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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164126Apollonij Pergæi

PROPOSITIO XXXVI.

IN ſectione elliptica quatuor lineæ

158[Figure 158] breuiſſimæ, vt B D, F I, G K,
H L, non conueniunt omnes in vno
puncto.

Alioquin ſit occurſus in E, & prius ſit
B D perpendicularis ſuper A C, tranſi-
ens per D centrum ſectionis; & quia E
eſt occurſus duarum breuiſſimarum B D,
1135. huius. F I, & B E tranſit per centrum; igitur

159[Figure 159] G K non eſt linea breuiſſima, quod eſt
contra hypotheſim. Si vero nullus eorũ
tranſit per centrum, educamus per cen-
trum D O perpendicularem ad A C; qua-
re duæ breuiſſimæ F I, G K conueniunt
intra angulum A D O (34. ex 5.) ſimi-
liter H L, M N breuiſſimæ occurrunt in-
tra angulum C D O (34. ex 5.) ſed cõ-
ueniunt in E, quod eſt abſurdum; igitur
quatuor lineæ breuiſſimæ non cõueniunt in vno puncto; quod erat oſten-
dendum.

PROPOSITIO XXXVII. XLVI.

IN coniſectione A B, cuius centrum D duci non poſſunt-duæ
lineæ maximæ in ellipſi, neque duæbreuiſſimæ in omnibus
ſectionibus, vt A E, A F ad vnum punctum A circumferentiæ
ſectionis terminatæ.

Educamus A G perpendicularem ad axim B E. Si itaque ſectio fue-
rit parabole, fiet E G æqualis F G, quia quælibet earum eſt æqualis di-
midio erecti (13. ex 5.) ſi vero fuerit hyperbole, aut ellipſis, fiet D G
ad G E, vt D G ad G F; quia quælibet earum eſt, vt proportio figuræ
(14. 15. ex 5.) igitur G F æqualis eſt G E, quod eſt abſurdum. Simi-
liter ſi B G fuerit minor duarum axium ellipſis, & fuerint A E, A F
rami maximi oſtendetur, quod G F æqualis ſit G E (23. ex 5.) Patet
igitur, vt dictum eſt, quod ex vno puncto ſectionis educi non poſſunt
ad axim illius duæ lineæ maximæ, neque breuiſſimæ, & hoc erat oſten-
dendum.

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