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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291253Conicor. Lib. VI.
Notæ in Propoſit. XXIX.
ET faciamus ſuper E K triangulum ſimile triangulo A B C, & c. 11a mirum, fiat angulus K E L æqualis angulo A, & angulus L fiat æqualis angulo B.
Ergo L K, quæ eſt latus trianguli tranſeuntis per axim E G para llelũ
22b eſt E G, &
c. Legi debet, vt in textu videre eſt. Hoc conſtat ex conſtructio-
ne;
nam duo anguli alterni G E K,, & L K E æquales ſunt eidem angulo C.
Et propterea planum, in quo eſt ſectio D E
33C339[Figure 339] F producit in cono ſectionem parabolicam, &
c.
Quoniam planum circuli, cuius diameter E K
perpendiculare eſt ad planum trianguli L E K:
igi-
tur ſi ducatur planum N F O æquidiſtans circulo E
K ſecans planum D E F in recta linea D G F, erit
quoque circulus, &
perpendicularis ad planum triã-
guli per axim L E K:
ſed ex conſtructione planum
D E F perpendiculare quoque erat ad idem trian-
gulum per axim E L K;
igitur D F communis ſectio
eorundem planorum perpendicularis quoque erit ad
idem planum L N O, &
efficiet angulos rectos cum
diametro circuli N O, &
cum E G, quæ in eodẽ pla-
no exiſtunt, &
cũ illo conueniunt in puncto G; ſuntq; E G, & L O parallelæ: igitur
4411. lib. 1. planum ſectionis D E F producit neceſſariò in cono L N O producto parabolam.
Igitur H E ad E L, quæ eſt æqualis ipſi L K eamdem proportionem,
55d habet, quàm quadratum E K ad quadratum K L, &
c. Quoniam conus
L E K ſimilis eſt cono recto A B C erit quoque rectus:
& propterea duo latera
trianguli per axim E L, &
L K æqualia erunt inter ſe, & ideo E K ad K L,
atque ad E L eandem proportionem habebit, &
c.
Et dico, quod ſectio D E F non reperitur in alio cono ſimili cono A
66e B C, cuius vertex ſit ex parte plani ſectionis præter hunc conum, &
c.
Ideſt. Nullus alius conus rectus continebit eandem parabolam D E F, qui ſit
ſinilis cono A B C, &
vertex E parabole magis, aut minus recedat à vertice
coni, quàm E L.
Ergo E M eſt indirectum ipſi E L, & c. Quia D G baſis ſectionis conicæ
77f perpendicularis eße debet ad G O, &
ad G E, & ideo ad triangulum per axim
vtriuſque coni recti L E K, &
M E I; & conueniunt plana eorundem trian-
gulorum in E G axi conicæ ſectionis geniti ab eis;
ergo dicta triangula in eo-
dem plano exiſtunt per rectas E G, &
G O ducto; & in vtroquè cono triangu-
lorum per axes latera L K, &
M I parallela ſunt eidem axi E G paraboles:
ergo L K, M I parallelæ ſunt inter ſe, & anguli L, & M æquales ſunt pro-
pter ſimilitudinem triangulorum per axes in conis ſimilibus:
igitur L E, & M
E ſunt quoq;
parallelæ, & conueniunt in E vertice paraboles; ergo in directum
ſunt conſtitutæ.

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