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LIBER
PRIMUS.
PRIMUS.
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.
Cas.2. Quod ſi PR& PTponantur in data ratione ad invi
cem, tum ſimili ratiocinio regrediendo, ſequetur eſſe rectangulum
DEXDFad rectangulum DGXDHin ratione data, adeoque
punctum D(per Lemma XVIII.) contingere Conicam ſectionem
tranſeuntem per puncta A, B, C, P. Q.E.D.
cem, tum ſimili ratiocinio regrediendo, ſequetur eſſe rectangulum
DEXDFad rectangulum DGXDHin ratione data, adeoque
punctum D(per Lemma XVIII.) contingere Conicam ſectionem
tranſeuntem per puncta A, B, C, P. Q.E.D.
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