230 THE PRINCIPLES OF SCIENCE. [CHAP. 



such a law ; and I then ascertain its truth, by proving 

 deductively from the rules of decimal numeration, that any 

 number ending in five must be made up of multiples of 

 five, and must therefore be itself a multiple. 



To make this more plain, let the reader now examine 

 the numbers 



7> *7> 37, 47, fy> 97- 

 Th ey all end in 7 instead of 5, and though not at equal 

 intervals, the intervals are the same as in the previous 

 case. After consideration, the reader will perceive that 

 these numbers all agree in being prime numbers, or mul- 

 tiples of unity only. May we then infer that the next, or 

 any other number ending in 7, is a prime number? 

 Clearly not, for on trial we find that 27, 57, 117 are not 

 primes. Six instances, then, treated empirically, lead us 

 to a true and universal law in one case, and mislead us in 

 another case. We ought, in fact, to have no confidence in 

 any law until we have treated it deductively, and have 

 shown that from the conditions supposed the results ex- 

 pected must ensue. No one can show from the principles 

 of number, 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, 



4i, 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*- + x + 41, putting for # in succes- 

 sion the values, o, I, 2, 3, 4, &c. We seem always to 

 obtain a prime number, and the induction is apparently 

 strong, to the effect that this expression always will 

 give primes. Yet a few more trials disprove this false con- 

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

 or 41 x 41. Such a failure could never have happened, 

 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. Babbage pointed out, in his Ninth Bridgewater 



