172445ET HYPERBOLÆ QUADRATURA.
F D M K, P L M D.
dico triangulum A F L eſſe medium arith-
meticum inter parallelogramma F D M K, P L M D. Grego-
rius à S. Vincentio in Lib. de Hyperbola demonſtrat triangu-
lum A F L eſſe æquale trapezio D F L M, ſed manifeſtum eſt
trapezium D F L M eſſe medium arithmeticum inter paral-
lelogramma F D M K, P L M D; & ideo patet propo-
ſitum.
meticum inter parallelogramma F D M K, P L M D. Grego-
rius à S. Vincentio in Lib. de Hyperbola demonſtrat triangu-
lum A F L eſſe æquale trapezio D F L M, ſed manifeſtum eſt
trapezium D F L M eſſe medium arithmeticum inter paral-
lelogramma F D M K, P L M D; & ideo patet propo-
ſitum.
PROP. XXVII. THEOREMA.
Iisdem poſitis:
ducatur A I rectam F L bifariam dividens in
11TAB.
XLVIII.
fig. 4. I & hyperbolam interſecans in puncto G, fiatque trape-
zium ſectori circumſcriptum A F G L, quod dico eſſe me-
dium geometricum inter parallellogramma F D M K, P L M D.
ex demonſtratis Gregorii à S. Vincentio evidens eſt trape-
zium A F G L æquale eſſe rectilineo D F G L M. & quoniam
A G I recta ſecat rectam F L bifariam in I, ex ejuſdem Gre-
gorii à S. Vincentio Lib. de hyperbola, manifeſtum eſt rectas
L M, G H, FD, eſſe continuè proportionales in eadem ratione
cum tribus continuè proportionalibus A D, A H, A M. aſym-
ptoto A O per punctum G ducatur parallela recta R G S
rectis F D, M K, occurrens in punctis R, S. quoniam rectæ
F D, G H, L M, ſunt continuè proportionales, erit dividen-
do & permutando F R ad S L ut G H ad L M: & quoniam
rectæ M A, H A, D A, ſunt continuè proportionales, erit
etiam dividendo & permutando M H ad H D hoc eſt S G ad
G R ut H A ad D A, vel ut G H ad L M; & proinde F R
eſt ad S L ut S G ad G R, cumque anguli F R G, G S L, ſint
æquales ob parallelas F R, S L, erunt triangula F R G, G L S,
æqualia; & proinde parallelogrammum R D M S æquale eſt
rectilineo D F G L M ſeu trapezio A F G L; ſed parallelo-
grammum R D M S eſt medium geometricum inter parale-
logramma P D M L, F D M K, quoniam eandem habentia alti-
tudinem eorum baſes nempe L M, S M, K M, ſunt continuè
proportionales; & ideo trapezium A F G L eſt medium
11TAB.
XLVIII.
fig. 4. I & hyperbolam interſecans in puncto G, fiatque trape-
zium ſectori circumſcriptum A F G L, quod dico eſſe me-
dium geometricum inter parallellogramma F D M K, P L M D.
ex demonſtratis Gregorii à S. Vincentio evidens eſt trape-
zium A F G L æquale eſſe rectilineo D F G L M. & quoniam
A G I recta ſecat rectam F L bifariam in I, ex ejuſdem Gre-
gorii à S. Vincentio Lib. de hyperbola, manifeſtum eſt rectas
L M, G H, FD, eſſe continuè proportionales in eadem ratione
cum tribus continuè proportionalibus A D, A H, A M. aſym-
ptoto A O per punctum G ducatur parallela recta R G S
rectis F D, M K, occurrens in punctis R, S. quoniam rectæ
F D, G H, L M, ſunt continuè proportionales, erit dividen-
do & permutando F R ad S L ut G H ad L M: & quoniam
rectæ M A, H A, D A, ſunt continuè proportionales, erit
etiam dividendo & permutando M H ad H D hoc eſt S G ad
G R ut H A ad D A, vel ut G H ad L M; & proinde F R
eſt ad S L ut S G ad G R, cumque anguli F R G, G S L, ſint
æquales ob parallelas F R, S L, erunt triangula F R G, G L S,
æqualia; & proinde parallelogrammum R D M S æquale eſt
rectilineo D F G L M ſeu trapezio A F G L; ſed parallelo-
grammum R D M S eſt medium geometricum inter parale-
logramma P D M L, F D M K, quoniam eandem habentia alti-
tudinem eorum baſes nempe L M, S M, K M, ſunt continuè
proportionales; & ideo trapezium A F G L eſt medium