FALLACIES OF CONFUSION. 



535 



terminology. I refer to the famous 

 dispute respecting the vis viva, the 

 history of which is given at large in 

 Professor Playfair's Dissertation. The 

 question was whether the force of a 

 moving body was proportional (its 

 mass being given) to its velocity sim- 

 ply, or to the square of its velocity ; 

 and the ambiguity was in the word 

 Force. "One of the efifects," says 

 Playfair, " produced by a moving 

 body is proportional to the square of 

 the velocity, while another is propor- 

 tional to the velocity simply ; " from 

 whence clearer thinkers were subse- 

 quently 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 question was, with 

 which of the two effects the term 

 force should be, or could most con- 

 veniently 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 logi- 

 cal puzzle of Achilles and the Tortoise, 

 a puzzle which has been too hard for 

 the ingenuity or patience of many 

 philosophers, and which no less a 

 thinker than Sir William Hamilton 

 considered as insoluble, as a sound 

 argument though leading to a palpable 

 falsehood. The fallacy, as Hobbes 

 hinted, lies in the tacit assumption 

 that whatever is infinitely divisible is 

 infinite ; but the following solution 

 (to the invention of which I have 

 no claim) is more precise and satis- 

 factory. 



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 sup- 

 pose them to be at fii-st separated by 



an interval of a thousand feet : when 

 Achilles has run these thousand feet, 

 the tortoise will have got on a hun- 

 dred ; 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 conclu- 

 sion means, for any length of time 

 that can be supposed ; but in the pre- 

 mises, " 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 bean end to the 

 subdivisions of the distance, nor conse- 

 quently to those of the time in which 

 it is performed. But an unlimited 

 number of subdivisions may be made 

 of that which is itsdf 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 re- 

 mains 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 

 altogether. It proves, in short, that 

 to pass through this finite space re- 

 quires a time which is infinitely divi- 

 sible, but not an in finite time — thecon- 

 founding 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 obvious 

 and familiar one of a right and the 

 adjective right) are extracted from a 

 forgotten paper of my own, in a perio- 

 dical : — 



" Speaking morally, you are said to 

 have a right to do a thing, if all per- 

 sons are morally bound not to hinder 

 you from doing it. But, in another 

 sense, to have a right to do a thing is 

 the opposite of having no right to do 

 it, i.e. of being under a moral obliga- 

 tion to forbear doing it. In this 

 sense, to say that you have a right to 

 do a thing means that you may do it 



