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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188150Apollonij Pergæi illo, & ad angulos rectos, ſi intelligatur ſuperficies B I G, ſuperpoſita ſuperfi-
ciei B I H, itaut axis ſuper axim cadat, atque vertex B ſit communis neceſ-
ſario punctum I commune erit, atque recta I G cadet ſuper I H, cum anguli G
I B, & H I B recti ſint, atque punctum G cadet in H, propter æqualitatem
duarum ordinatim applicatarum I G, I H: eadem ratione quælibet alia puncta
ſectionis G B inter G, & B ſumpta cadent ſuper B H; & ideo portio ſectionis
conicæ G B congruet portioni B H, & eidem æqualis erit. Simili modo conſtat,
portionem G C æqualem eße portioni H A, & ſic

201[Figure 201] ſuperficies ipſæ. Quod verò portio H A non con-
gruat alicui alteri ſegmento C K præter G C, con-
ſtat ex eo, quod ſi portiones K C, & A H ſibi mu-
tuò congruunt, vt nimirum punctum C ſuper H, &
punctum K ſuper A cadat: & concipiatur punctũ
C idem ac N, & K idem ac O, & portio O N L
æqualis immo eadem ſectio K C B, & illius axis
L M omnino idem ac axis B D: tunc quidem (ex
precedenti prop. 6.) ſectiones ipſæ A B, & K B, ſeu O L æquales erunt, & ſi-
bi mutuò congruentes: & propterea H B cadet ſuper portionem maiorem C B
ſeu ei æqualem N B L (cum H B æqualis oſtenſa ſit ipſi G B) & ideo vertices
B, & L duarum axium B D, & L M in duabus ſectionibus A B, & K B ſeu
O N L inæqualibus non conuenient: quapropter in duabus congruentibus, ſeu in
eadem ſectione duo axes B D, & L M exiſtent, quod eſt abſurdum, quia eſt
contra propoſ: 48. libri 2.

Notæ in Propoſit. IX.

MAnifeſtum eſt ex demonſtratis, quod portiones ſectionum æqua-
11a lium non congruunt, & c. Sicuti in propoſ. 7. dictum eſt, quod duæ
portiones non æqualiter à vertice axis diſtantes ſibi mutuò congruere nõ poſſunt,
ita hic in duabus quibuslibet æqualibus coniſectionibus idem verificari oſtendi-
tur, quod nimirum duæ portiones cuiuslibet ſectionis conicæ, vel duarum æqua-
lium ſectionum inæqualiter à vertice axis diſtantes non ſint congruentes. Hoc
autem alia ratione demonſtrare ſuperuacaneum non erit, cum demonſtratio, quæ
in textu Arabico corrupto affertur non omnino ſufficiens videatur, ſed prius
oſtendendum eſt.

LEMMAI.

IN duabus æqualibus coniſectionibus A B C, & D E F, quarum
axes A G, D H deſcribere duos circulos æquales contingentes coni-
cas ſectiones, quorum is, qui propinquior eſt vertici extrinſecùs, reli-
quus verò intrinſecùs ſectionem tangat.

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