2O MONISM IN ARITHMETIC. 



tion, the operation of third degree, the two laws are inapplicable, 

 and the result of their inapplicability is that operations of a still 

 higher degree than the third form no possible advancement of pure 

 arithmetic. The product of b factors a is not equal to the product 

 of a factors b; that is, the law of commutation does not hold. The 

 only two different integers for which a to the b th power is equal to b 

 to the a th power are 2 and 4, for 2 to the 4"' power is 16, and 4 to 

 the second power also is 16. So, too, the law of association as a 

 general rule does not hold. For it is hardly the same thing whether 

 we take the (b c j h power of a or the c th power of a b . 



From the definition of involution follow the usual rules for reck- 

 oning with powers, of which we shall only mention one, namely, 

 that the (b cj h power of a is equal to the result of the division of 

 a to the b th power by a to the c* h power. If we put here c equal to 

 b, we are obliged, by the principle of no exception, to put a to the 

 O' ;z power equal to i; a new result not contained in the original no- 

 tion of involution, for that implied necessarily that the exponent 

 should be a result of counting. Again, if we make b smaller than c 

 we obtain a negative exponent, which we should not know how to 

 dispose of if we did not follow our monistic law of arithmetic. Ac- 

 cording to the latter, a to the (b c] th power must still remain equal 

 to a b divided by a c even when b is smaller than c. Whence follows 

 that a to the minus d th power is equal to i divided by a to the 

 d th power, or to take specific numbers, that 3 to the minus 2"^ power 

 is equal to \. 



At this point, perhaps, the reader will inquire what a raised'to a 

 fractional power is. But this can be explained only when we have 

 discussed the inverse processes of involution, to which we now pass. 



If a 6 = f, we may ask two questions : first, what the base is 

 which raised to the b th power gives c\ the second, what the exponent 

 of the power is to which a must be raised to produce c. In the first 

 case we seek the base, and term the operation which yields this re- 

 sult evolution; in the second case we seek the exponent and call the 

 operation which yields this exponent, the finding of the logarithm. 

 In the first case, we write v / c = a (which we read, the b th root of c is 

 equal to a), and call c the radicand, b the exponent of the root, and a 



