Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/100

 terms AD4 and AD3 there were interpoed the eries AD13/6, AD11/5, AD9/4, AD7/3, AD5/2, AD8/3, AD11/4, AD14probably a misprint for AD^14&#47;5 [sic], AD17/6, &c. And again, between any two angles of this eries, a new eries of intermediate angles may be interpoed, differing from one another by infinite intervals. Nor is nature confin'd to any bounds.

Thoe things which have been demontrated of curve lines and the uperficies which they comprehend, may be easily applied to the curve uperficies and contents of olids. These lemmas are premied, to avoid the tediounes of deducing perplexed demontrations ad aburdum, according to the method of the ancient geometers. For demontrations are more contracted by the method of indiviibles: but because the hypotheis of indiviibles eems omewhat harsh, and therefore that method is reckoned les geometrical, I chose rather to reduce the demontrations of the following propoitions to the firt and lat ums and ratio's of nacent and evanecent quantities, that is, to the limits of those ums and ratio's; and o to premie, as hort as I could, the demontrations of those limits. For hereby the ame thing is perform'd as by the method of indiviibles; and now thoe principles being demontrated, we may use them with more afety. Therefore if hereafter, I hould happen to conider quantities as made up of particles, or hould ue little curve lines for right ones; I would not be undertood to mean indiviibles, but evanecent diviible quantities; not the ums and ratio's of determinate parts, but always the limits of ums and ratio's: and that the force of such demontrations 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 evanecent quantities; becaue the