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

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Igitur duo plana tranſeuntia per K L, T V eleuata ſuper triangulum. H F I ad angulos rectos producunt in cono H F I duas ſectiones hypor-
bolicas, quarum axes L M, V X, & inclinati ipſarum L K, V T, & ſingulì eorum ad ſuos erectos ſunt, vt D B ad B E; ergo figuræ trium. ſectionum ſunt ſimiles, & æquales; & propterea duæ ſectiones, qua-
rum axes ſunt L M, V X ſunt æquales ſectioni A B, & c. Ex textu men-
doſo expungi debent ſuperuacanea aliqua verba, ſicut in contextu habetur. Non enim verum eſt, quod duæ tantummodo hyperbole æquales eidem A B duci
poſſunt in cono recto H F I, vertices habentes in lateribus H F, & F I, ſed
quatuor inter ſe æquales eße poßunt; nam ſuper latus F H duci poſſunt duæ
hyperbole, quarum axes tranſuerſi K L æquales ſint ipſi B D, & æquidiſtan-
tes ſint rectis lineis F N, & F S. Quod ſic oſtendetur. Quoniam recta linea
Q R ducta eſt parallela ipſi H I erunt duo arcus circuli intercepti H Q, I R
æquales inter ſe; & ideo duo anguli ad peripheriam H F Q, & I F R æquales
erunt inter ſe; poſita autem fuit K L æqualis, & parallela ipſi F N; igitur
duo anguli alterni K L F, & H F N æquales ſunt inter ſe: pari ratione; quia
reliqua K L ducta eſt parallela ipſi F S, erit angulus externus S F I æqualis
interno, & oppoſito, & ad eaſdem partes L K F; & ideo duo triangula L F K
habent angulum F, communem, & duos angolos in ſingulis triangulis K, & L æquales; igitur ſunt æquiangula, & ſimilia, & , vt antea dictum eſt, fieri
poſſunt duæ rectæ lineæ K L æquales eidem D B, & inter ſe: ſi igitur per duas
rectas lineas K L ducantur plana perpendicularia ad planum trianguli per axim
H F I, eſſicientur in cono recto duæ hyperbole, quarum bini axes tranſuerſi K L
ſunt æquales: & quia, propter parallelas H I, Q R, eſt F N ad N Q ſeu qua-
dratum F N ad rectangulum F N Q vt F S æd S R ſeu vt quadratum F S ad
rectangum F S R; ſed rectangulum H N I æquale eſt rectangulo F N Q, & rectangulum H S I æquale eſt rectangulo F S R: ergo quadratum F N ad re-
ctangulum H N I eandem proportionem habet, quàm quaàratum F S ad rectã-
gulum H S I; eſtque latus tranſuerſum K L ad ſuum latus rectum, vt quadra-
tum F N ad rectangulum H N I, pariterque latus tranſuerſum K L alterius
ſectionis ad ſuum latus rectum eſt vt quadratum F S ad rectangulum H S I:

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