1
SECTIO III.
PROPOSITIO XI. PROBLEMA VI.
Revolvatur corpus in Ellipſi: requiritur Lex vis centripetæ tenden
tis ad umbilicum Ellipſeos.
tis ad umbilicum Ellipſeos.
Eſto Ellipſeos umbilicus S.Agatur SPſecans Ellipſeos
tum diametrum DKin E,tum ordinatim applicatam Qvin
x,& compleatur parallelogrammum QxPR.Patet EPæqua
lem eſſe ſemiaxi ma
21[Figure 21]
jori AC,eo quod
acta ab altero Ellip
ſeos umbilico Hli
nea HIipſi ECpa
rallela, (ob æquales
CS, CH) æquentur
ES, EI,adeo ut EP
ſemiſumma ſit ipſa
rum PS, PI,id eſt
(ob parallelas HI,
PR& angulos æqua
les IPR, HPZ)
ipſarum PS, PH,
quæ conjunctim axem
totum 2ACadæ
quant. Ad SPde
mittatur perpendicularis QT,& Ellipſeos latere recto principali
(ſeu (2BC quad./AC)) dicto L,erit LXQRad LXPvut QRad
Pv,id eſt ut PEſeu ACad PC; & LXPvad GvPut Lad
Gv; & GvPad Qv quad.ut PC quad.ad CD quad; & (per Corol.
2 Lem. VII.) Qv quad.ad Qx quad,punctis Q& Pcoeuntibus,
eſt ratio æqualitatis; & Qx quad.ſeu Qv quad.eſt ad QT quad.
ut EP quad.ad PF quad,id eſt ut CA quad.ad PF quad.ſive (per
Lem XII.) ut CD quad.ad CB quad.Et conjunctis his omnibus ratio
nibus, LXQRfit ad QT quad.ut ACXLXPCq.XCDqueſeu 2CBque
XPCq.XCDquead PCXGvXCDq.XCBqueſive ut 2PCad Gv.
tum diametrum DKin E,tum ordinatim applicatam Qvin
x,& compleatur parallelogrammum QxPR.Patet EPæqua
lem eſſe ſemiaxi ma
21[Figure 21]jori AC,eo quod
acta ab altero Ellip
ſeos umbilico Hli
nea HIipſi ECpa
rallela, (ob æquales
CS, CH) æquentur
ES, EI,adeo ut EP
ſemiſumma ſit ipſa
rum PS, PI,id eſt
(ob parallelas HI,
PR& angulos æqua
les IPR, HPZ)
ipſarum PS, PH,
quæ conjunctim axem
totum 2ACadæ
quant. Ad SPde
mittatur perpendicularis QT,& Ellipſeos latere recto principali
(ſeu (2BC quad./AC)) dicto L,erit LXQRad LXPvut QRad
Pv,id eſt ut PEſeu ACad PC; & LXPvad GvPut Lad
Gv; & GvPad Qv quad.ut PC quad.ad CD quad; & (per Corol.
2 Lem. VII.) Qv quad.ad Qx quad,punctis Q& Pcoeuntibus,
eſt ratio æqualitatis; & Qx quad.ſeu Qv quad.eſt ad QT quad.
ut EP quad.ad PF quad,id eſt ut CA quad.ad PF quad.ſive (per
Lem XII.) ut CD quad.ad CB quad.Et conjunctis his omnibus ratio
nibus, LXQRfit ad QT quad.ut ACXLXPCq.XCDqueſeu 2CBque
XPCq.XCDquead PCXGvXCDq.XCBqueſive ut 2PCad Gv.