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

Apollonij Pergæi A B C, & D E F ſimilia, & ſimiliter poſita: poſtea in plano per B C, M K
ducto, diametris B C, & E F, fiant duo circuli B K C, E L F, qui ſint ba-
ſes duorum conorum, quorum vertices ſint A, & D, & in eorum ſuper ficie-
bus planum per H I, M K ductum efficiat ſectiones K H M, & L X S: Dico
eas eſſe quæſitas. Quoniam duo triangula A B C, D E F ſimilia, & ſimiliter
poſita in eodem ſunt plano, pariterque duo circuli baſium in vno plano exiſtunt; ergo duo coni A B C, & D E F ſimiles erunt; poſtea quia triangula A B C,
& D E F ſimilia ſunt, & communis ſectionum diameter H X I æque inclina-
tur ad coincidentes baſes M K, S L, & axi communi A D G æquidiſtat, & in angulis æqualibus intercipiunt G I communem portionem baſium triangulorum
ſimilium per axes; igitur hyperbolæ K H M, & L X S æquales ſunt, & ſimi-
les inter ſe, & earum figuris æqualia ſunt quadrata ex dupla interceptæ G I
deſcripta. Secundò quia ( propter parallelas A O, & B C ) triangula H O A,
ctangulum B G C eandem proportionem habebit, quàm quadratum H O ad qua-
dratum O A, ſeu quàm latus tranſuerſum ad rectum figuræ Z; ſed vt quadra-
tum A G ad rectangulum B G C, ita eſt latus tranſuerſum ad rectum hyperbo-
les K H M; igitur duæ hyperbolæ Z, & K H M, habent figurarum latera,
porportionalia; ſuntq; prædictæ figuræ æquales cum ſint æquales quadratis ex du-
plis ipsarum A O, & interceptæ G I: quæ ſunt æquales in parallelogrammo G
O, & habent angulos à diametris, & baſibus contenti, æquales inter ſe: erunt
hyperbolæ K H M, & Z æquales, & ſimiles inter ſe: & propterea ſectio L X S,
quæ ſimilis, & æqualis oſtenſa eſt ipſi K H M, erit quoque æqualis, & ſimilis
eidem ſectioni Z. Tertiò, quia in duobus conis ſimilibus, & ſimiliter poſitis
circa communem axim A D G, ſuperficies nunquàm conueniunt, propterea,
quod latera A B, & D E, à quibus generantur in tota reuolutione inter fc,

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