Newton, Isaac - Philosophia naturalis principia mathematica

Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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PROPOSITIO LXXXI. PROBLEMA XLI.

Stantibus jam poſitis, menſuranda est AreaABNA.

A puncto P ducatur recta PHSphæram tangens in H,& ad
axem PABdemiſſa normali HI,biſecetur PIin L;& erit
(per Prop. 12, Lib. 2. Elem.) PEqæquale PSq + SEq+
2PSD.Eſt autem SEqſeu SHq(ob ſimilitudinem triangu
lorum SPH, SHI) æquale rectangulo PSI.Ergo PEqæquale
eſt contento ſub PS& PS+SI+2SD,hoc eſt, ſub PS&
2LS+2SD,id eſt, ſub PS& 2LD.Porro DE quadæquale
eſt SEq-SDq,ſeu SEq -LSq+2SLD-LDq,id eſt,
2SLD-LDq-ALB.Nam LSq-SEqſeu LSq-SAq

121[Figure 121]
(per Prop. 6, Lib. 2. Elem.) æquatur rectangulo ALB.Scriba
tur itaque 2SLD -LDq -ALBpro DEq; & quantitas
(DEqXPS/PEXV), quæ ſecundum Corollarium quartum Propoſitionis
præcedentis eſt ut longitudo ordinatim applicatæ DN,reſolvet
ſeſe in tres partes (2SLDXPS/PEXV)-(LDqXPS/PEXV)-(ALBXPS/PEXV):
ubi ſi pro V ſcribatur ratio inverſa vis centripetæ, & pro PE me
dium proportionale inter PS& 2LD; tres illæ partes evadent
ordinatim applicatæ linearum totidem curvarum, quarum areæ per
Methodos vulgatas innoteſcunt. que E. F.

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