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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168130Apollonij Pergæi oſtendetur ex dictis, quod lineæ maximæ mutuò ſe ſecant inter diame
trum, & rectum, & c. Textũ corrigi debere maniſeſtum eſt ex dictis ſuperius

11b

Quia D Q ad Q I eſt, vt D O ad O

166[Figure 166] G, & c. In eadem figura propoſitionis 32.
præcedentis perſiciatur conſtructio, vt priùs
quia duæ K M, H N ſunt breuiſsimæ li-
22Pr. 15-
huius. neæ; ergo M R ad R D, nec non N P ad
P D eandem proportionem habent, ſcilicet
eam quàm habent latus rectum ad tranſuer-
ſum, ſeu eandem quàm habet ſemierectus
3315. lib. I. E B ad ſemiaxim B D; eſt verò C A ad eius
latus rectum, ſeu D A ad A F, vt E B ad
B D; igitur tam M R ad R D, quàm N P
ad P D eandem proporiionem habent, quàm
D A ad A F; ſed propter parallelas C D, R
K, P H, cſt M K ad K I, vt M R ad R D;
pariterque N H ad H G eandem proportionẽ
habet, quàm N P ad P D; atque propter pa-
rallelas D B, Q K, O H eſt D Q, ad Q I
vt M K ad K I, & D O ad O G eſt vt N H
ad H G; ergo tam D Q ad Q I, quàm D
O ad O G eandem proportionem habent, quàm D A ad A F, ſeu quàm axis mi-
4420. 21. 22.
huius. nor A C ad ſuum erectum, & propterea tam K I, quàm H G eſt ramus maxi-
mus; igitur ſi duæ lineæ breuiſſimæ H G, & K I producantur quouſque axim
minorem ſecent in punctis G, & I efficientur rami omnium maximi. Poſtea quia
D Q ad Q I, eſt vt D O ad O G; permutando D Q ad D O eandem propor-
tionem habebit, quàm Q I ad O G; & permutando, & comparando antecedentes ad
differentias terminorum erit D Q ad D I, vt D O ad D G: eſtque D Q minor
quàm D O; igitur Q I minor eſt, quàm O G; pariterque D I minor eſt, quàm
D G; & propterea punctum I cadit inter exim B D, & ramum H G; eſtque
etiam potentialis K Q propinquior & parallela axi maiori, & ideo maior re-
motiore H O; igitur punctum K cadit inter axim B D, & ramum H G; &
propterea ramus K I ſecat ramum H G in puncto L inter puncta H, & G:
5536. huius. ſed duæ breuiſsimæ K M, H N ſe ſecant vltra axim B D: igitur occurſus L
cadit intra angulum B D C ab axibus compræhenſum. Tandem quia K I ſecat
H G inter puncta G, & H; ergo efficit angulum externum K I A maio-
rem interno, & oppoſito G: & propterea ramus K I propinquior vertici B,
quàm H G efficiet cum axe minore C A angulum A I K maiorem.

Notæ in Propoſit. XXXV.

SI tranſeat per centrum ellipſis vna duarum breuiſſimarum; vtique ra-
66a mi, & c. Hæc propoſitio parum differt à 54. & 55. buius, vbi oſtenſum
eſt, quod ſi duo rami E B, E G breuiſecantes ex eodem concurſu E ad ellipſim
A B ducuntur, quilibet alius ramus E F, extra breuiſecantes poſitus, cadet ſu-
pra breuiſsimam ex puncto F ad axim A C ductam: hic vero ſupponuntur

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