1Sed, punctis Q& Pcoeuntibus, æquantur 2PC& Gv.Ergo & his pro
portionalia LXQR& QT quad.æquantur. Ducantur hæc æqualia in
(SPq/QR) & fiet LXSPqueæquale (SPq.XQTq/QR). Ergo (per Corol. 1
& 5 Prop. VI.) vis centripeta reciproce eſt ut LXSPqueid eſt, reci
proce in ratione duplicata diſtantiæ SP. Q.E.I.
portionalia LXQR& QT quad.æquantur. Ducantur hæc æqualia in
(SPq/QR) & fiet LXSPqueæquale (SPq.XQTq/QR). Ergo (per Corol. 1
& 5 Prop. VI.) vis centripeta reciproce eſt ut LXSPqueid eſt, reci
proce in ratione duplicata diſtantiæ SP. Q.E.I.
Cum vis ad centrum Ellipſeos tendens, qua corpus Pin Ellipſi
illa revolvi poteſt, ſit (per Corol. I Prop. X) ut CPdiſtantia cor
poris ab Ellipſeos centro C; ducatur CEparallela Ellipſeos tan
genti PR:& vis qua corpus idem P,circum aliud quodvis Ellip
ſeos punctum Srevolvi poteſt, ſi CE& PSconcurrant in E,erit ut
(PE cub./SPq) (per Corol. 3 Prop. VII,) hoc eſt, ſi punctum Sſit umbili
cus Ellipſeos, adeoque PEdetur, ut SPqreciproce. Q.E.I.
illa revolvi poteſt, ſit (per Corol. I Prop. X) ut CPdiſtantia cor
poris ab Ellipſeos centro C; ducatur CEparallela Ellipſeos tan
genti PR:& vis qua corpus idem P,circum aliud quodvis Ellip
ſeos punctum Srevolvi poteſt, ſi CE& PSconcurrant in E,erit ut
(PE cub./SPq) (per Corol. 3 Prop. VII,) hoc eſt, ſi punctum Sſit umbili
cus Ellipſeos, adeoque PEdetur, ut SPqreciproce. Q.E.I.
Eadem brevitate qua traduximus Problema quintum ad Parabo
lam, & Hyperbolam, liceret idem hic facere: verum ob dignita
tem Problematis & uſum ejus in ſequentibus, non pigebit caſus ce
teros demonſtratione confirmare.
lam, & Hyperbolam, liceret idem hic facere: verum ob dignita
tem Problematis & uſum ejus in ſequentibus, non pigebit caſus ce
teros demonſtratione confirmare.
PROPOSITIO XII. PROBLEMA. VII.
Sunto CA, CBſemi-axes Hyperbolæ; PG, KDdiametri con
jugatæ; PF, Qtperpendicula ad diametros; & Qvordinatim
applicata ad diametrum GP.Agatur SPſecans cum diametrum
DKin E,tum ordinatim applicatam Qvin x,& compleatur pa
rallelogrammum QRPx.Patet EPæqualem eſſe ſemiaxi tranſ
verſo AC,eo quod, acta ab altero Hyperbolæ umbilico Hlinea
HIipſi ECparallela, ob æquales CS, CH,æquentur ES, EI;
adeo ut EPſemidifferentia ſit ipſarum PS, PI,id eſt (ob pa
rallelas IH, PR& angulos æquales IPR, HPZ) ipſarum PS,
PH,quarum differentia axem totum 2ACadæquat. Ad SPde
mittatur perpendicularis QT.Et Hyperbolæ latere recto princi
pali (ſeu (2BCq/AC)) dicto L,erit LXQRad LXPvut QRad Pv,
id eſt, ut PEſeu ACad PC; Et LXPvad GvPut Lad
jugatæ; PF, Qtperpendicula ad diametros; & Qvordinatim
applicata ad diametrum GP.Agatur SPſecans cum diametrum
DKin E,tum ordinatim applicatam Qvin x,& compleatur pa
rallelogrammum QRPx.Patet EPæqualem eſſe ſemiaxi tranſ
verſo AC,eo quod, acta ab altero Hyperbolæ umbilico Hlinea
HIipſi ECparallela, ob æquales CS, CH,æquentur ES, EI;
adeo ut EPſemidifferentia ſit ipſarum PS, PI,id eſt (ob pa
rallelas IH, PR& angulos æquales IPR, HPZ) ipſarum PS,
PH,quarum differentia axem totum 2ACadæquat. Ad SPde
mittatur perpendicularis QT.Et Hyperbolæ latere recto princi
pali (ſeu (2BCq/AC)) dicto L,erit LXQRad LXPvut QRad Pv,
id eſt, ut PEſeu ACad PC; Et LXPvad GvPut Lad