9456Apollonij Pergæi
Sit ſectio ellipſis
11b A C B tranſuerſa A
70[Figure 70]
B, &
c.
Lego;
Sit ſe-
ctio ellipſis A C B, &
axis maior A B, cen-
trum D, & perpendi-
cularis E F ſecans a-
xim in F inter cen-
trũ ellipſis D, & ver-
ticem A.
11b A C B tranſuerſa A
ctio ellipſis A C B, &
axis maior A B, cen-
trum D, & perpendi-
cularis E F ſecans a-
xim in F inter cen-
trũ ellipſis D, & ver-
ticem A.
Et ducamus per
22c punctum E ſectionẽ
hyperbolicam E M
C circa duas eius continentes, & c. Ideſt circa duas asymptotos I L, I H per
E deſcribatur hyperbole E M C, quæ ſecet axim A B æquidiſtantem alteri asym-
3312. & 13.
lib. 2. ptoton in aliquo puncto vt in M; oſtendetur punctum M ſuper ellipſis centrum
D cadere.
22c punctum E ſectionẽ
hyperbolicam E M
C circa duas eius continentes, & c. Ideſt circa duas asymptotos I L, I H per
E deſcribatur hyperbole E M C, quæ ſecet axim A B æquidiſtantem alteri asym-
3312. & 13.
lib. 2. ptoton in aliquo puncto vt in M; oſtendetur punctum M ſuper ellipſis centrum
D cadere.
Ergo E H prima in proportione in IH ſubſequentem, nempe G F ſub-
44d ſequens ipſam M G quartam, æquale eſt ſubſequenti D G ſecundæ in,
I G nempe F H tertiam. Ergo punctum N, & c. Textus corruptus ſic reſti-
tui poſſe cenſeo; Ergo E H prima proportionalium in H I, nempe G F quartam
æquale eſt D G ſecundæ in I G, nempe F H tertiam, & c. Propterea quod E H ad
F H, atque D G ad G F poſitæ fuerunt, vt latus tranſuerſum ad rectum; ergo re-
ctangulum ſub D G, & H F, ſeu I G, extremis quatuor proportionalium, æqua-
le eſt rectangulo ſub intermedijs E H, & F G, ſeu H I, eſt que punctum E in,
hyperbola E M C cuius aſymptoti K I, L I; ergo punctum D in eadem hyperbola
exiſtit; ſed erat prius in ellipſis diametro A B, ſcilicet in centro; quare in eorum
communi ſectione exiſtet: erat autem punctum M communis ſectio hyperboles
E C, & axis ellipſis A B; igitur puncta M, & D coincidunt, & hyperbole E D C
tranſit per centrũ ſectionis ellipticæ A C B, & ideo hyperbole E D C, quæ in infinitũ
558. lib. I. extendi, & dilatari poteſt neceſſario ſecabit finitam ellipſim alicubi, vt in C.
44d ſequens ipſam M G quartam, æquale eſt ſubſequenti D G ſecundæ in,
I G nempe F H tertiam. Ergo punctum N, & c. Textus corruptus ſic reſti-
tui poſſe cenſeo; Ergo E H prima proportionalium in H I, nempe G F quartam
æquale eſt D G ſecundæ in I G, nempe F H tertiam, & c. Propterea quod E H ad
F H, atque D G ad G F poſitæ fuerunt, vt latus tranſuerſum ad rectum; ergo re-
ctangulum ſub D G, & H F, ſeu I G, extremis quatuor proportionalium, æqua-
le eſt rectangulo ſub intermedijs E H, & F G, ſeu H I, eſt que punctum E in,
hyperbola E M C cuius aſymptoti K I, L I; ergo punctum D in eadem hyperbola
exiſtit; ſed erat prius in ellipſis diametro A B, ſcilicet in centro; quare in eorum
communi ſectione exiſtet: erat autem punctum M communis ſectio hyperboles
E C, & axis ellipſis A B; igitur puncta M, & D coincidunt, & hyperbole E D C
tranſit per centrũ ſectionis ellipticæ A C B, & ideo hyperbole E D C, quæ in infinitũ
558. lib. I. extendi, & dilatari poteſt neceſſario ſecabit finitam ellipſim alicubi, vt in C.
Et producamus per E C lineam, &
c.
Et producamus per E C rectam li-
66e neam, quæ occurrat continentibus in L, K, & ſecet axim ellipſis in P.
66e neam, quæ occurrat continentibus in L, K, & ſecet axim ellipſis in P.
Erit G F æqualis O N, quare F O, &
c.
Quia duæ rectæ lineæ A O, L K
ſecantur à parallelis I L, F E, C N, K O proportionaliter, & ſunt K C, L E
æquales, ergo O N, F G inter ſe æquales erunt, & addita communiter N F erit
778. lib. 2. F O æqualis N G; Et quoniam E H ad H F eſt vt E K ad K P (propter pa-
rallelas K I, O A) nempe vt F O, ſeu ei æqualis G N ad O P (propter paral-
lelas E F, O K) ſed eandem proportionẽ habet D G ad G F, quàm E H ad H F;
ergo G N ad O P eandem proportionem habet quàm D G ad G F, & compa-
rando homologorum differentias D N ad N P erit vt D G ad G F, ſeu vt latus
88Lem. 3.
10. huius. tranſuerſum ad rectum; & ideo C P eſt breuiſsima.
ſecantur à parallelis I L, F E, C N, K O proportionaliter, & ſunt K C, L E
æquales, ergo O N, F G inter ſe æquales erunt, & addita communiter N F erit
778. lib. 2. F O æqualis N G; Et quoniam E H ad H F eſt vt E K ad K P (propter pa-
rallelas K I, O A) nempe vt F O, ſeu ei æqualis G N ad O P (propter paral-
lelas E F, O K) ſed eandem proportionẽ habet D G ad G F, quàm E H ad H F;
ergo G N ad O P eandem proportionem habet quàm D G ad G F, & compa-
rando homologorum differentias D N ad N P erit vt D G ad G F, ſeu vt latus
88Lem. 3.
10. huius. tranſuerſum ad rectum; & ideo C P eſt breuiſsima.
Quia in ſequenti propoſitione 57;
&
in alijs adhibetur propoſitio non adhuc
demonſtrata; nimirum poſita C P linea breuiſsima, pariter que I D ſemiſsi axis
recti minoris etiam breuiſsima (ex II. huius) quæ occurrant vltra axim in,
M deducuntur ea omnia, quæ in propoſitionibus 51. & 52. ex hypotheſi
demonſtrata; nimirum poſita C P linea breuiſsima, pariter que I D ſemiſsi axis
recti minoris etiam breuiſsima (ex II. huius) quæ occurrant vltra axim in,
M deducuntur ea omnia, quæ in propoſitionibus 51. & 52. ex hypotheſi