402 



INDUCTION. 



ftirive at that conclusion (aa all know 

 who remember how they first learned 

 it) by adding a single unit at a time : 

 5+1=6, therefore 5 + 1 + 1 = 6 

 + 1 = 7: and again 2=1 + 1, there- 

 fore 5 + 2 = 5 + 1 + 1 = 7. 



§ 6. Innumerable as are the true 

 propositions which can be formed con- 

 cerning particular numbers, no ade- 

 quate 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 numeri- 

 cal 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 num- 

 bers ; not of all things under the con- 

 dition of being divided in a particular 

 way, but of all things under the con- 

 dition 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 con- 

 cerning numbers relate to their modes 

 of formation from other numbers, and 

 yet that there are propositi^ ns which 

 are true of all numbers. But this 

 very paradox leads to the real prin 

 ciple of generalisation 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 multi- 

 plying it into itself, and sixteen is 

 formed from four by the same pro- 

 cess. Thus there arises a classifica- 

 tion of modes of formation, or, in the 

 language commonly used by mathe- 

 maticians, a classification of Func- 

 tioiu. ^y 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 opera- 

 tions which form simple functions, or 

 by successive repetitions of some one 

 of those operations. The simple func- 

 tions of any number x are all reducible 

 to the following forms : x -V a, x—a, 



a X, - X*, v^ 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 = x). All 

 other functions of x are formed by put- 

 ting some one or more of the simple 

 functions in the place of x or a, and 

 subjecting them to the same elemen- 

 tary operations. 



In order to carry on general reason- 

 ings on the subject of Functions, we 

 require a nomenclature enabling us 

 to express any two numbers by names 

 which, without specifying what par- 

 ticular 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 expres- 

 sions a, b, n, and (a + 6)«, 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 func- 

 tion F will be of any function of that 

 number. For example, a binomial 

 a + b is a function of its two parts a 

 and 6, and the parts are, in their 

 turn, functions of a + 6 ; now (a + A)»^ 

 is a certain function of the binomial ; 

 what function will this be of a and 6, 

 the two parts? The answer to this 



