IO LIMITS AND FLUXIONS 



ultimate velocity. And there is a like lìmit in ali 

 quantities and proportions that begin and cease to 

 be. And since such limits are certain and definite, 

 to determine the same is a problem strictly geo- 

 metrica!. But whatever is geometrical we may be 

 allowed to use in determining and demonstrating 

 any other thing that is likewise geometrical. 



15. '' It may be also argued, that if the ultimate 

 ratios of evanescent quantities are given, their 

 ultimate magnitudes will be also given ; and so ali 

 quantities will consist of indivisibles, which is.con- 

 trary to what Euclid has demonstrated concern- 

 ing incommensurables, in the tenth book of bis 

 Elements. But this objection is founded on a false 

 supposition, for those ultimate ratios with which 

 quantities vanish are not truly the ratios of ultimate 

 quantities, but the limits to which the ratios of 

 quantities, decreasing without end, always con- 

 verge ; and to which they may approach nearer 

 than by any difference, but can never go beyond, 

 nor attain to, unless the quantities are diminished 

 indefinitely. This will appear more evident in 

 quantities indefinitely great. If two quantities, 

 whose difference is given, are augmented continu- 

 ally, their ultimate ratio will be given, to wit, the 

 ratio of equality ; but the ultimate or greatest 

 quantities themselves, of which that is the ratio, 

 will not therefore be given. If then in what foUows, 

 for the more easy apprehension of things, I shall 

 ever mention quantities the least possihle, or evanes- 

 cent, or ultimate, beware lest you understand quan- 



