad, 

 and a second similar substitution gives us 



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 number 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 



axaxa = bxbxb, 

 or a 3 = b 3 . 



The truth will hold of roots, that is to say, 



y^"= */ 



e Todhunter's ' Algebra,' 3rd ed. p. 40. 



