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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289251Conicor. Lib. VI. Inde demonſtrabitur, quod H E ad E I habebit neceſſario eandem pro-
portionem, quàm O e ad e Z; quod eſt abſurdum, quia haberet eandem
proportionem, quàm O N ad N X. Quapropter non continet illam ter-
tius alius conus ſimilis cono A B C.

Supponamus iam, quadratum B Q ad quadratum Q A maiorem pro-
portionem habere, quàm H E ad E I. Dico, exhiberi non poſſe conum
11i ſimilem cono A B C, qui contineat ſectionem D E F. Alioquin conti-
neat illam conus, cuius vertex eſt R, & demonſtrabitur, quod O V ad
V R ſit, vt H E ad E I, quæ habet minorem proportionem, quàm qua-
dratum B Q ad quadratum Q A, quæ oſtenſa eſt eadem, quàm O N ad
N L; ergo O V ad V R; nempe O N ad N X minorem, proportionem
habet, quàm eadẽ O N ad N L, quod eſt abſurdum. Non igitur conti-
nebit ſectionem D E F conus ſimilis cono A B C. Vt propoſitũ fuerat.

PROPOSITIO XXXI.

SIt tandem ſectio elliptica A B C, eiuſque tranſuerſus axis A C, &
22a erectus A D, & in plano perpendiculariter erecto ad ſectionis pla-
num A B C, fiat ſuper A C ſegmentum circuli, quod capiat angulum.

337[Figure 337] æqualem angulo F, eumque bifariam diuidamus in H, & iungamus A H,
C H, & ex H educamus H I, quæ ſecet circulum in K, & occurrat ſub-
33Lem. 10.
huius. tenſæ extra circulum in I; ſitque H I ad I K, vt A C ad A D: & e-
ducamus H L M eaſdem conditiones habens; & iungamus C K, A K,
ducaturque K N parallela A C, & A N parallela H I, quæ ſecet K C
44b in O. Quia H I in I K (quod eſt æquale ipſi C I in A I ad quadratum
I K) eſt vt A C ed A D; & proportio C I in A I ad quadratum I K
componitur ex ratione C I ad I K, nempe K N ad N O (propter

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