322 INDUCTION. 



no difference between all known sections of the cone and all 

 sections, since a cone demonstrably cannot be intersected by 

 a plane except in one of these four lines. It would be diffi- 

 cult, therefore, to refuse to the proposition arrived at, the name 

 of a generalization, since there is no room for any generaliza- 

 tion beyond it. But 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, though 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 de- 

 monstration (as formerly observed*) does not prove directly the 

 general theorem ; it proves only that the conclusion, which the 

 theorem asserts generally, is true of the particular triangle 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 pro- 

 position. 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 

 ABC, but for the same reason which proved it to be true 

 of ABC. If this were to be called Induction, an appropriate 

 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 par- 

 ticular 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 convic- 

 tion 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 on some of the par- 

 ticular cases included in it. A mathematician, when he has 



* Supra, p. 214. 



