82 THE PRINCIPLES OF SCIENCE. [CHAP 



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, when- 

 ever we bring both these Laws of Thought into explicit 

 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 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 inferiority 

 to the direct method, which should prevent our using it 

 except when obliged. But there are many 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 Eratos- 

 thenes' Sieve is the only mode by which we can select the 

 prime numbers. 1 It bears a strong analogy to the indirect 

 method here to be described. \Ve can prove that the side 

 and diameter of a square are incommensurable, but only in 

 the negative or indirect manner, by showing that the con- 

 trary supposition inevitably leads to contradiction. 2 Many 

 other demonstrations in various branches of the mathe- 

 matical sciences proceed upon a like method. Now, if 

 there is only one important truth which must be, and can 

 only be, proved indirectly, we may say that the process is a 

 necessary and sufficient one, and the question of its com- 

 parative excellence or usefulness is not worth discussion. 

 As a matter of fact I believe that nearly half our logical 

 conclusions rest upon its employment. 



1 SeeHorsley, Philosophical Transactions, 1772 ; vol. Ixii. p. 327. 

 Montucla, Histoire des Mathematuiuei, vol. i p. 239. Penny 

 Cyclopaedia, article " Eratosthenes." 



* Euclid, Book x. Prop. 117. 



