1in f,conveniant autem hæ tangentes in axe TQad Y; & ſi
MLdeſignet ſpatium quod Luna in Circulo revolvens, interea
dum deſcribit arcum PM,urgente & impellente vi prædicta
3IT,motu tranſverſo deſcribere poſſet, & mldeſignet ſpatium
quod Luna in Ellipſi revolvens eodem tempore, urgente etiam vi
3IT,deſcribere poſſet; & producantur LP& lpdonec occurrant
plano Eclipticæ in G& g; & jungantur FG& fg,quarum FG
producta ſecet pf, pg& TQin c, e& Rreſpective, & fgpro
ducta ſecet TQin r: Quoniam vis 3ITſeu 3PKin Circulo
eſt ad vim 3ITſeu 3pKin Ellipſi, ut PKad pK,ſeu ATad
aT; erit ſpatium MLvi priore genitum, ad ſpatium mlvi po
ſteriore genitum, ut PKad pK,id eſt, ob ſimiles figuras
PYKp& FYRc,ut FRad cR.Eſt autem MLad FG(ob
ſimilia triangula PLM, PGF) ut PLad PG,hoc eſt (ob
parallelas Lk, PK, GR) ut plad pe,id eſt, (ob ſimilia trian
gula plm, cpe) ut lmad ce; & inverſe ut LMeſt ad lm,ſeu
FRad cR,ita eſt FGad ce.Et propterea ſi fgeſſet ad ceut
fYad cY,id eſt, ut frad cR(hoc eſt, ut frad FR& FRad cR
conjunctim, id eſt, ut fTad FT& FGad ceconjunctim,) quo
niam ratio FGad ceutrinque ablata relinquit rationes fgad FG
& fTad FT,foret fgad FGut fTad FT; atque adeo anguli,
quos FG& fgſubtenderent ad Terram T,æquarentur inter ſe.
Sed anguli illi (per ca quæ in præcedente Propoſitione expoſui
mus) ſunt motus Nodorum, quo tempore Luna in Circulo ar
cum PM,in Ellipſi arcum pmpercurrit: & propterea motus
Nodorum in Circulo & Ellipſi æquarentur inter ſe. Hæc ita ſe
haberent, ſi modo fgeſſet ad ceut fYad cY,id eſt, ſi fgæqua
lis eſſet (ceXfY/cY). Verum ob ſimilia triangula fgp, cep,eſt fg
ad ceut fpad cp; ideoque fgæqualis eſt (ceXfp/cp); & propterea
angulus, quem fgrevera ſubtendit, eſt ad angulum priorem, quem
FGſubtendit, hoc eſt, motus Nodorum in Ellipſi ad motum
Nodorum in Circulo, ut hæc fgſeu (ceXfp/cp) ad priorem fgſeu
(ceXfY/cY), id eſt, ut fpXcYad fYXcp,ſeu fpad fY& cYad cp,
hoc eſt, ſi phipſi TNparallela occurrat FPin h,ut Fhad FY
& FYad FP; hoc eſt, ut Fhad FPſeu Dpad DP,adeoque
ut area Dpmdad aream DPMd.Et propterea, cum area po-
MLdeſignet ſpatium quod Luna in Circulo revolvens, interea
dum deſcribit arcum PM,urgente & impellente vi prædicta
3IT,motu tranſverſo deſcribere poſſet, & mldeſignet ſpatium
quod Luna in Ellipſi revolvens eodem tempore, urgente etiam vi
3IT,deſcribere poſſet; & producantur LP& lpdonec occurrant
plano Eclipticæ in G& g; & jungantur FG& fg,quarum FG
producta ſecet pf, pg& TQin c, e& Rreſpective, & fgpro
ducta ſecet TQin r: Quoniam vis 3ITſeu 3PKin Circulo
eſt ad vim 3ITſeu 3pKin Ellipſi, ut PKad pK,ſeu ATad
aT; erit ſpatium MLvi priore genitum, ad ſpatium mlvi po
ſteriore genitum, ut PKad pK,id eſt, ob ſimiles figuras
PYKp& FYRc,ut FRad cR.Eſt autem MLad FG(ob
ſimilia triangula PLM, PGF) ut PLad PG,hoc eſt (ob
parallelas Lk, PK, GR) ut plad pe,id eſt, (ob ſimilia trian
gula plm, cpe) ut lmad ce; & inverſe ut LMeſt ad lm,ſeu
FRad cR,ita eſt FGad ce.Et propterea ſi fgeſſet ad ceut
fYad cY,id eſt, ut frad cR(hoc eſt, ut frad FR& FRad cR
conjunctim, id eſt, ut fTad FT& FGad ceconjunctim,) quo
niam ratio FGad ceutrinque ablata relinquit rationes fgad FG
& fTad FT,foret fgad FGut fTad FT; atque adeo anguli,
quos FG& fgſubtenderent ad Terram T,æquarentur inter ſe.
Sed anguli illi (per ca quæ in præcedente Propoſitione expoſui
mus) ſunt motus Nodorum, quo tempore Luna in Circulo ar
cum PM,in Ellipſi arcum pmpercurrit: & propterea motus
Nodorum in Circulo & Ellipſi æquarentur inter ſe. Hæc ita ſe
haberent, ſi modo fgeſſet ad ceut fYad cY,id eſt, ſi fgæqua
lis eſſet (ceXfY/cY). Verum ob ſimilia triangula fgp, cep,eſt fg
ad ceut fpad cp; ideoque fgæqualis eſt (ceXfp/cp); & propterea
angulus, quem fgrevera ſubtendit, eſt ad angulum priorem, quem
FGſubtendit, hoc eſt, motus Nodorum in Ellipſi ad motum
Nodorum in Circulo, ut hæc fgſeu (ceXfp/cp) ad priorem fgſeu
(ceXfY/cY), id eſt, ut fpXcYad fYXcp,ſeu fpad fY& cYad cp,
hoc eſt, ſi phipſi TNparallela occurrat FPin h,ut Fhad FY
& FYad FP; hoc eſt, ut Fhad FPſeu Dpad DP,adeoque
ut area Dpmdad aream DPMd.Et propterea, cum area po-
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