432 INDUCTION. 



law, or of other laws capable of being deduced from it. This is our war- 

 rant for all calculations. We believe that five and two are equal to seven, 

 on the evidence of this inductive law, combined with the definitions of those 

 numbers. We arrive at that conclusion (as all know who remember how 

 they first learned it) by adding a single unit at a time : 5 + 1 = 6, therefore 

 6 + 1 + 1 = 6 + 1=7; and again 2 = 1 + 1, therefore 6 + 2 = 5 + 1 + 1 = 7. 



§ 6. Innumerable as are the true propositions which can be formed con- 

 cerning particular numbers, no adequate conception 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 numer- 

 ical truths. It is true that even these are co-extensive 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 particular number, but of all numbers ; not of all things under the condi- 

 tion of being divided in a particular way, but of all things under the condi- 

 tion 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 numbers. Two different 

 numbers can not 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 form- 

 ed 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 sever- 

 al of the operations which form simple functions, or by successive repeti- 

 tions of some one of those operations. The simple functions of any num- 



^ 



ber X are all reducible to the following forms: cc+a, x—a, ax, -, x", \^x, 



log. X (to the base a), and the same expressions varied by putting x for a 

 and a for x, wherever that substitution would alter the value : to which, 

 perhaps, ought to be added sin x, and arc (sin=ic). All other functions 

 of X are formed by putting some one or more of the simple functions in 

 the place of x or a, and subjecting them to the same elementary operations. 



In order to carry on general reasonings on the subject of Functions, we 

 require a nomenclature enabling us to express any two numbers by names 

 which, without specifying what particular numbers they are, shall show 

 what function each is of the other ; or, in other words, shall put in evi- 

 dence their mode of formation from one another. The system of general 

 language called algebraical notation does this. The expressions a and 

 a' -{-3a denote, the one any number, the other the number formed from it 

 in a particular manner. The expressions a, b, n, and («+&)", denote any 

 three numbers, and a fourth which is formed from them in a certain mode. 



The following may be stated as the general problem of the algebraical 

 calculus : F being a certain function of a given number, to find what funo- 



