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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page |< < (126) of 458 > >|
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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