Prof. De Morgan on Indirect Demonstration. 437 



seems that the contrapositive proposition is often more accessible 

 than the positive one, because we know more about the negative 

 terms than about the positive ones : and we have to proceed 

 from the more known to the less known. It is surely no great 

 Avonder, and no cause of complaint against the nature of things, 

 that we should sometimes find ourselves in a position in which 

 we can only proceed to comparison of equals by previous com- 

 parison of unec-[uals. On the contrary, it seems clear to me that 

 it should rather be matter of surprise that we are not obliged 

 to do something yet more specific in the way of departure from 

 consideration of equality. 



The relations of magnitude (ratios) are infinite in number. If 

 there were a person well versed in the truths of geometry and 

 arithmetic,but wholly ignorant of their systematic derivation from 

 each other, and if this person were informed that he must proceed 

 to study demonstration, he woidd imagine that his earliest in- 

 strument would be —ratio in all its varieties. He would be 

 surprised when he was told that, for a considerable time, he 

 would not be required to subdivide ratio into more than three 

 cases, ratio of equality, and the two forms of ratio of inequality 

 without any specification of the degree of inequality. But per- 

 haps he would be more surprised if he were told that, after this 

 renunciation of the difi"erent modes of inequality, geometers were 

 still unsatisfied whenever they had to reason from inequality to 

 equality. And if he were a logician, though by my supposition 

 one who had not applied his logic in mathematics, he would be 

 most surprised to know that geometers never made the con- 

 trapositive conversion of the universal affirmative except by an 

 indirect demonstration, and laid the blame on the essential cha- 

 racter of geometry, instead of laying it on their own neglect of 

 the study of the pure laws of thought, as they apply in geometry 

 and everything else. 



I have been led to offer these remarks at this particular time 

 by Mr. Sylvestei"'s paper contained in your last Number, as to 

 which I agree almost entirely with all that is Mr. Sylvester's 

 own, and differ only as to the view of the indirect proof which 

 he holds in common with most other geometers. I cannot 

 answer his invitation or challenge, because he will perhajis insist 

 upon my passing from the contrapositive to the positive form 

 only by an indirect demonstration. But I claim to see identity 

 in Every A is B and every not-B is not-A, by a process of 

 thought prior to syllogism : and, proving that the inequality of 

 the nearer segments makes the inequality of the remoter ones 

 follow, I conclude that the equality of the remoter ones makes 

 the equality of the nearer ones follow, as a new logical form of 

 the preceding conclusion, identical with it in meaning. Of 



