Full text: Pergaeus, Apollonius: Apollonii Pergaei Conicorvm Lib. V. VI. VII. paraphraste Abalphato Asphahanensi

Conicor. Lib. VI. inter E, & I exiſtit; ergo recta linea I K poſita intra conicũ ſegmentum E K I
ſupra eius baſim E I cadit; & ideo ei parallela E O cadit infra eandem ſeg-
menti conici baſim E I, & propterea occurret ipſi H L intra coniſectionem, & infra punctum L in ſectione poſitum, vt in O; & ideo O S maior erit, quàm,
S L. Et quoniam S E, & R K ſunt inter ſe parallelæ ( quia eidem A C æqui-
diſtant) pariterque E O, & K I factæ ſunt parallelæ, atque S O, & R I (ex
hypotheſi) æquidiſtantes erant; igitur duo triangula E S O, & K R I ſimilia
ſunt inter ſe, & eorũ latera homologa E S, & K R æqualia ſunt inter ſe (quiæ
in parallelogrãmis C S, & G R latera E S, R K æqualia ſunt oppoſitis C H, G
F inter ſe æqualibus, ex hypotheſi) igitur reliqua latera homologa S O, & R I
æqualia ſunt inter ſe; & propterea R I differentia æquidiſtantium F I, G K ad
partes centri A, & asymptoti A B vlterius tendentium, maior erit, quàm S L,
quæ portio eſt ipſius S O, & eſt differentia æquidiſtantium H L, & C E alte-
rius ſegmenti H C. Quod erat oſtendendum.

234.1.

0248-02

Ex conſtructione, & demonſtratione huius propoſitionis colligitur, quod ſi à
duobus punctis eiuſdem asymptoti A C ad hyperbolen ducantur duæ rectæ lineæ
inter ſe parallelæ; illa, quæ ad partes centri A, & asymptoti A B vlterius ten-
dit, maior eſt reliqua. Nam recta linea K R, asymptoto A C parallela cadit ex-
tra ſectionem, & ideo ſecat interceptam parallelam F I, quæ erit maior, quàm
F R, ſeu G K; igitur F I ad partes centri A vlterius tendens maior eſt quali-
bet alia parallela G K ad partes oppoſitas tendente. Eadem ratione F I maior
erit quàm H L, & H L maior, quàm C E. Vnde patet propoſitum.

234.1.

COROL
LAR.

Si fuerint duæ hyperbolæ A B, & D E æquales, & ſimiles ad eaſ-
dem partes cauæ, quarum centra H, & L, & aſymptoti G H I, & K L M, nec non axes A H, & D L ſint parallelæ inter ſe, & rectæ
lineæ B E, & C F ab hyperbolis interceptæ parallelæ fuerint rectæ H
L centra coniungenti; erunt B E, & C F æquales ipſi H L, & in-
ter ſe.

234.1.

PROP.3.
Addit.

Si autem parallelæ ſint alicui rectæ lineæ L f diuidenti angulum K L
H contentum à recta linea L H cen- tra coniungente, & interiore aſympto- to L K, in qua B E, & C F poſitæ ſunt: Dico B E vlterius tendentem. ad partes reliquæ aſymptoti L M ma- iorem eſſe, quàm C F.

234.1.

Dd 2 0249-01
Figure 1. Dd 2

Si vero B E, & C F parallelæ
ſint alicui rectæ lineæ H g diuidenti
angulum L H G à recta linea L H
centra coniungente, & eadem aſym-
ptoto H G contentum: Dico B E vl-
terius tendentẽ ad partes reliquæ aſym-
ptoti H I minorem eſſe, quàm C F.

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