INDUCTIONS IMPROPERLY SO CALLED. 323 



calculated a sufficient number of the terms of an algebraical 

 or arithmetical series to have ascertained what is called the law 

 of the series, does not hesitate to fill up any number of the 

 succeeding terms without repeating the calculations. But I 

 apprehend he only does so when it is apparent from a priori 

 considerations (which might be exhibited in the form of 

 demonstration) that the mode of formation of the subsequent 

 terms, each from that which preceded it, must be similar to 

 the formation of the terms which have been already calculated. 

 And when the attempt has been hazarded without the sanction 

 of such general considerations, there are instances on record in 

 which it has led to false results. 



It is said that Newton discovered the binomial theorem 

 by induction ; by raising a binomial successively to a certain 

 number of powers, and comparing those powers with one 

 another until he detected the relation in which the algebraic 

 formula of each power stands to the exponent of that power, 

 and to the two terms of the binomial. The fact is not im- 

 probable : but a mathematician like Newton, who seemed to 

 arrive per saltum at principles and conclusions that ordinary 

 mathematicians only reached by a succession of steps, certainly 

 could not have performed the comparison in question without 

 being led by it to the d priori ground of the law ; since any 

 one who understands sufficiently the nature of multiplication 

 to venture upon multiplying several lines of symbols at one 

 operation, cannot but perceive that in raising a binomial to a 

 power, the coefficients must depend on the laws of permuta- 

 tion and combination : and as soon as this is recognised, the 

 theorem is demonstrated. Indeed, when once it was seen that 

 the law prevailed in a few of the lower powers, its identity 

 with the law of permutation would at once suggest the con- 

 siderations which prove it to obtain universally. Even, 

 therefore, such cases as these, are but examples of what I 

 have called Induction by parity of reasoning, that is, not 

 really Induction, because not involving inference of a general 

 proposition from particular instances. 



3. There remains a third improper use of the term 

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