436 Prof. De IMorgan on Indirect Demonstration. 



indirect process contains postulates of reasoning as difficult as 

 the required transformation, if not of its very nature. Euclid 

 would not apply the indirect process to prove the conclusion 

 about a space-area : but he does apply it when the area is what 

 logicians call the extent of a term. When A is entirely within B, 

 species within genus, he never admits that all the notions outside 

 the genus are outside the species, without an indirect demonstra- 

 tion. From Eveiy not-B is not-A he produces Every A is B, 

 thus : — If it be possible, let this A be iiot-B, but every not-B 

 is not-A, therefore this A is not-A, which is absurd : whence 

 every A is B. He might as well argue into the conclusion of 

 a common syllogism from the premises, as thus ; — Every A is B, 

 this is an A, therefore it is a B ; for if not let it be not-B, then 

 one not-B is A, but every A is B, therefore not-B is B, which 

 is absurd, &c. Here it is manifest that our reasoning takes 

 fully as much for granted as the direct transition from premises 

 to conclusion : we take syllogism for granted in proving syllo- 

 gism. Euclid does more : he takes syllogism for granted in 

 proving the antesyllogistic conversion of propositions. This 

 does well for beginners, to whom simple affirmative syllogism is 

 more familiar than conversion by contraposition : but I am now 

 speaking to mathematicians who examine the laws of thought. 



It is au easily ascertained fact, that really inchrect demon- 

 stration is uncommon in geometry, except as a (to a logician) 

 unnecessary help to coutrapositive directness of proof. Take 

 for example. Book I. Prop. 6. A non-isosceles triangle is un- 

 equally angled (at the base). Now i. 4 is, in one of its contra- 

 positive forms, as follows. Two sides severally equal to two 

 sides, with unequal areas, have unequal angles contained. Euclid's 

 construction instantly brings out of a non-isosceles triangle two 

 triangles with two sides severally equal to two sides, and areas in 

 the relation of whole and part. Hence follows that a non- 

 isosceles triangle is unequally angled at the base : to the logician 

 this is identical with Euclid's form, Equal angles at the base give 

 equal sides : the geometer who is not a logician is helped over 

 this last step by the addition of an indirect demonstration. 



Seeing that this so-called indirect proof, then, is in its indirect 

 part seldom anything except the demonsti'ation of the passage 

 from coutrapositive to positive, for the benefit of those to whom 

 this step of pure logic is of uneasy transition, we may ask how 

 the necessity for the coutrapositive form is to be explained ? The 

 refutation of contradiction is viewed by some geometers as a kind 

 of lame and imperfect proof. It is, indeed, mostly superfluous ; 

 but it is rather a crutch proof than a lame proof, when applied 

 only to help in the conversion of a proposition. With reference, 

 however, to the unavoidable entrance of both the direct forms, it 



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