354 INDUCTION. 



2. A second process which requires to be distin- 

 guished from Induction, is one to which mathemati- 

 cians sometimes give that name : and which so far resem- 

 bles Induction properly so called, that the propositions it 

 leads to are really general propositions. For example, 

 when we have proved, with respect to the circle, that 

 a straight line cannot meet it in more than two points, 

 and when the same thing has been successively proved 

 of the ellipse, the parabola, and the hyperbola, 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 some- 

 what, 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 demonstration (as we formerly observed*) does 

 not prove directly the general theorem ; it proves only 

 that the conclusion, which the theorem asserts gene- 

 rally, is true of the particular triangle or circle exhi- 

 bited 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 proposition. 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 A B C. If this were to be 



Supra, p. 256. 



