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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186148Apollonij Pergæi

Sit coniſectio A B C, & eius axis B D, & ſu-

194[Figure 194] mantur in ſectione puncta G, C, ab eis educã-
tur duæ ordinationes G H, C A occurrentes axi
in I, D. Dico, quod B G congruit B H, & G
C ipſi H A, & ſuperſicies B D C ſuperficiei B
D A, & ſegmentum B G C ſegmento B H A.
Quoniam axis B D bifariam diuidit G H, A C
in I, D, vtique G I ipſi I H congruet, & D C
11b ipſi D A, & duo puncta G, C ſuper duobus
punctis H, A cadent, & portio ſectionis conicæ
G C ſuper portionem H A, & G B ſuper H B:

195[Figure 195] Et dico, quod portio H A non congruit
alicui alteri portioni, quàm G C: ſi enim
poſſibile eſt cõgruat portioni C K, & por-
tio H B congruet portioni, quæ continua-
tur ipſi K C; ergo cadet B ex H B non ſu-
per B ex C G B; quia portio H B non eſt
æqualis portioni C B; & propterea incidet
axis B D in alium locum, eſſentque eidem
ſectioni plures axes: quod eſt abſurdum;
(51. 52. ex 2.) igitur non cadit H A niſi
2248. lib. 2. ſuper C G. Vt fuerat propoſitum.

PROPOSITIO IX.

M Anifeſtum eſt ex demoſtratis, quod portiones ſectionum
33a æqualium non congruunt ſibi inuicem, niſi earum di-
ſtantiæ à verticibus ſint æquales.

Oſtenſum enim eſt ſibi non congruere, quarum diſtantiæ à verticibus
non ſunt æquales, quia portio H A, ſi caderet ſuper portionem C K, &
earum diſtantiæ à B non eſſent æquales, conſequitur, quod in hyperbola
ſint duo axes, & in ellipſi tres axes: quod eſt abſurdum (51. 52. 53.
4448. lib. 2 ex 2.)

196[Figure 196]

Si autem in ellipſi cadit axis A E tranſuer-
55b

197[Figure 197] ſus ſuper axim rectum illius, vtique differunt
inter ſe, & non ſibi inuicem congruunt ſectio-
nes.

Conſtat etiam, quod in ſectionibus inæ-
qualibus, vt A B C, D E F portio vnius ea-
rum non congruit portioni alterius.

Alioqui congruet B A ipſi D E, & congrue-
ret etiam E F ipſi B C (6. ex 6.) eſſetque ſe-
ctio C B A æqualis ſectioni F E D: at ſuppo-
ſuimus, non eſſe æquales, quod eſt abſurdum:

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