404 FALLACIES. 



the premise, the proposition appeals directly and in concrete 

 language to the incapacity of the human imagination for con 

 ceiving a minimum. Viewed in this light, it becomes a case 

 of the a priori fallacy or natural prejudice, that whatever 

 cannot be conceived cannot exist. Every fallacy of Confusion 

 (it is almost unnecessary to repeat) will, if cleared up, become 

 a fallacy of some other sort ; and it will be found of deductive 

 or ratiocinative fallacies generally, that when they mislead, 

 there is mostly, as in this case, a fallacy of some other 

 description lurking under them, by virtue of which chiefly it 

 is that the verbal juggle, which is the outside or body of this 

 kind of fallacy, passes undetected. 



Euler s Algebra, a book otherwise of great merit, but full, 

 to overflowing, of logical errors in respect to the foundation 

 of the science, contains the following argument to prove that 

 minus multiplied by minus gives plus, a doctrine the oppro 

 brium of all mere mathematicians, and which Euler had not a 

 glimpse of the true method of proving. He says, minus mul 

 tiplied 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 premise requires proof, as 

 much as the conclusion : nor can it be proved, except by that 

 more comprehensive view of the nature of multiplication, and 

 of algebraic processes in general, which would also supply a 

 far better proof of the mysterious doctrine which Euler is here 

 endeavouring to demonstrate. 



A striking instance of reasoning in a circle is that of some 

 ethical writers, 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 



