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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10062Apollonij Pergæi

PROPOSITIO LX.

D Einde perpendicularis egrediens ex

78[Figure 78]11a C cadat ad centrum D ſectionis A B
hyperboles, & ponamus C E ad E D, vt
proportio figuræ, & producamus ex E ad
ſectionem rectã lineam E B, quæ parallela
ſit D E, producaturque C B, quæ occur-
rat axi in G. Et quia C E ad E D, nempe
22b C B ad B G, nempe D H ad H G eſt, vt
proportio figuræ; erit G B linea breuiſſima
(nona ex quinto) quod erat oſtenden-
dum.

PROPOSITIO LXI.

S It poſtea punctum C, &

79[Figure 79] perpendicularis C F, &
F remotius à vertice ſectio-
nis, quàm ſit centrum, & po-
namus C E ad E F, vt eſt
proportio figuræ, & ſimiliter
D G ad G F, & ex E pro-
ducamus E H, quæ ſit paral-
lela ipſi F A, & ex G, D.
ad illam G I, D K, quæ ſint
parallelæ ipſi C F; & duca-
mus ſectionem hyperbolen
334 lib. 2. tranſeuntem per D, quam
contineant I H, I G, quæ occurret ſectioni A B ſimiliter in B; Itaque
44a per B, C producamus lineam, quæ occurrat axi F A in L, & ipſi E H
in M. Dico, quod B L eſt linea breuiſſima. quia ducta perpendiculari
55b H N, C E ad E F, ſeu ad K D, eſt vt D G ad G F, nempe vt K I ad
I E, & propterea E C in E I erit æquale rectangulo D I ſubſequenti
(octaua ex ſecundo) nempe rectangulo B I conſequenti; Ergo C E in
6612. lib. 2. E I eſt æquale B H in H I, & propterea B H ad C E, nempe H M ad
M E eſt, vt E I ad I H; ergo H I, nempe N G æqualis eſt E M, & ideo
L F ad E M, nempe ad N G eſt, vt C F ad E C, nempe D F ad D G,
quia quælibet earum aſſignata eſt, vt proportio figuræ; ergo L F ad N
G eſt, vt D F ad D G; itaq; comparando homologorum differentias L
D ad D N, vt D F ad D G; & per conuerſionem rationis, & poſtea
diuidendo D N ad N L erit, vt D G, ad G F, quæ eſt vt propor-
tio figuræ; Ergo B L eſt linea breuiſſima ( nona ex quinto ) & hoc erat
oſtendendum.

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