JURIN V. ROBIN S AND PEMBERTON 97 



based by Robins upon the following two defini- 

 tions and certain general propositions annexed to 

 them : 



1. Definition : *' . . . we . . . define an ulti- 

 mate magnitude to be the limit, to which a varying 

 magnitude can approach within any degree of 

 nearness whatever, though it can never be made 

 absolutely equal to it." 



Here for the first time is the stand taken openly, 

 clearly, explicitly, that a variable (say the peri- 

 meter of a polygon inscribed in a circle) can never 

 reach its limit (the circumference). The gain from 

 the standpoint of debating is very great ; a regular 

 inscribed polygon vvhose sides are steadily doubling 

 at set intervals of time, say, every half second, 

 presents a picture to the imagination which is 

 comparatively simple. The hopeless attempt of 

 imagining the limit as reached need not be made. 

 But this great gain is made at the expense of 

 generality. Robins descends to a very special type 

 of variation which is not the variation encountered 

 in ordinary mechanics ; it is an exceedingly artificial 

 variation. According to Robins's definition, Achilles 

 never caught the tortoise. It would not be difficult 

 to assume a time rate in the doubling of the sides 

 of a polygon inscribed in a circle, so that the cir- 

 cumference is 7'eached. Thus, let the first doubling 

 of the number of sides take place in i second, the 

 second doubling in \ a second, the third m \ 2. 

 second, and so on. It is easy to see that under 

 this mode of variation the polygons do reach the 



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