VII.] INDUCTION. 147 



the comparative numbers of known cryptogams which 

 are and are not celhilar. Thus the first step in every 

 induction will consist in accurately summing up the 

 number of instances of a particular phenomenon which 

 have fallen under our observation. Adams and Leverrier, 

 ior instance, must have inferred that the undiscovered 

 ]>lanet Neptune would obey Bode's law, because all the 

 planets known at that time obeyed it. On what principles 

 the passage from the known to the apparently unknown 

 is warranted, must be carefully discussed in the next sec- 

 tion, and in various parts of this work. 



It would be a great mistake, however, to suppose that 

 Perfect Induction is in itself useless. Even when the 

 enumeration of objects belonging to any class is complete, 

 ami admits of no inference to unexamined objects, the 

 statement of our knowledge in a general proposition is a 

 process of so much importance that we may consider it 

 necessary. In many cases we may rendf^r our investiga- 

 tions exhaustive ; all the teeth or bones of an animal ; all 

 the cells in a minute vegetable organ ; all the caves in a 

 mountain side ; all the strata in a geological section ; all 

 the coins in a newly found hoard, may be so completely 

 scrutinized that we may make some general assertion 

 concerning them without fear of mistake. Every bone 

 might be proved to contain phosphate of lime ; every cell 

 to enclose a nucleus ; every cave to hide remains of extinct 

 animals ; every stratum to exhibit signs of marine origin ; 

 every coin to be of lioman 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 Vi" oases where induction is 

 naturally and necessarily Imntea 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, thiity 

 edges, find twenly corners ; for by the {)rinciples of geometry 

 we learn tliat 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 endess 

 variety of perfect inductions might be made; we can show 

 that no ininiber less than sixty jjossesses so many divisors, 

 and the like is true of 3G0 ; ior it does not require a great 

 amount of labour to ascertain and count all the divisors 



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