MONISM IN ARITHMETIC. 21 



the root. In the second case, we write log a ^ = (which we read, the 

 logarithm of c to the base a is equal to b\ and call c the logarithmand 

 or number, a the base of the logarithm, and b the logarithm. 



While, owing to the validity of the law of commutation in addi- 

 tion and multiplication, the two inverse processes of those opera- 

 ions are identical so far as computation is concerned, here in the 

 case of involution the two inverse operations are in this regard es- 

 sentially different, for in this case the law of commutation does not 

 hold. 



From the definitional formulae for evolution and the finding of 

 logarithms, namely, 



(j/'O* = c, and (a) lo **' = c, 



follow, by the application of the laws of involution, the rules for 

 computation with roots and logarithms. These rules we pass over 

 here, only remarking, first, that for the present I/ 7 ' c has meaning 

 only when c is the b th power of some number already defined ; and, 

 secondly, that for the present also log rt <r has meaning only when c 

 can be produced by raising the number a to some power which is a 

 number already defined. In the phrase "has only meaning for the 

 present" is contained a possibility of new extensions of the domain 

 of number. But before we pass to those extensions we shall first 

 make use of the idea of evolution just defined to extend the notion 

 of power also to cases in which the exponent is a fractional number. 



According to the original definition of involution, a b was mean- 

 ingless except where b was a result of counting. But afterwards, 

 even powers which had for their exponents zero or a negative integer 

 could be invested with meaning. Now we have to consider the 

 arithmetical combination "a raised to the fractional power A" The 

 principle of no exception compels us to give to the arithmetical com- 

 bination "a to the -*- th power" a significance such that all the rules 

 of computation will hold with respect to it. Now, one rule that 

 holds is, that the ;;/'* power of the n th power of a is equal to the 

 (;X)' A power of a. Consequently, the q th power of a raised to the 

 '* power must be equal to a raised to a power whose exponent U 

 equal to p times q. But the last-mentioned product gives, according 

 to the definition of division, the number p. Consequently the sym- 



