356 INDUCTION. 



terms of the binomial. The fact is not improbable : 

 but a mind like Newton's, which 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 venture upon multi- 

 plying several lines of figures or symbols at one 

 operation, cannot but perceive that in raising a bino- 

 mial to a power, the coefficients must depend upon 

 the laws of permutation and combination : and as 

 soon as this is recognised, the theorem is demon- 

 strated. 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 examples 

 of what I have called induction by parity of reasoning, 

 that is, not really induction, because not involving 

 any inference of a general proposition 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, to no 

 ordinary degree, confused by it, and because the con- 

 fusion is exemplified in the most recent and most 

 elaborate treatise on the inductive philosophy which 

 exists in our language. The error in question is that 



* I am happy to be able to refer, in confirmation of this view 

 of what is called induction in mathematics, to the highest English 

 authority on the philosophy of algebra, Mr. Peacock. See pp. 

 107-8 of his profound Treatise on Algebra. 



