166 THE PRINCIPLES OF SCIENCE. 



assertion concerning them without fear of mistake. Every 

 bone might be proved to consist of phosphate of lime ; 

 every cell to enclose a nucleus; every cave to contain 

 remains of extinct animals ; every stratum to exhibit signs 

 of marine origin ; every coin to be of Roman manufacture. 

 These are cases where our investigation is limited to a 

 definite portion of matter, or a definite area on the earth s 

 surface. 



There is another class of cases where induction is 

 naturally and necessarily limited to a definite number of 

 alternatives. Of the regular solids we can say without 

 the least doubt that no one has more than twenty faces, 

 thirty edges, and twenty corners ; for by the principles 

 of geometry we learn that there cannot exist more than 

 five regular solids, of each of which we easily observe 

 that the above statements are true. In the theory of 

 numbers, an endless variety of perfect inductions might 

 be made ; we can show that no number less than sixty 

 possesses so many divisors, and the like is true of 360 e , 

 for it does not require any very great amount of labour to 

 ascertain and count all the divisors of numbers up to sixty 

 or 360. Similarly I can assert that between 60,041 and 

 60,077 no prime number occurs, because the exhaustive 

 examination of those who have constructed tables of prime 

 numbers proves it to be so. 



In matters of human appointment or history, we can 

 frequently have a complete limitation to the numbers of 

 instances to be included in an induction. We might show 

 that none of the other kings of England reigned so long as 

 George III ; that Magna Charta has not been repealed by 

 any subsequent statute ; that the propositions of the third 

 book of Euclid treat only of circles ; that no part of the 

 works of Galen mentions the fourth figure of the syl- 



Wallie s Treatise of Algebra (1685), p. 22. 



