1ac denique per punctum Qagatur LRquæ ipſi SPparallela
ſit & occurrat tum circulo in Ltum tangenti PZin R.Et
ob ſimilia triangula ZQR, ZTP, VPA; erit RP quad.hoc
eſt QRLad QT quad.ut AV quad.ad PV quad.Ideoque
(QRLXPV quad./AV quad.) æquatur QT quad.Ducantur hæc æqualia in
(SP quad./QR) &, punctis P& Qcoeuntibus, ſcribatur PVpro RL.
Sic fiet (SP quad.XPV cub./AV quad.) æquale (SP quad.XQT quad./QR) Ergo (per
Corol.1 & 5 Prop.VI.) vis centripeta eſt reciproce ut (SPqXPV cub./AV quad)
id eſt, (ob datum AV quad.) reciproce ut quadratum diſtantiæ ſeu
altitudinis SP& cubus chordæ PVconjunctim. Q.E.I.
ſit & occurrat tum circulo in Ltum tangenti PZin R.Et
ob ſimilia triangula ZQR, ZTP, VPA; erit RP quad.hoc
eſt QRLad QT quad.ut AV quad.ad PV quad.Ideoque
(QRLXPV quad./AV quad.) æquatur QT quad.Ducantur hæc æqualia in
(SP quad./QR) &, punctis P& Qcoeuntibus, ſcribatur PVpro RL.
Sic fiet (SP quad.XPV cub./AV quad.) æquale (SP quad.XQT quad./QR) Ergo (per
Corol.1 & 5 Prop.VI.) vis centripeta eſt reciproce ut (SPqXPV cub./AV quad)
id eſt, (ob datum AV quad.) reciproce ut quadratum diſtantiæ ſeu
altitudinis SP& cubus chordæ PVconjunctim. Q.E.I.
Idem aliter.
Ad tangentem PRproductam demittatur perpendiculum SY,
& ob ſimilia triangula SYP, VPA; erit AVad PVut SPad
SY,ideoque (SPXPV/AV) æquale SY,& (SP quad.XPV cub./AV quad.) æquale
SY quad.XPV.Et propterea (per Corol.3 & 5 Prop.VI.) vis centri
peta eſt reciproce ut (SPqXPV cub./AVq) hoc eſt, ob datam AV,reci
proce ut SPqXPV cub. que E. I.
& ob ſimilia triangula SYP, VPA; erit AVad PVut SPad
SY,ideoque (SPXPV/AV) æquale SY,& (SP quad.XPV cub./AV quad.) æquale
SY quad.XPV.Et propterea (per Corol.3 & 5 Prop.VI.) vis centri
peta eſt reciproce ut (SPqXPV cub./AVq) hoc eſt, ob datam AV,reci
proce ut SPqXPV cub. que E. I.
Corol.1. Hinc ſi punctum datum Sad quod vis centripeta ſem
per tendit, locetur in circumferentia hujus circuli, puta ad V; erit
vis centripeta reciproce ut quadrato cubus altitudinis SP.
per tendit, locetur in circumferentia hujus circuli, puta ad V; erit
vis centripeta reciproce ut quadrato cubus altitudinis SP.
Corol.2. Vis qua corpus Pin cir
17[Figure 17]
culo APTVcircum virium centrum
Srevolvitur, eſt ad vim qua corpus
idem Pin eodem circulo & eodem
tempore periodico circum aliud quod
vis virium centrum Rrevolvi poteſt,
ut RP quad.XSPad cubum rectæ SG
quæ a primo virium centro Sad or
bis tangentem PGducitur, & diſtan
tiæ corporis a ſecundo virium centro
parallela eſt. Nam, per conſtructionem hujus Propoſitionis, vis
prior eſt ad vim poſteriorem, ut RPqXPT cub.ad SPqXPV cub.
17[Figure 17]culo APTVcircum virium centrum
Srevolvitur, eſt ad vim qua corpus
idem Pin eodem circulo & eodem
tempore periodico circum aliud quod
vis virium centrum Rrevolvi poteſt,
ut RP quad.XSPad cubum rectæ SG
quæ a primo virium centro Sad or
bis tangentem PGducitur, & diſtan
tiæ corporis a ſecundo virium centro
parallela eſt. Nam, per conſtructionem hujus Propoſitionis, vis
prior eſt ad vim poſteriorem, ut RPqXPT cub.ad SPqXPV cub.