Corol.3. Vis, qua corpus Pin Orbe quocunque circum virium
centrum Srevolvitur, eſt ad vim qua corpus idem Pin eodem
orbe eodemque tempore periodico circum aliud quodvis virium
centrum Rrevolvi poteſt, ut SPXRPqcontentum utique ſub di
ſtantia corporis a primo virium centro S& quadrato diſtantiæ ejus
a ſecundo virium centro Rad cubum rectæ SGquæ a primo vi
rium centro Sad orbis tangentem PGducitur, & corporis a ſe
cundo virium centro diſtantiæ RPparallela eſt. Nam vires in
hoc Orbe, ad ejus punctum quodvis P,eædem ſunt ac in Circulo
ejuſdem curvaturæ.
centrum Srevolvitur, eſt ad vim qua corpus idem Pin eodem
orbe eodemque tempore periodico circum aliud quodvis virium
centrum Rrevolvi poteſt, ut SPXRPqcontentum utique ſub di
ſtantia corporis a primo virium centro S& quadrato diſtantiæ ejus
a ſecundo virium centro Rad cubum rectæ SGquæ a primo vi
rium centro Sad orbis tangentem PGducitur, & corporis a ſe
cundo virium centro diſtantiæ RPparallela eſt. Nam vires in
hoc Orbe, ad ejus punctum quodvis P,eædem ſunt ac in Circulo
ejuſdem curvaturæ.
PROPOSITIO. VIII. PROBLEMA. III.
Moveatur corpus in CirculoPQA: ad hunc effectum requiritur Lex
vis centripetæ tendentis ad punctum adeo longinquumS, ut lineæ
omnesPS, RS ad id ductæ, pro parallelis haberi poſſint.
vis centripetæ tendentis ad punctum adeo longinquumS, ut lineæ
omnesPS, RS ad id ductæ, pro parallelis haberi poſſint.
A Circuli centro Cagatur ſemidiameter CAparallelas iſtas
perpendiculariter ſecans in M&
18[Figure 18]
N,& jungatur CP.Ob ſimilia
triangula CPM, PZT& RZQ
eſt CPqad PMqut PRqad
QTq& ex natura Circuli PRq
æquale eſt rectangulo QRX√RN+QN&c.
ſive coeuntibus punctis P, Qrect
angulo QRX2PM.Ergo eſt
CPqad PM quad.ut QRX2PM
ad QT quad.adeoque (QT quad./QR)
æquale (2PM cub./CP quad.), & (QT quad.XSP quad./QR) æquale (2PM cub.XSP qu./CP quad.)
Eſt ergo (per Corol. 1 & 5 Prop. VI.) vis centripeta reciproce ut
(2PMcub.XSP quad./CP quad.) hoc eſt (neglecta ratione determinata (2SP quad./CP quad.))
reciproce ut PM cub. que E. I.
perpendiculariter ſecans in M&
18[Figure 18]N,& jungatur CP.Ob ſimilia
triangula CPM, PZT& RZQ
eſt CPqad PMqut PRqad
QTq& ex natura Circuli PRq
æquale eſt rectangulo QRX√RN+QN&c.
ſive coeuntibus punctis P, Qrect
angulo QRX2PM.Ergo eſt
CPqad PM quad.ut QRX2PM
ad QT quad.adeoque (QT quad./QR)
æquale (2PM cub./CP quad.), & (QT quad.XSP quad./QR) æquale (2PM cub.XSP qu./CP quad.)
Eſt ergo (per Corol. 1 & 5 Prop. VI.) vis centripeta reciproce ut
(2PMcub.XSP quad./CP quad.) hoc eſt (neglecta ratione determinata (2SP quad./CP quad.))
reciproce ut PM cub. que E. I.