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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163125Conicor. Lib. V. P D; ergo B E ſemiſsis erecti ad B D ſemiſsim tranſuerſi eſt, vt N P ad
P D, & ideo N H eſt breuiſsima linearum egredientium ex N (10.
ex 5.) & fic oſtendetur, quod ſi K I fuerit maximus, erit K M breuiſſima.

PROPOSITIO XXXIII. XXXIV.

EContra oſtendetur, quod duæ breuiſſimæ, ſi producantur
11a ad partes ſuarum originum vſque ad axim minorem rectũ
ellipſis, fient duo maximi; & lineæ maximæ mutuò ſe ſecant in-
ter tranſuerſum, & rectum in eadem parte, & quod continent
cum menſura angulos, quorum proximior vertici ſectiouis ma-
ior eſt.

Quia D Q ad Q I eſt, vt D O ad O G, quia quælibet earum eſt, vt
22b D A ad A F (22. ex 5.) diuidendo, & permutando, fiet D Q minor ad
D O maiorem, vt D I ad D G; ergo D I minor eſt, quàm D G, & K Q
maior, quàm H O; quare angulus I maior eſt, quàm G; igitur H G, K I,
ſe mutuo ſecantes, conueniunt in L.

Et conſtat, quod occurſus duarum breuiſsimarum (ſi producantur ver-
ſus ſuam originem) erit intra angulum contentum à duabus medietati-
bus axium ellipſis B D, D C ſupra vnum eorum, nempe punctum L ca-
dit intra angulum B D C. Quoniam breuiſsimæ N H, M K ſe mutuò ſe-
cant, ſi producantur ad partes ſuæ originis (28. ex 5.) occurrent vtique
extra B D, & intra A G (33. ex 5.) & hoc erat oſtendendum.

PROPOSITIO XXXV.

SI per centrũ ellipfis tranſierit vna
33a

157[Figure 157] duarum breuiſſimarum, vtique
rami egrediẽtes ab eorum occurſu ad
ſectionis quadrantem alterius breuiſſi-
mæ habebunt proprietates expoſitas
in propoſitionibus 54. & 55.

In ellipſi A B C ſit punctum E occur-
ſus duarum breuiſſimarum B D, C I, &
centrum ſectionis D: & ex E educamus E F, quæ ſecet tranſuerſum a-
xim in H. Dico, quod H F nõ eſt breuiſſima, & quod breuiſſima egre-
diens ex F abſcindit ex ſagitta A C cum A lineam maiorem, quàm A
H. Quoniam G I eſt breuiſſima; igitur F H, ſi eſſet quoque breuiſſima,
4434. Huius. occurreret ipſi G I intra angulum A D E: ſed non occurrit ei, niſi in E,
ergo F H non eſt breuiſſima; & quia F E non cadit inter duas breuiſe-
cantes E B, E G; ergo breuiſſima, egrediens ex F, abſcindit ex ſagitta
lineam maiorem, quàm A H (54. ex 5.) quod erat oſtenden@um.

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