8 LIMITS AND FLUXIONS 



siquando facili rerum imaginationi ^ consulens, 

 dixero quantitates quam minimas, vel evanescentes 

 vel ultimas ; cave intelligas quantitates magnitudine 

 determinatas, sed cogita semper diminuendas sine 

 limite." 



Translation by Robert Thorp : 



13. "Those things which have been demonstrated 

 of curve lines, and the surfaces which they compre- 

 hend, are easily applied to the curve surfaces and 

 contents of solids. But I premised these lemmas 

 to avoid the tediousness of deducing long demon- 

 strations to an absurdity, according to the method 

 of the ancient geometers. For demonstrations are 

 rendered more concise by the method of indivisibles. 

 But, because the hypothesis of indivisibles is some- 

 what harsh, and therefore that method is esteemed 

 less geometrical, I chose rather to reduce the 

 demonstrations of the following propositions to the 

 prime and ultimate sums and ratios of nascent and 

 evanescent quantities ; that is, to the limits of those 

 sums and ratios : and so to premise the demonstra- 

 tions of those limits, as briefly as 1 could. For 

 hereby the same thing is performed, as by the 

 method of indivisibles ; and those principles being 

 demonstrated, we may now use them with more 

 safety. Therefore, if hereafter I shall happen to 

 consider quantities, as made up of particles, or shall 

 use little curve lines for right ones, I would not be 

 understood to mean indivisible, but evanescent 



' In the third edition " conceptui-" takes the place of "imaginationi." 



