1
LEMMA XX.
Si Parallelogrammum quodvisASPQ angulis duobus oppoſitisA &
P tangit ſectionem quamvis Conicam in punctisA &P; &, lateri
bus unius angulorum illorum infinite productisAQ, AS, occurrit
eidem ſectioni Conicæ inB &C; a punctis autem occurſuumB &
C ad quintum quodvis ſectionis Conicæ punctumD agantur rec
tæ duæBD, CD occurrentes alteris duobus infinite productis pa
rallelogrammi lateribusPS, PQ inT &R: erunt ſemper abſciſſæ
laterum partesPR &PT adinvicem in data ratione. Et contra, ſi
partes illæ abſciſſæ ſunt ad invicem in data ratione, punctumD tan
get Sectionem Conicam per puncta quatuorA, B, C, P tranſeuntem.
P tangit ſectionem quamvis Conicam in punctisA &P; &, lateri
bus unius angulorum illorum infinite productisAQ, AS, occurrit
eidem ſectioni Conicæ inB &C; a punctis autem occurſuumB &
C ad quintum quodvis ſectionis Conicæ punctumD agantur rec
tæ duæBD, CD occurrentes alteris duobus infinite productis pa
rallelogrammi lateribusPS, PQ inT &R: erunt ſemper abſciſſæ
laterum partesPR &PT adinvicem in data ratione. Et contra, ſi
partes illæ abſciſſæ ſunt ad invicem in data ratione, punctumD tan
get Sectionem Conicam per puncta quatuorA, B, C, P tranſeuntem.
Cas.1. Jungantur BP, CP& a puncto Dagantur rectæ duæ
DG, DE,quarum prior
45[Figure 45]
DGipſi ABparallela ſit &
occurrat PB, PQ, CAin
H, I, G; altera DEparal
lela ſit ipfi AC& occurrat
PC, PS, ABin F, K, E:
& erit (per Lemma XVII.) re
ctangulum DEXDFad re
ctangulum DGXDHin ra
tione data. Sed eſt PQad
DE(ſeu IQ) ut PBad HB,
adeoque ut PTad DH; &
viciſſim PQad PTut DEad DH.Eſt & PRad DFut RC
ad DC,adeoque ut (IGvel) PSad DG,& viciſſim PRad PS
ut DFad DG; & conjunctis rationibus fit rectangulum PQXPR
ad rectangulum PSXPTut rectangulum DEXDFad rectan
gulum DGXDH,atque adeo in data ratione. Sed dantur PQ
& PS& propterea ratio PRad PTdatur. Q.E.D.
DG, DE,quarum prior
45[Figure 45]DGipſi ABparallela ſit &
occurrat PB, PQ, CAin
H, I, G; altera DEparal
lela ſit ipfi AC& occurrat
PC, PS, ABin F, K, E:
& erit (per Lemma XVII.) re
ctangulum DEXDFad re
ctangulum DGXDHin ra
tione data. Sed eſt PQad
DE(ſeu IQ) ut PBad HB,
adeoque ut PTad DH; &
viciſſim PQad PTut DEad DH.Eſt & PRad DFut RC
ad DC,adeoque ut (IGvel) PSad DG,& viciſſim PRad PS
ut DFad DG; & conjunctis rationibus fit rectangulum PQXPR
ad rectangulum PSXPTut rectangulum DEXDFad rectan
gulum DGXDH,atque adeo in data ratione. Sed dantur PQ
& PS& propterea ratio PRad PTdatur. Q.E.D.