190 



INDUCTION. 



example) we miffht have inferred, not 

 that all the planets, but that all 

 planets, shine by reflected light : the 

 former is no induction ; the latter 

 is an induction, and a bad one, 

 being disproved 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 In- 

 duction 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 cannot 

 meet it in more than two points, and 

 when the same thing has been suc- 

 cessively proved of the ellipse, the 

 parabola, and the hyperbola, it may 

 be laid down as an universal property 

 of the sections of the cone. The dis- 

 tinction drawn in the two previous 

 exaiDples can have no place here, 

 there being no difference between all 

 known sections of the cone and aU 

 sections, since a cone demonstrably 

 cannot 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 

 oi a generalisation, since there is no 

 room for any generalisation 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 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 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 ex- 



* Supra, p. 125. 



hibited 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 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 called In- 

 duction, an appropriate name for it 

 would be, induction by parity of rea- 

 soning. But the term cannot properly 

 belong to it ; the characteristic quality 

 of Induction is wanting, since the 

 truth obtained, though really general, 

 is not beloved 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 mathe- 

 matics, some examples of so-called 

 Induction, in which the conclusion 

 does bear the appearance of a gene- 

 ralisation grounded on some of the 

 particular cases included in it. A 

 mathematician, when he has calcu- 

 lated 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 calcula- 

 tions. But I apprehend he only does 

 so when it is apparent from a priori 

 considerations (which might be ex- 

 hibited in the form of demonstration) 

 that the mode of formation of the sub- 

 sequent 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 with- 

 out the sanction of such general con-., 

 siderationsi, there are instances on 



