266 THE PRINCIPLES OF SCIENCE. [CHAP. 



(3) Observing whether the consequences agree with the 

 particular facts under consideration. 



In very simple cases of inverse reasoning, hypothesis 

 may seem altogether needless. To take numbers again as 

 a convenient illustration, 1 have only to look at the series, 



i, 2, 4, 8, 1 6, 32, &c., 



to know at once that the general law is that of geo- 

 metrical progression ; I need no successive trial of various 

 hypotheses, because I am familiar with the series, and have 

 long since learnt from what general formula it proceeds. 

 In the same way a mathematician becomes acquainted 

 with the integrals of a number of common formulas, so 

 that he need not go through any process of discovery. 

 But it is none the less true that whenever previous reason- 

 ing does not furnish the knowledge, hypotheses must be 

 framed and tried (p. 124). 



There naturally arise two cases, according as the nature 

 of the subject admits of certain or only probable deductive 

 reasoning. Certainty, indeed, is but a singular case of 

 probability, and the general principles of procedure are 

 always the same. Nevertheless, when certainty of infer- 

 ence is possible, the process is simplified. Of several 

 mutually inconsistent hypotheses, the results of which 

 can be certainly compared with fact, but one hypothesis 

 can ultimately be entertained. Thus in the inverse logical 

 problem, two logically distinct conditions could not yield 

 the same series of possible combinations. Accordingly, 

 in the case of two terms we had to choose one of six 

 different kinds of propositions (p. 136), and in the case of 

 three terms, our choice lay among 192 possible distinct 

 hypotheses (p. 140). Natural laws, however, are often 

 quantitative in character, and the possible hypotheses are 

 then infinite in variety. 



When deduction is certain, comparison with fact is 

 needed only to assure ourselves that we have rightly 

 selected the hypothetical conditions. The law establishes 

 itself, and no number of particular verifications can add 

 to its probability. Having once deduced from the prin- 

 ciples of algebra that the difference of the squares of two 

 numbers is equal to the product of their sum and dif- 

 ference, no number of particular trials of its truth will 

 render it more certain. Oil the other hand, no finite 



