FALLACIES OF CONFUSION. 393 



Playfair, "produced by a moving body is proportional to the 

 square of the velocity, while another is proportional 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 conveniently be, asso- 

 ciated. But the disputants were by no means aware that this 

 was all ; they thought that force was one thing, the produc- 

 tion 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 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 

 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 premises 

 " 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 



