368 INDUCTION. 



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 fi'om the same numbers ; but they may be formed in the same 

 manner from different numbers ; as nine is formed from three by mul- 

 Uplying 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 num- 

 ber, 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 foiined 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"', ai — J log. X (to the baso a), and the same expressions 



a V X ^ 



varied by putting x for a and a for x, wherever that substitution would 

 alter the value : to which perhaps we ought tO; add (with M. Comte) 

 sin X, and arc (sin = x). 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 -]3articular numbers they are, 

 shall show what fimction each is of the other ; or, in other words, shall 

 put in evidence their mode of formation from one another. The sys- 

 tem 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, h, n, and [a + Z»)", 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 alge- 

 braical calculus : F being a certain function of a given number, to find 

 what function F will be of any function of that number. For example, 

 a binomial a-{-b is a function of its two parts a and b, and the parts 

 are, in their turn, functions of a + ^: now (a + J)" is a certain function 

 of the binomial; what fimction will. this be of a and b, and the two 

 parts % The answer to this question is the binomial theorem. The 



formula {a + b)" = a" -| a"~^b ■\ — ^-5- a"~^b'^ -{-, &c., shows in what 



manner the number which is formed by multiplying a-{-b into itself 

 n times, might be formed without that process, directly fi-om a, b, and n. 

 And of this nature are all the theorems of the science of number. 

 They assert the identity of the result of different modes of foimation. 

 They affirm that some mode of formation from a-, and some mode of 

 formation from a certain function of x, produce the same number. 



Besides these general theorems or formulae, what remains in the 

 algebraical calculus is the resolution of equations. But the resolution 

 of an equation is also a theorem. If the equati on be x^ + ax — b, the 

 resolution of this equation, viz., x= — ^a± '^ i«^ + i, isa general 



