PHILOSOPHY OF INDUCTIVE INFERENCE. 265 



principles of number, no one can show that numbers 

 ending in 7 should be primes. 



From the history of the theory of numbers some good 

 examples of false induction can be adduced. Taking the 

 following series of prime numbers 



41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, &c., 

 it will be found that they all agree in being values of 

 the general expression x 2 + x 4- 4 1 , putting for x in succes- 

 sion the values, o, i, 2, 3, 4, &c. We thus seem always 

 to obtain a prime number, and the induction is apparently 

 very strong, to the effect that this expression always will 

 give primes. Yet a few more trials will disprove this false 

 conclusion. Put x = 40, and we obtain 40 x 40 + 40 + 41, 

 or 41 x 4 1 . Now such a failure could never have hap- 

 pened, had we shown any deductive reason why x 2 + x + 41 

 should give primes. 



There can be no doubt that what here happens with 

 forty instances, might happen with forty thousand or 

 forty million instances. An apparent law never once 

 failing up to a certain point may then suddenly break 

 down, so that inductive reasoning, as it has been described 

 by some writers, can give no sure knowledge of what is to 

 come. Mr. Babbage admirably pointed out, in his Ninth 

 Bridgewater 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 progres- 

 sion 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 sure knowledge of even one other instance. 



General mathematical theorems have indeed been dis- 



