XL] PHILOSOPHY OF INDUCTIVE INFERENCE. 231 



Treatise, that a machine could be constructed to give a 

 perfectly regular series of numbers through a vast series 

 of steps, and yet to break the law of progression suddenly 

 at any required point. No number of particular cases as 

 particulars enables us to pass by inference to any new case. 

 It is hardly needful to inquire here what can be inferred 

 from an infinite series of facts, because they are never 

 practically within our power ; but we may unhesitatingly 

 accept the conclusion, that no finite number of instances 

 can ever prove a general law, or can give us certain know- 

 ledge of even one other instance. 



General mathematical theorems have indeed been dis- 

 covered by the observation of particular cases, and may 

 again be so discovered. We have Newton's own state- 

 ment, to the effect that he was thus led to the all-impor- 

 tant Binomial Theorem, the basis of the whole structure 

 of mathematical analysis. Speaking of a certain series of 

 terms, expressing the area of a circle or hyperbola, he says : 

 " I reflected that the denominators were in arithmetical 

 progression; so that only the numerical co-efficients of 

 the numerators remained to be investigated. But these, 

 in the alternate areas, were the figures of the powers of 

 the number eleven, namely 1 1, 1 1 1 , 1 1 2 , 1 1 3 , 1 1 4 ; that is, 

 in the first I ; in the second I, I ; in the third I, 2, I ; in 

 the fourth i, 3, 3, I ; in the fifth I, 4, 6, 4, i. 1 I inquired, 

 therefore, in what manner all the remaining figures could 

 be found from the first two ; and I found that if the first 

 figure be called m, all the rest could be found by the 

 continual multiplication of the terms of the formula 

 -o x -,, x m-a x m-3 x &c , 



It is pretty evident, from this most interesting statement, 

 that Newton, having simply observed the succession of the 

 numbers, tried various formulae until he found one which 

 agreed with them all. He was so little satisfied with this 

 process, however, that he verified particular results of his 

 new theorem by comparison with the results of common 



1 These are the figurate numbers considered in pages 183, 187, &c. 



2 Comtnercium, Epistolicum. Epistola ad Oldenburgum, Oct. 24, 

 1676. Horsley's Works of Newton, vol. iv. p. 541. See De Morgan 

 in Penny Cyclopaedia, art. " Binomial Theorem," p. 412. 



