INDUCTIONS IMPROPERLY SO CALLED. 213 



gebraical ov 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 calculations. But I apprehend he only does so when 

 it is apparent from a priori considerations (which might be exhibited in 

 the form of demonstration) that the mode of formation of the subsequent 

 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 without the sanction of such general considerations, there 

 are instances on record in which it has led to false results. 

 . It is said that Newton discovei'ed the binomial theorem by induction ; 

 by raising a binomial successively to a certain number of powers, and 

 comparing 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 

 principles 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 a priori 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, can not 

 but perceive that in raising a binomial to a power, the co-efficients must 

 depend on the laws of permutation and combination : and as soon as this 

 is recognized, 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, therefore, such cases as these, are but ex- 

 jramples of what I have called Induction by parity of reasoning, that is, not 

 I really Induction, because not involving inference of a general proposition 

 I from particular instances. 



§ 3. There remains a third improper use of the term Induction, which it 

 is of real importance to clear up, because the theory of Induction has been, 

 in no ordinary degree, confused by it, and because the confusion is exem- 

 plified in the most recent and elaborate treatise on the inductive philosophy 

 which exists in our language. The error in question is that of confound- 

 ing a mere description, by general terms, of a set of observed phenomena, 

 with an induction from them. 



Suppose that a phenomenon consists of pai'ts, 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 (amounting 

 for many purposes to a necessity) in obtaining a representation of the phe- 

 nomenon as a whole, by combining, or as we may say, piecing these de- 

 tached fragments together. A navigator sailing in the midst of the ocean 

 discovers land : he can not 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 completely round it: he then pro- 

 nounces it an island. • Now there was no particular time or place of ob- 

 servation 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 or three 

 words the whole of what he so observed. But is there any thing of the 

 nature of an induction in this process? Did he infer any thing that had 

 not been observed, from something else which had ? Certainly not. He 



