250212Apollonij Pergæi
Rectæ lineæ parallelæ B E, C F ſe-
288[Figure 288]
cent æquidiſtantes aſymptotos H G,
L K in punctis N, O, P, Q. De-
bent autem coniſectiones in eodem pla-
no collocari ſicuti aliæ omnes, quæ in.
ſequentibus propoſitionibus 4. 5. 6. 7.
8. & 9. vſurpantur ſemper in vno
plano poſitæ intelligi debent.
L K in punctis N, O, P, Q. De-
bent autem coniſectiones in eodem pla-
no collocari ſicuti aliæ omnes, quæ in.
ſequentibus propoſitionibus 4. 5. 6. 7.
8. & 9. vſurpantur ſemper in vno
plano poſitæ intelligi debent.
Et primo duæ rectæ B E, C F paralle-
læ ſint rectæ lineæ H L centra coniungen-
ti. Quoniam hyperbolæ A B, D E æqua-
les ſunt, & congruentes; atque æquidiſtan-
tes asymptoti H N, L P æque inclinan-
tur ad æquales ſemiaxes tranſuerſos H
A, & L D; & ſegmenta asymptotorum H N, L P æqualia ſunt in paralle-
logrammo H P, nec non duo anguli H N B, & L P E æquales ſunt inter ſe, pro-
pter parallelas asymptotos: igitur duæ figuræ A H N B A, & D L P E D æquales
erunt, & congruentes: quapropter interpoſitæ rectæ lineæ N B & P E congruẽ-
tes, & æquales erunt; & addita vel ablata communi B P, erit N P æqualis
B E: eſt verò N P æqualis H L, eo quod H P parallelogrammum eſt; igitur
intercepta B E æqualis eſt rectæ lineæ H L centra coniungenti. Eadem ratione
quælibet alia intercepta C F parallela ipſi H L eidem æqualis oſtendetur: qua-
propter duæ interceptæ æquidiſtantes B E, & C F inter ſe æquales erunt.
læ ſint rectæ lineæ H L centra coniungen-
ti. Quoniam hyperbolæ A B, D E æqua-
les ſunt, & congruentes; atque æquidiſtan-
tes asymptoti H N, L P æque inclinan-
tur ad æquales ſemiaxes tranſuerſos H
A, & L D; & ſegmenta asymptotorum H N, L P æqualia ſunt in paralle-
logrammo H P, nec non duo anguli H N B, & L P E æquales ſunt inter ſe, pro-
pter parallelas asymptotos: igitur duæ figuræ A H N B A, & D L P E D æquales
erunt, & congruentes: quapropter interpoſitæ rectæ lineæ N B & P E congruẽ-
tes, & æquales erunt; & addita vel ablata communi B P, erit N P æqualis
B E: eſt verò N P æqualis H L, eo quod H P parallelogrammum eſt; igitur
intercepta B E æqualis eſt rectæ lineæ H L centra coniungenti. Eadem ratione
quælibet alia intercepta C F parallela ipſi H L eidem æqualis oſtendetur: qua-
propter duæ interceptæ æquidiſtantes B E, & C F inter ſe æquales erunt.
Secundo B E, C F parallelæ ſint alicui rectæ lineæ L f diuidenti angulum K
L H; ideoque P L f N, & Q L f O parallelogramma erunt: ſecetur L T æqua-
289[Figure 289]
lis H N, atque L V æqualis H O;
ducan-
turque T X, V Z parallelæ ipſis N B, O
C ſecantes reliquam hyperbolen in X, Z;
eritque ( vt in prima parte oſtenſum eſt)
T X æqualis N B, atque V Z æqualis O C.
Et ſiquidem B E, C F cadunt infra cen-
tra H, L ad partes G, K, cadent quoque
infra L f eis parallelam per L ductam in-
fra centrum H incidentem, & ideo N f,
ſeu ei æqualis P L in parallelogrãmo P f
minor erit, quàm H N; eſtque L T æqua-
lis H N; igitur L P minor erit, quàm L T ; & propterea punctum P propin-
quius erit centro L, quàm T: Eadem ratione oſtendetur, quod punctum Q pro-
pinquius ſit centro L, quàm V, & P propinquius centro quàm Q; ergo quatuor
11Def. add. æquidiſtantium P E, Q F, T X, V Z cadentium infra centrum ad partes K,
duæ P E, T X vlterius ad partes centri, vel asymptoti L M tendunt, quàm,
duæ Q F, V Z. At ſi B E, C F ſecent rectã lineam centra coniungentem inter
duo centra H, & L, manifeſtum eſt puncta P, & Q cadere ſupra centrum L,
atque duo puncta N, & O cadere infra centrnm H alterius hyperboles, cumque
L T ſecta ſit æqualis ipſi H N ad eaſdem partes; pariterque L V æqualis
L H; ideoque P L f N, & Q L f O parallelogramma erunt: ſecetur L T æqua-
turque T X, V Z parallelæ ipſis N B, O
C ſecantes reliquam hyperbolen in X, Z;
eritque ( vt in prima parte oſtenſum eſt)
T X æqualis N B, atque V Z æqualis O C.
Et ſiquidem B E, C F cadunt infra cen-
tra H, L ad partes G, K, cadent quoque
infra L f eis parallelam per L ductam in-
fra centrum H incidentem, & ideo N f,
ſeu ei æqualis P L in parallelogrãmo P f
minor erit, quàm H N; eſtque L T æqua-
lis H N; igitur L P minor erit, quàm L T ; & propterea punctum P propin-
quius erit centro L, quàm T: Eadem ratione oſtendetur, quod punctum Q pro-
pinquius ſit centro L, quàm V, & P propinquius centro quàm Q; ergo quatuor
11Def. add. æquidiſtantium P E, Q F, T X, V Z cadentium infra centrum ad partes K,
duæ P E, T X vlterius ad partes centri, vel asymptoti L M tendunt, quàm,
duæ Q F, V Z. At ſi B E, C F ſecent rectã lineam centra coniungentem inter
duo centra H, & L, manifeſtum eſt puncta P, & Q cadere ſupra centrum L,
atque duo puncta N, & O cadere infra centrnm H alterius hyperboles, cumque
L T ſecta ſit æqualis ipſi H N ad eaſdem partes; pariterque L V æqualis