96 THE PRINCIPLES OF SCIENCE. 



are such invariable and necessary conditions of all thought, 

 that they need not be specially laid down. The Law of 

 Contradiction is a further condition of all thought and of 

 all logical symbols ; it enables, and in fact obliges, us to 

 reject from further consideration all terms which imply 

 the presence and absence of the same quality. Now, 

 whenever we bring both these Laws of Thought into ex- 

 plicit action by the method of substitution, we employ the 

 Indirect Method of Inference. It will be found that we 

 can treat not only those arguments already exhibited 

 according to the direct method, but we can also include an 

 infinite multitude of other arguments which are incapable 

 of solution by any other means. 



Some philosophers, especially those of France, have 

 held that the Indirect Method of Proof has a certain infe- 

 riority to a direct method, which should prevent our using 

 it except when obliged. But there are an unlimited 

 number of truths which we can prove only indirectly. 

 We can prove that a number is a prime only by the 

 purely indirect method of showing that it is not any of the 

 numbers which have divisors, and the remarkable process 

 known as Eratosthenes' Sieve is the only mode by which 

 we can select the prime numbers a . It bears a strong 

 analogy to the indirect method here to be described. We 

 can also prove that the side and diameter of a square are 

 incommensurable, but only in the negative or indirect 

 manner, by showing that the contrary supposition con- 

 stantly and inevitably leads to contradiction b . Many other 

 demonstrations in various branches of the mathematical 

 sciences rest upon a like method. Now if there is only 

 one important truth which must be, and can only be 



a See Horsley, ' Philosophical Transactions/ 1772; vol. Ixii. p. 327. 

 Montucla, ' Histoire des Mathematiques,' vol. i. p. 239. ' Penny 

 Cyclopaedia,' article Eratostfienes. 



b Euclid, Book x. Prop. 117. 



