196 SCIENTIFIC METHOD IN PHILOSOPHY 



i and 2, between 2 and 3, and so on. If a property of 

 numbers is hereditary with respect to this relation, then 

 if it belongs to (say) 100, it must belong also to all finite 

 numbers greater than 100 ; for, being hereditary, it 

 belongs to 10 1 because it belongs to 100, and it belongs 

 to 102 because it belongs to 101, and so on where the 

 " and so on " will take us, sooner or later, to any finite 

 number greater than 100. Thus, for example, the 

 property of being greater than 99 is hereditary in the 

 series of finite numbers ; and generally, a property is 

 hereditary in this series when, given any number that 

 possesses the property, the next number must always 

 also possess it. 



It will be seen that a hereditary property, though it 

 must belong to all the finite numbers greater than a given 

 number possessing the property, need not belong to all 

 the numbers less than this number. For example, the 

 hereditary property of being greater than 99 belongs to 

 100 and all greater numbers, but not to any smaller 

 number. Similarly, the hereditary property of being 

 called Jones belongs to all the descendants (in the direct 

 male line) of those who have this property, but not to 

 all their ancestors, because we reach at last a first Jones, 

 before whom the ancestors have no surname. It is 

 obvious, however, that any hereditary property possessed 

 by Adam must belong to all men ; and similarly any 

 hereditary property possessed by o must belong to all 

 finite numbers. This is the principle of what is called 

 " mathematical induction." It frequently happens, when 

 we wish to prove that all finite numbers have some 

 property, that we have first to prove that o has the 

 property, and then that the property is hereditary, i.e. 

 that, if it belongs to a given number, then it belongs to 

 the next number. Owing to the fact that such proofs 



