REMAINING LAWS OF NATURE. 169 



all objects that are divisible into four equal parts, and 

 all objects are either actually or ideally so divisible. 

 But the propositions which compose the science of 

 algebra are true, not of a particular number, but of 

 all numbers ; not of all things under the condition of 

 being divided in a particular way, but of all things 

 under the condition of being divided in any way of 

 being designated by a number at all. 



Since it is impossible for different numbers to have 

 any of their modes of formation completely in common, 

 it looks like a paradox to say, that all propositions 

 which can be made concerning numbers relate to their 

 modes of formation from other numbers, and yet that 

 there are propositions which are true of all numbers. 

 But this very paradox leads to the real principle of 

 generalization concerning the properties of numbers. 

 Two different numbers cannot be formed in the same 

 manner from the same numbers ; but they may be 

 formed in the same manner from different numbers; as 

 nine is formed from three by multiplying it into 

 itself, and sixteen is formed from four by the same 

 process. Thus there arises a classification of modes 

 of formation, or, in the language commonly used by 

 mathematicians, a classification of Functions. Any 

 number, considered as formed from any other number, 

 is called a function of it; and there are as many kinds 

 of functions as there are modes of formation. The 

 simple functions are by no means numerous, most 

 functions being formed by the combination of several 

 of the operations which form simple functions, or by 

 successive repetitions of some one of those operations. 

 The simple functions of any number x are all 

 reducible to the following forms : x + a, x-a, ax, 



, x n , a , , log. x (to the base a), and the same 



a 



