INDUCTIONS IMPROPERLY SO CALLED. 



191 



record in which it has led to false 

 results. 



It is said that Newton discovered 

 the binomial theorem by induction ; 

 by raising a binomial successively to 

 a certain number of powers, and com- 

 paring those powers with one another 

 until he detected the relation in which 

 the algebraic formula 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 

 mathematician like Newton, who 

 seemed to arrive per saltum at prin- 

 ciples and conclusions that ordinary 

 mathematicians only reached by a 

 succession of steps, certainly could 

 not have performed the comparison 

 in question without being led by it to 

 the d priat^i ground of the law ; since 

 any one who understands sufficiently 

 the nature of multiplication to ven- 

 ture upon multiplying several lines of 

 symbols at one operation, cannot but 

 perceive that in raising a binomial to 

 a power, the co-efficients must depend 

 on the laws of permutation and com- 

 bination, and as soon as this is recog- 

 nised, the theorem is demonstrated. 

 Indeed, when once it was seen that 

 the law prevailed in a few of the 

 lower powers, its identity with the 

 law of permutation would at once 

 suggest the considerations which prove 

 it to obtain universally. Even, there- 

 fore such cases as these, are but ex- 

 amples of what I have called Induc- 

 tion by parity of reasoning, that is, 

 not really Induction, because not 

 involving inference of a general pro- 

 position from particular instances. 



§ 3. There remains a third impro- 

 per use of the term Induction, which 

 it is of real importance to clear up, 

 because the theory of Induction has 

 been, in no ordinary degiee, confused 

 by it, and because tlie confusion is 

 exemplified in the most recent and 

 elaborate treatise on the inductive 

 philosophy which exists in our lan- 

 guage. The error in question is that 

 of confounding a mere description, by 

 general terms, of a set of observed 



phenomena, with an induction from 

 them. 



Suppose that a phenomenon con- 

 sists of parts, and that these parts 

 are only capable of being observed 

 separately, and as it were piecemeal. 

 When the observations have been 

 made, there is a convenience (amount- 

 ing for many purposes to a necessity) 

 in obtaining a representation of the 

 phenomenon as a whole, by combining, 

 or, as we may say, piecing these de- 

 tached fragments together. A navi- 

 gator sailing in the midst of the ocean 

 discovers land : he cannot at first, or 

 by any one observation, determine 

 whether it is a continent or an island ; 

 but he coasts along it, and after a few 

 days finds himself to have sailed com- 

 pletely round it : he then pronounces 

 it an island. Now there was no par- 

 ticular time or place of observation 

 at which he could perceive that this 

 land was entirely surrounded by 

 water ; he ascertained the fact by a 

 succession of partial observations, and 

 then selected a general expression 

 which summed up in two ot three 

 words the whole of what he so ob- 

 served. But is there anything of the 

 nature of an induction in this process? 

 Did he infer anything that had not 

 been observed, from something else 

 which had ? Certainly not. He had 

 observed the whole of what the pro- 

 position asserts. That the land in 

 question is an island, is not an infer- 

 ence from the partial facts which the 

 navigator saw in the course of his 

 circumnavigation ; it is the facts 

 themselves ; it is a summary of those 

 facts ; the description of a complex 

 fact, to which those simpler ones are 

 as the parte of a whole. 



Now there is, I conceive, no differ- 

 ence in kind between this simple 

 operation, and that by which Kepler 

 ascertained the nature of the plane- 

 tary orbite ; and Kepler's operation, 

 all at least that was characteristic in 

 it, was not more an inductive act 

 than that of our supposed navigator. 



The object of Kepler was to deter- 

 mine the real path described by each 



