212 INDUCTION. 



we should have examined without exception every known species. It is 

 the number and nature of the instances, and not their being the whole of 

 those which happen to be known, that makes them sufficient evidence to 

 prove a general law : while the more limited assertion, which stops at all 

 known animals, can not be made unless we have rigorously verified it in 

 every species. In like manner (to retui'n to a former example) we might 

 have inferred, not that all the planets, but that all planets, shine by reflect- 

 ed light : the former is no induction ; the latter is an induction, and a bad 

 one, being dispi'oved by the case of double stars — self-luminous bodies 

 which are properly planets, since they revolve round a centre, 



§ 2. There are several processes used in mathematics which require to 

 be distinguished from Induction, being not unfrequently called by that 

 name, and being so far similar to Induction properly so called, that the 

 propositions they lead to are really general propositions. For example, 

 when we have proved with respect to the circle, that a straight line can 

 not 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 a universal property of the sections of the cone. The 

 distinction drawn in the two previous examples can have no place here, 

 there being no difference between all known sections of the cone and all 

 sections, since a cone demonstrably can not be intersected by a plane ex- 

 \ cept in one of these four lines. It would be difficult, therefore, to refuse 

 \ to the proposition arrived at, the name of a generalization, since there is 

 I no room for any generalization beyond it. But there is no induction, be- 

 1 cause there is no inference : the conclusion is a mere summing up of what 

 k» I was asserted in the various propositions from which it is drawn. A case 

 somewhat, though not altogether, similar, is the proof of a geometrical theo- 

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

 '^ lin the imagination, the demonstration (as formei'ly observed*) does not 

 ,<^ 'prove directly the general theorem; it proves only that the conclusion, 

 J^ which the theorem asserts generally, is true of the particular triangle or 

 nT 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 

 l^ropositions susceptible of being thus proved, and embody them in a uni- 

 versal 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 ABC. If this were to be icalled In- 

 duction, an appropriate name for it would be, induction by parity of rea- 

 soning. But the 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 particular instances. 



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

 duction, in which the conclusion does bear the appearance of a generaliza- 

 tion grounded on some of the particular cases included in it. A mathe- 

 matician, when he has calculated a sufficient number of the terms of an al- 



* Supra, p. 145. 



