REMAINING LAWS OF NATURE. 151 



learned it) by adding a single unit at a time : 5 + 1 - G, there 

 fore 5 + 1+1 = + 1 = 7: and again 2=1 + 1, therefore 5 + 2 

 = 5+1 + 1 = 7. 



G. Innumerable as are the true propositions which can 

 be formed concerning particular numbers, no adequate con 

 ception could be gained, from these alone, of the extent of the 

 truths composing the science of number. Such propositions 

 as we have spoken of are the least general of all numerical 

 truths. It is true that even these are coextensive with all 

 nature : the properties of the number four are true of 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 par 

 ticular 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 is a kind 

 of 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 num 

 bers. 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 classifica 

 tion 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 



