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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277239Conicor. Lib. VI.

322[Figure 322] ipſi K L, non eſſet eidem æqualis.) His poſitis ſi educatur ex F linea ipſi
patallela cadet inter F G, F H, aut inter F I, F G; ſitque F N; igitur
1112. lib. 1. quadratum F N ad I N in N H eſt, vt D B ad B E: quod eſt abſurdum;
quia quadratum F N maius eſt, quàm quadratum F G, & N H in N I
minus eſt, quàm quadratum G H.

Poſtea habeat quadratum F G ad quadratum G H minorem propor-
tionem quàm babet D B ad B E; & circumſcribamus circa triangulum.
H F I circulum ; & producamus F G quouſque occurrat circuli circum-
ferentię in O; ergo quadratum F G ad quadratum G H, nempe ad F G
in G O habet minorem proportionem, quàm D B ad B E: & ponamus
F G ad G P, vt D B ad B E ; & per P ducamus P Q parallellam H I ;
& coniungamus F R, F Q; quæ occurrant H I in S, N: quare D B ad
B E eſt, vt F G ad G P, quæ eſt, vt F N ad N Q; nempe vt quadra-
tum F N ad F N in N Q æquale ipſi I N in N H, atque vt quadra-
tum F S ad F S in S R, nempe vt quadratum F S ad I S in S H; & edu-
camus T V, K L, quæ ſubtendant duos angulos H F K, I F T, & ſint
22c parallelæ ipſis F N, & F S, & æquales ipſi D B; igitur duo plana per K
33d L, T V extenſa ſuper triangulum H F I ad angulos rectos eleuata, pro-
ducunt in cono H F I ſectiones hyperbolicas, quarum axes L M, V X,
& inclinati ipſarum L K, T V, & ſinguli earum ad ſuos erectos eandem
proportionem habent, quàm D B ad B E, & propterea figuræ ſectionum
442. huius. ſimiles ſunt, & æquales, ideoque ſectiones, quarum axes ſunt L M, V
X ſunt æquales ſectioni A B.

Nec reperitur ſectio præter iam dictas, cuius vertex ſit ſuper aliquam
55e duarum linearum H F, F I, & ſit æqualis ſectioni A B. Quia ſi reperiri
poſſet, caderet eius axis in planum trianguli H F I, illiuſque axi educa-
tur parallela F Z a, quæ non cadet ſuper F R, neque ſuper F Q, eritq;
quadratum F Z ad I Z in Z H, quod eſt æquale ipſi F Z in Z a, nempe
F Z ad Z a eandem proportionem haberet, quàm D B ad B E; ſed D
B ad B E eſt, vt F G ad G P, nempe F Z ad Z b; ergo proportio F

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