457Conicor. Lib. V.
dente in hyperbola, &
deficiente in ellipſi rectangulo F K H ſimile ei, quod la-
teribus recto, & tranſuerſo continetur, ſcilicet G A E, & eſt A F ſemiſsis la-
teris recti, igitur quadratum B G æquale eſt ſummæ in hyperbole, & differen-
tiæ in ellipſi rectanguli G A F bis ſumpti, & rectanguli F K H, quod eſt æqua-
le duplo trianguli F K H: ſed quadrilaterum A G H F æquale eſt aggregato in
hyperbola, & differentiæ in ellipſi rectanguli G A F, & trianguli F K H, ergò
quadratum B G æquale eſt duplo quadrilateri A G H F, ſeù diſſerentiæ triangu-
lorum D A F, & D G H.
teribus recto, & tranſuerſo continetur, ſcilicet G A E, & eſt A F ſemiſsis la-
teris recti, igitur quadratum B G æquale eſt ſummæ in hyperbole, & differen-
tiæ in ellipſi rectanguli G A F bis ſumpti, & rectanguli F K H, quod eſt æqua-
le duplo trianguli F K H: ſed quadrilaterum A G H F æquale eſt aggregato in
hyperbola, & differentiæ in ellipſi rectanguli G A F, & trianguli F K H, ergò
quadratum B G æquale eſt duplo quadrilateri A G H F, ſeù diſſerentiæ triangu-
lorum D A F, & D G H.
Notæ in Propoſitionem
ſecundam.
ſecundam.
SEcunda propoſitio facilè ex prima deducitur;
nam, quando ordinata B G H I tranſit per cen-
trum D ellipſis; tunc tria puncta G, D, H conue-
niunt, & triangulum D G H euaneſcit, & ideò
differentia trianguli D A F, & trianguli D G H
nullum ſpatium habentis, erit triangulum ipſum
D A F.
nam, quando ordinata B G H I tranſit per cen-
trum D ellipſis; tunc tria puncta G, D, H conue-
niunt, & triangulum D G H euaneſcit, & ideò
differentia trianguli D A F, & trianguli D G H
nullum ſpatium habentis, erit triangulum ipſum
D A F.
Notæ in Propoſitionem
tertiam.
tertiam.
IN tertia propoſitione ſimilitèr, quandò ordinata
B H G I cadit infrà centrum D ellipſis, tunc
ducta C L parallela ipſi A E, erunt duo triangula
D A F, & D C L æqualia inter ſe, cum ſint ſimi-
lia, & latera homologa D A, D C ſint æqualia,
quia ſunt ſemiaxes; proptereà differentia triangu-
lorum D G H, & D A F, ſeù D C L erit trapezium
C G H L, quod ſubduplum eſt quadrati ordinatæ
B G.
B H G I cadit infrà centrum D ellipſis, tunc
ducta C L parallela ipſi A E, erunt duo triangula
D A F, & D C L æqualia inter ſe, cum ſint ſimi-
lia, & latera homologa D A, D C ſint æqualia,
quia ſunt ſemiaxes; proptereà differentia triangu-
lorum D G H, & D A F, ſeù D C L erit trapezium
C G H L, quod ſubduplum eſt quadrati ordinatæ
B G.
SECTIO SECVNDA
Continens propoſitiones IV. V. VI. Apollonij.
COmparata eſt minima ramorum egredientium ex ſua origine
(4) in parabola (5) & hyperbola (6) pariterque in ellipſi (ſi
comparata fuerit portio maioris duorum axium, & tunc maxi-
mus eſt reſiduum tranſuerſi axis.) Reliquorum verò
(4) in parabola (5) & hyperbola (6) pariterque in ellipſi (ſi
comparata fuerit portio maioris duorum axium, & tunc maxi-
mus eſt reſiduum tranſuerſi axis.) Reliquorum verò