176 INDUCTION. 



bola, it may be laid down. as an universal property of the sections of 

 the cone. In this example there is no induction, because there is no 

 inference : the conclusion is a mere summing up of what was asserted 

 in the various propositions from which it is drawn. A case somewhat, 

 thouo-h not altogether, similar, is the proof of a geometrical theorem 

 by means of a diagram. Whether the diagram be on paper or only 

 in the imagination, the demonstration (as we formerly observed*) does 

 not prove directly the general theorem ; it proves only that the con- 

 clusion, which the theorem asserts generally, is true of the particular 

 ti-iangle or circle exhibited in the diagram : but since we perceive that 

 in the same way in which we have proved it of that circle, it might 

 also be proved of any other circle, we gather up into one general 

 expression all the singular propositions susceptible of being thus 

 proved, and embody them in an universal jjrojDosition. Having shown 

 that the three angles of the triangle ABC are together equal to two 

 right angles, we conclude that this is true of every other triangle, not 

 because it is true of A B C, but for the same reason which proved it 

 to be true of A B C. If this were to be called Induction, an appro- 

 priate name for it would be, Induction by parity of reasoning. But 

 the term cannot 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 

 gi'ound of our conviction in the particular instances. 



There are nevertheless, in mathematics, some examples of so-called 

 induction, in which the conclusion 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 ascer- 

 tained what is called the Icnv 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 fi-om 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 fonnation 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 upon record in which it has led to false results. 



It is said that Newton discovered the binomial theorem by induc- 

 tion ; 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 foi-mula of each power stands to the 

 exponent of that power, and to the two terms of the binomial. The 

 fact is not improbable : but a mind like Newton's, which seemed to 

 aiTive per saltuvi at principles and conclusions that ordinary mathe- 

 maticians only reached by a succession of steps, certainly could not 

 have performed the comparison in question without being led by it to 

 the a priori ground of the law ; since any one who understands suf- 

 ficiently the nature of multiplication to venture upon multiplying 

 several lines of figures or symbols at one operation, cannot but perceive 



* Supra, p. 127, 128. 



