terms AD4 and AD3 there were interpoſed the ſeries AD13/6, AD11/5, AD9/4, AD7/3, AD5/2, AD8/3, AD11/4, AD14, AD17/6, &c. And again, between any two angles of this ſeries, a new ſeries of intermediate angles may be interpoſed, differing from one another by infinite intervals. Nor is nature confin'd to any bounds.
Thoſe things which have been demonſtrated of curve lines and the ſuperficies which they comprehend, may be easily applied to the curve ſuperficies and contents of ſolids. These lemmas are premiſed, to avoid the tediouſneſs of deducing perplexed demonſtrations ad abſurdum, according to the method of the ancient geometers. For demonſtrations are more contracted by the method of indiviſibles: but because the hypotheſis of indiviſibles ſeems ſomewhat harsh, and therefore that method is reckoned leſs geometrical, I chose rather to reduce the demonſtrations of the following propoſitions to the firſt and laſt ſums and ratio's of naſcent and evaneſcent quantities, that is, to the limits of those ſums and ratio's; and ſo to premiſe, as ſhort as I could, the demonſtrations of those limits. For hereby the ſame thing is perform'd as by the method of indiviſibles; and now thoſe principles being demonſtrated, we may use them with more ſafety. Therefore if hereafter, I ſhould happen to conſider quantities as made up of particles, or ſhould uſe little curve lines for right ones; I would not be underſtood to mean indiviſibles, but evaneſcent diviſible quantities; not the ſums and ratio's of determinate parts, but always the limits of ſums and ratio's: and that the force of such demonſtrations always depends on the method lay'd down in the foregoing lemma's.
Perhaps it may be objected, that there is no ultimate proportion of evaneſcent quantities; becauſe the
