INDUCTIONS IMPROPERLY SO CALLED. 355 



called Induction, an appropriate name for it would be, 

 Induction by parity of reasoning. But tbe term can- 

 not properly belong to it ; the characteristic quality 

 of Induction is wanting, since the truth obtained, 

 though really general, is not believed on the evidence 

 of particular instances. We do not conclude that all 

 triangles have the property because some triangles 

 have, but from the ulterior demonstrative evidence 

 which was the ground of our conviction in the parti- 

 cular instances. 



There are nevertheless, in mathematics, some 

 examples of so-called induction, in which the conclu- 

 sion does bear the appearance of a generalization 

 grounded upon some of the particular cases included 

 in it. A mathematician, when he has 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 demonstra- 

 tion) 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 con- 

 siderations, there are instances upon record in which 

 it has led to false results. 



It is said that Newton discovered the binomial 

 theorem by induction ; by raising a binomial succes- 

 sively 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 



2 A 2 



