THE INDUCTIVE OR INVERSE METHOD. 309 



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. On the other hand, no finite 

 number of particular verifications of a supposed law will 

 render that law certain. In short; certainty belongs only 

 to the deductive process, and to the teachings of direct 

 intuition ; and as the conditions of nature are not given 

 by intuition, we can only be certain that w T e have got a 

 correct hypothesis when, out of a limited number con- 

 ceivably possible, we select that one which alone agrees 

 with the facts to be explained. 



In geometry and kindred branches of .mathematics, 

 deductive reasoning is conspicuously certain, and it would 

 often seem as if the consideration of a single diagram 

 yields us certain knowledge of a general proposition. 

 But in reality all this certainty is of a purely hypothetical 

 character. Doubtless if we could ascertain that a sup- 

 posed circle was a true and perfect circle, w could be 

 certain concerning a multitude of its geometrical pro- 

 perties. But geometrical figures are physical objects, and 

 the senses can never assure us as to their exact forms. 

 The figures really treated in Euclid's ' Elements' are 

 imaginary, and we never can verify in practice the con- 

 clusions which we draw with certainty in inference; 

 questions of degree and probability enter. 



Passing now to subjects in which deduction is only 

 probable, it ceases to be possible to adopt one hypothesis 

 to the exclusion of the others. We must entertain at the 

 same time all conceivable hypotheses, and regard each 



