Apollonius <Pergaeus> - Apollonii Pergaei Conicorvm Lib. V. VI. VII. paraphraste Abalphato Asphahanensi : nunc primum editi ; additvs in calce Archimedis assvmptorvm liber, ex codibvs arabicis mss Abrahamus Ecchellensis Maronita latinos reddidit, Jo. Alfonsvs Borellvs curam in geometricis versione contulit & [et] notas vberiores in vniuersum opus adiecit

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10971Conicor. Lib. V.

PROPOSITIO LXX.

P Oſtea in ellipſi iungamus E H, A H, & C

90[Figure 90] ſit extremitas axis recti; erit A H minor
quàm E H (11. ex 5.) & angulus EGH, nempe
11c A G F maior erit, quàm A G H, ſeu E G F,
ergo E F minor eſt, quàm F A, & hoc erat
propoſitum.

PROPOSITIO LXXI.

P Atet ex hoc, quod ſi producantur ex duo-
22d bus punctis contactus in ellipſi perpendi-
culares E M, A L, & fuerit E M minor,
exempli gratia, tunc tangens educta ab eius
extremitate minor quoque eſt, quemadmodum demonſtrauimus, & hoc
erat oſtendendum.

Notæ in Propoſit. LXVIII. LXIX. LXX.
& LXXI.

S I occurrant duæ tangentes alicui fectioni A B C, aut circulo, vt ſunt,
33a& c. Ideſt ſi coniſectionem A B C contingant duæ rectæ A F, E F in pun-
ctis A, & E concurrentes in F, erit portio tangentis inter occurſum, & conta-
ctum vertici B proximiorem intercepta, minor ea, quæ inter occur ſum, & re-
motiorem à vertice contactum continetur: oportet autem in ellipſi B verticem,
eſſe axis maioris. Expungo verba, aut circulo, tanquam erronea, & incaute
ab aliquo textui ſuperaddita. Circulum enim tangentes ab eodem puncto ductæ
inæquales eſſe nequeunt.

Et ducamus A D in parabola, & hyperbola, & c. Et ducamus A D in
44b parabola, & hyperbola perpendicularem ſuper axim B D, ſecantem eum in D,
atque G F H in I; cumque in parabola diameter F G I ſit parallela axi B D,
erit angulus A I G rectus æqualis interno, & oppoſito ad eaſdem partes, angu-
lo D; in hyperbola vero cum triangulum H D I ſit rectangulum in D, erit ex-
ternus A I G obtuſus, eſtque in triangulo G I A angulus externus A G F maior
interno, & oppoſito A I G, recto in parabola, & obtuſo in hyperbola; erit quo-
que angulus F G A obtuſus in parabola, & hyperbola.

Et angulus E G H, & c. Zuia F H eſt diameter ſecans bifariam E A in
55c6630. ex 2.
Com. G; ergo triangula E G H, & A G H habent àuo latera ægualia E G, A G, &
7711. huius. G H, commune; eſtque H E, vertici B axis maioris ellipſis propinquior, maior
remotiore H A; ergo angulus E G H maior erit angulo A G H; eſtque angulus
A G F æqualis E G H maiori, & E G F æqualis minori A G H; igitur angulus
A G F maior eſt angulo E G F, & latera circa inæquales angulos ſunt æqualia
ſingula ſingulis, ergo tangens A F maior eſt, quàm E F.

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