308 THE PRINCIPLES OF SCIENCE. 



(2) Deducing consequences from that law. 



(3) Observing whether the consequences agree with the 

 particular facts under consideration. 



In very simple cases of inverse reasoning, hypothesis 

 may sometimes seem altogether needless. Thus, to take 

 numbers again as a convenient illustration, I have only 

 to look at the series, 



i, 2, 4, 8, 16, 32, &c., 



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

 metrical progression ; I need no successive trial of vari- 

 ous 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 

 formulae, so that we have no need to go through any pro- 

 cess of discovery. But it is none the less true that when- 

 ever previous reasoning does not furnish the knowledge, 

 hypotheses must be framed and tried. (See p. 142.) 



There naturally arise two different cases, according as 

 the nature of the subject admits of certain or only pro- 

 bable 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 inference is possible the process is simplified. 

 Of several mutually inconsistent hypotheses, the results of 

 which can be certainly compared with fact, but one hypo- 

 thesis can ultimately be entertained. Thus in the inverse 

 logical problem, two logically distinct conditions could not 

 yield the same series of possible combinations. Accord- 

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

 seven different kinds of propositions, or in the case of 

 three terms, our choice lay among 192 possible distinct 

 hypotheses (pp. 1 54- 1 64 ). Natural laws, however, are often 

 quantitative in character, and the possible hypotheses are 

 then infinite in variety. 



