146 THE PRINCIPLES OF SCIENCE. [CHAP. 



will in time be disclosed between the sciences of logic 

 and mathematics. 



Distinction between Perfect and Imperfect Induction. 



We cannot proceed with advantage before noticing the 

 extreme difference which exists between cases of perfect 

 and those of imperfect induction. We call an induction 

 perfect when all the objects or events which can possibly 

 come under the class treated have been examined. But 

 in the majority of cases it is impossible to collect together, 

 or in any way to investigate, the properties of all portions 

 of a substance or of all the individuals of a race. The 

 number of objects would often be practically infinite, and 

 the greater part of them might be beyond our reach, in 

 the interior of the earth, or in the most distant parts of 

 the Universe. In all such cases induction is imperfect, 

 and is affected by more or less uncertainty. As some 

 writers have fallen into much error concerning the func- 

 tions and relative importance of these two branches of 

 reasoning, I shall have to point out that 



1. Perfect Induction is a process absolutely requisite, 



both in the performance of imperfect induction and 

 in the treatment of large bodies of facts of which 

 our knowledge is complete. 



2. Imperfect Induction is founded on Perfect Induction, 



but involves another process of inference of a 



widely different character. 



It is certain that if I can draw any inference at all 

 concerning objects not examined, it must be done on the 

 data afforded by the objects which have been examined. 

 If I judge that a distant star obeys the law of gravity, 

 it must be because all other material objects sufficiently 

 known to me obey that law. If I venture to assert that 

 all ruminant animals have cloven hoofs, it is because all 

 ruminant animals which have come under my notice have 

 cloven hoofs. On the other hand, I cannot safely say 

 that all cryptogamous plants possess a purely cellular 

 structure, because some cryptogamous plants, which have 

 been examined by botanists, have a partially vascular 

 structure. The probability that a new cryptogam will be 

 cellular only can be estimated, if at all, on the ground of 



