22 MONISM IN ARITHMETIC. 



bol a to the -~' A power is so constituted that its q th power is equal to 

 a to the/"' power; i. e., it is equal to the q th root of a p . Similarly, 

 we find that the symbol "a to the minus * th power" must be put 

 equal to i divided by the q th root of a to the p th power, if we are to 

 reckon with this symbol as we do with real powers. Again, just as 

 a to the b th power is invested with meaning when b is a fractional 

 number, so some meaning harmonious with the principle of no ex- 

 ception must be imparted to the b th root of c where b is a positive or 

 negative fractional number. For example, the three-fourths"' root 

 of 8 is equal to 8 to the |- power, that is, to the cube root of 8 to the 

 4/ A power, or 16. 



The principle underlying arithmetic now also compels us to 

 give to the symbol the "b th root of <r" a meaning when c is not the 

 b th power of any number yet defined. First, let c be any positive 

 integer or fraction. Then always to be able to. reckon with the 

 b' h root of c in the same way that we do with extractible roots, we 

 must agree always to put the b th power of the b th root of c equal to 

 c for example, (V 3) 2 always exactly equal to 3. A careful inspec- 

 tion of the new symbols, which we will also call numbers, shows, that 

 though no one of them is exactly equal to a number hitherto defined, 

 yet by a certain extension of the notions greater and less, two num- 

 bers of the character of numbers already defined may be found for 

 each such new number, such that the new number is greater than the 

 one and less than the other of the two, and that further, these two 

 numbers may be made to differ from each other by as small a quan- 

 tity as we please. For example, 



3 = m = 2 ry < 3 < 3| = -y- = (f)'. 



The number 3, as we see, is here included between two limits which 

 are the third powers of two numbers ^ and ^ whose difference is y 1 ^. 

 We could also have arranged it so that the difference should be 

 equal to T J^, or to any specified number, however small. Now, in- 

 stead of putting the symbol " less than" between (J) 3 and 3, and 

 between 3 and (f ) 3 , let us put it between their third roots ; for ex- 

 ample, let us say : 



I < ^3~< f , meaning by this that (f ) 3 < 3 < (f) 3 . 

 In this sense we may say that the new numbers always lie between 



