152 INDUCTION. 



operations. The simple functions of any number x are all 



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reducible to the following forms : x + a, x a, a x, -, x a } \/ #, 



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=#). 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 evidence their 

 mode of formation from one another. The system of general 

 language called algebraical notation does this. The expres- 

 sions a and a 2 + 3a denote, the one any number, the 

 other the number formed from it in a particular manner. 

 The expressions a, b } n, and (a + b) n , denote any three num- 

 bers, 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 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 + 1 : now (a + b) n is a certain function of the 

 binomial; what function will this be of a and &, the two 

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



The formula (a + b) n = a n + ? a n - l b + U '~ 1 a"- 2 6 2 + &c., shows 



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in what manner the number which is formed by multiplying 

 a + b into itself n times, might be formed without that 

 process, directly from 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 formation. They 

 affirm that some mode of formation from x, and some mode of 



