466 FALLACIES. 



ing argument to prove that minus multiplied by minus 

 gives plus, a doctrine the opprobrium of all mathema- 

 ticians who are not philosophers, and which Euler had 

 not a glimpse of the true method of proving. He 

 says, minus multiplied by minus cannot give minus ; 

 for minus multiplied by plus gives minus, and minus 

 multiplied by minus cannot give the same product as 

 minus multiplied by plus. Now one is obliged to ask. 

 why minus multiplied by minus must give any product 

 at all ? and if it does, why its product cannot be the 

 same as that of minus multiplied by plus ? for this 

 would seem, at the first glance, not more absurd 

 than that minus by minus should give the same as 

 plus by plus, the proposition which Euler prefers to it. 

 The premiss requires proof, as much as the conclusion : 

 nor can it be proved, except by that more comprehen- 

 sive view of the nature of multiplication, and of alge- 

 braic processes in general, which would also supply 

 a far better proof of the mysterious doctrine which 

 Euler is here endeavouring to demonstrate. 



A very striking instance of reasoning in a circle is 

 that of some ethical philosophers, who first take for 

 their standard of moral truth what, being the general, 

 they deem to be the natural or instinctive, sentiments 

 and perceptions of mankind, and then explain away 

 the numerous instances of divergence from their 

 assumed standard by representing them as cases in 

 which the perceptions are unhealthy. Some particular 

 mode of conduct or feeling is affirmed to be unnatural; 

 why? because it is abhorrent to the universal and 

 natural sentiments of mankind. Finding no such 

 sentiment in yourself, you question the fact; and the 

 answer is (if your antagonist is polite,) that you are 

 an exception, a peculiar case. But neither (say you) 

 do I find in the people of some other country, or of 



