Cyclomathesis : or, An easy introduction to the several branches of the mathematics; being principally designed for the instruction of young students, before they enter upon the more abtruse and difficult parts thereof . ce of gravity, as in thefe proportions. SECT. IV. In Prop 15.[And the decrements of thefe arcs arifing fromthe refiftance, will be as the fquares of the timesin which they are generated {\ ior fince the arcs areinfinitely fmall, the refiftance affects the fame as anuniform centripetal force would do,0 [ib. The decrement of the arch PQ^will be]for let q, t be the points it woul
Cyclomathesis : or, An easy introduction to the several branches of the mathematics; being principally designed for the instruction of young students, before they enter upon the more abtruse and difficult parts thereof . ce of gravity, as in thefe proportions. SECT. IV. In Prop 15.[And the decrements of thefe arcs arifing fromthe refiftance, will be as the fquares of the timesin which they are generated {\ ior fince the arcs areinfinitely fmall, the refiftance affects the fame as anuniform centripetal force would do,0 [ib. The decrement of the arch PQ^will be]for let q, t be the points it would arrive at, in thefame times, in free fpace, then Pq zz qt9 and PQ =1 Qr, thereforeQ^ z= qt —- Qr; and adding Q^, zQq — Q/ —Qr=/r; and Q^ — ~ir: but Q^ — ^R/ ; there-fore tr = ~R/ — Rr, and Qq = ~Rk The fame demonftration of this Prop, holdsgood, as well in afcending as in defcending motion. 41- [Cor. 1. The velocity, &c] for fuppofing ?Q^an arch of a circle whole center is f\ then if PQ^or PT be as the velocity in both curves, TQ willbe as the centripetal force in both, but in both curves TO = ^Q -, therefore ?f— rad. of the • 2tJ/circle ddcribed with the fame velocity. [Corf Fig. 34 35 . ju. Pmifjpias ~f>a J6. B. II. the PRINCIPIA. 5J [Cor. 4. the body will defcend, &c] for by cor. the velocity in P in a refitting medium is zz ve-locity in a circle at diftance SP in free fpace; butvelocity at diftance SP in free fpace : to velocity atdill. 4-SP in free fpace (by cor. 6. Prop. 4. B. I.) or Prop. 34. : : as j^p : t0 J r|p:: that is> •* ; v/2 •, therefore, &c. [Cor. 7, or as -JAS to AB] (for this fee my Geo-metrical Proportion, prop. 24.) In Prop. 16.[And the refiftance, &c] inftead of this propor-tion in prop. 15, viz. PQj QR :: SQ : \/bPxSQ, fubftitute this PQj QR : : (SQJ : SPr : :) SQj SQ + — SP — - SQ, (by prop. 24, of my Geo- 2 2 met. Proportion) that is, as the defcribing veloci-ties, &c. [Cor. 1.] For the refiftance : centripet
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