508 FALLACIES. 



word Force. " One of the effects," says Playfaiz', " produced by a 

 moving body is proportional to the square of the velocity, w^hile another 

 is proportional to tlie velocity simply :" fi-om w^hence clearer thinkers 

 were subsequently led to establish a double measure of the efficiency 

 of a moving power, one being called vis viva, and the other momentum. 

 About the facts, both parties were from the first agreed : the only ques- 

 tion was, with which of the two effects the term force should be, or 

 could most conveniently be, associated. But the disputants were by 

 no means aware that this was all ; they thought that force was one thing, 

 the production of effects another : and the question, by which set of 

 effects the force which produced both the one and the other should be 

 measured, was supposed to be a question not of terminology but of fact. 



The ambiguity of the word Infinite is the real fallacy in the amusing 

 logical puzzle of Achilles and the Tortoise, a puzzle which has been 

 too hard for the ingenuity or patience of many philosophers, and among 

 others of Dr. Thomas Brown, who considered the sophism as insoluble; 

 as a sound argument, though leading to a palpable falsehood: not seeing 

 that such an admission would be a reductio ad absurdum of the reason- 

 ing faculty itself The fallacy, as Hobbes hinted, lies in the tacit as- 

 sumption that whatever is infinitely divisible is infinite ; but the follow- 

 ing solution (to the invention of which I have no claim) is more precise 

 and satisfactory. 



The argument is, let Achilles run ten times as fast as the tortoise, 

 yet if the tortoise has the start, Achilles will never overtake him. For 

 suppose them to be at first separated by an interval of a thousand feet : 

 when Achilles has run these thousand feet, the tortoise will have got 

 on a hundred ; when Achilles has run those hundred, the tortoise will 

 have run ten, and so on for ever ; therefore Achilles may run for ever 

 without overtaking the tortoise. 



Now, the " for ever" in the conclusion, means, for any length of time 

 that can be supposed ; but in the premisses " ever" does not mean any 

 length of time : it means any number of subdivisions of time. It means 

 that we may divide a thousand feet by ten, and that quotient again by 

 ten, and so on as often as we please ; that there never needs be an end 

 to the subdivisions of the distance, nor consequently to those of the 

 time in which it is performed. But an unlimited number of subdivi- 

 sions may be made of that which is itself limited. The argument 

 proves no other infinity of duration than may be embraced within five 

 minutes. As long as the five minutes are not expired, what remains 

 of them may be divided by ten, and again by ten, as often as we like, 

 which is perfectly compatible with their being only five minutes alto- 

 gether. It proves, in short, that to pass through this finite space re- 

 quires a time which is infinitely divisible, but not an infinite time ; the 

 confounding of which distinction Hobbes had already seen to be the 

 gist of the fallacy. 



The following ambiguities of the word right (in addition to the ob- 

 vious and familiar one of a right and the adjective right) are abstracted 

 from a forgotten paper of my own, in a periodical work : — 



"Speaking morally, you are said to have a right to do a thing, if all 

 persons are morally bound not to hinder you from doing it. But, m 

 another sense, to have a right to do a thing, is the opposite of having 

 no right to do it, viz., of being under a moral obligation to forbeai- 

 from doing it. In this sense, to say that you have a right to do a 



