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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11981Conicor. Lib. V.

Deinde patebit, quemadmodum demonſtrauimus, & c. Quia D M fa-
11f cta eſt maior, quàm D B, & minor quàm D A, eſtque circuli radius D N
æqualis D M; ergo punctum M cadit intra coniſectionem, N vero extra ip-
ſam; & propterea circulus M L N ſectionem conicam ſecabit alicubi, vt in L,
& portio circuli M L intra coniſectionem A L incidet: rurſus ducatur radius
D L, & L G coniſectionem tangens in L erit, vt priùs angulus D L G acu-
2233. 34.
lib. 1. tus; & ideo L G cadit intra circulum L M, & propterea intra coniſectionem
A L, ſed eadem L G cadit extra ipſam, quia eam contingit in L, quod eſt ab-
ſurdum; quare ramus D A non eſt maior, quàm D B; ſed priùs neque illi
æqualis erat; igitur ramus terminatus D A minor eſt quolibet ramo ſecante
D B infra ipſum poſito, & propterea minimus erit omnium ſecantium.

Poſtea dico, quod D C maior eſt, quàm D B, & c. Demonſtratio ſe-
33g cundæ partis huius propoſitionis, quàm Apollonius innuit (quia conſtructione,
ac progreſſu ſimili ſuperiori perſici poteſt) hac ratione reſtituitur. Demonſtran-
dum eſt quemlibet ramum D B vertici A proximiorem eße minorem quolibet
ramo D C remotiore. Ducantur recta C P contingens ſectionem in C, & O B
tangens ſectionem in B, & recta B R perpendicularis ad ramum D B; & ſi
quidem ramus D C non concedatur maior, quàm D B, ſit primo ei æqualis, ſi
fieri poteſt, & centro D interuallo D C deſcribatur circulus C P R, qui tran-
ſibit per punctum B, ob æqualitatem radiorum D C, D B; & quia (ex Lem-
mate nono) angulus D C P verticem reſpiciens, eſt acutus, recta C P cadet
intra circulum C P R; ſed cadit extra coniſectionem, cum ſit contingens; igi-
tur portio circularis peripheriæ C P ducitur extra coniſectionem C Q B: rur-
ſus, quia angulus D B O eſt obtuſus (ex nono Lemmate, cum verticem A non reſpi-
ciat) ergo R B perpendicularis ad D B cadit intra coniſectionẽ, cum B O poſita ſit eã
contingens: cadit verò eadem B R extra circulum B R Q, cum ſit perpendicu-
laris ad circuli radium D B; igitur circuli portio B R intra coniſectionem ca-
det: ſed priùs eiuſdem circuli portio C P extra eandem ſectionem ducebatur;
igitur idem circulus ſecat coniſectionem alicubi, vt in Q, ducaturque denuo
ramus D Q, & Q O contingens ſectionem in Q; Vnde (ex nono Lemmate)
4433. 34.
lib. 1. angulus D Q O erit acutus; & propterea recta Q O intra circuli portionem;
Q R conſtituta intra coniſectionem cadet, quod eſt abſurdum; recta enim Q
O extra coniſectionem Q A cadit, quàm contingit in Q; non ergo ramus D
C æqualis eſt ipſi D B. Sit ſecundò D C minor, quàm D B (ſi fieri poteſt) ſe-
ceturque D T minor quàm D B, ſed maior quàm D C; & centro D interuallo
D T deſcribatur circulus T Q S; is quidem ad partes B cadet intra, ad par-
tes vero C extra coniſectionem; & propterea eam alicubi ſecabit, vt in Q;
& ducto ramo D Q, & Q O contingente ſectionem in Q, erit angulus D Q
55Lem. 9. O acutus, & ideo recta Q O cadet intra circulum T Q, & propterea intra
coniſectionem, quod eſt abſurdum; Q O enim cadit extra ſectionem Q A,
quàm contingit in Q; non ergo ramus D C minor eſt, quàm D B, ſed neque
æqualis priùs oſtenſus fuit; igitur quilibet ramus D B vertici A propinquior
minor eſt quolibet ramo remotiore D C, quod erat oſtendendum.

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