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

183[Figure 183] ſuper axim A G applicatur, ita vt vertices A,
C coincidant, neceſſariò ſectio C D cadet ſu-
per ſectionem A B alias aſſignari poſſet pun-
ctum eius D, extra ſectionem A B cadens.

Præterea ponamus duas ſectiones æqua-
11c les, & C F æqualis A E, & c. Textum cor-
ruptum ſic reſtituendum cenſeo. Præterea ſup-
ponamus, duas illas ſectiones æquales eſſe in-
ter ſe, & fiat C F æqualis A E, educamus ad
axes perpendiculares B E, D F, & c. Sic enim
diſtinguitur hypotheſis propoſitionis à conſtru-
ctione eius.

Ergo ſectio A B cadit ſuper ſectionem.
22d C D, & A E ſuper C F: alioqui eſſent ſe-
ctioni parabolicæ duo axes; ergo F cadit
ſuper E, & c. Quoniam (ex hypotheſi) ſectio-
nes A B, & C D æquales ſunt, facta intellectuali conuenienti ſuperpoſitione, ſi-
bi mutuò congruent, & vertex A cadet ſuper verticcm C. Dico iam, axim A
E cadere ſuper axim C F: alioquin in eadem parabola, ſcilicet in duabus pa-
rabolis ſibi congruentibus à communi vertice C, vel A, duo axes A E, & C F
ducerentur: quod eſt impoſſibile. Quare axis A E cadit ſuper axim C F.

Notæ in Propoſit. II.

SI fuerint figuræ duarum ſectionem hyperbolicarum, aut duarum elli-
33a pſium, vt duo plana G I, H K in A B, D E ſimiles, & æquales;
vtique duæ ſectiones æquales erunt: ſi vero duæ ſectiones ſint æquales
earum figuræ erunt æquales, ſimiles, & c. In duabus ſectionibus A B, &
D E ſumi debent figuræ G I, & H K, non qualeſcunque, ſed illæ, quæ ad axes
fiunt, nimirum debent eſſe G A, & H D axes inclinati, ſeu tranſuerſi, & A
I, atque D K eorum latera recta; tunc quidem, ſi figuræ axium G I, H K fue-
rint ſimiles, & æquales, conicæ ſectiones B A, D E æquales quoque oſtenduntur
in propoſitione. Quod verò particula illa (axium) deſideretur in textu propo-
ſitionis, conſtat ex primis verbis immediatè ſequentis conſtructionis. Inquit
enim. Quoniam ſi ponamus axim A M ſuper axim D O, & c.

Cumque G I, H K ſint duæ figuræ ſimiles, & æquales, pariterque
44b I P, K R; ergo duo plana A P, D R ſunt æqualia, & c. Quia rectangula
I P, G I circa communcm diametrum G I P conſiſtunt, erunt inter ſe ſimilia:
pariterque K R ſimile erit rectangulo K H: quare duo rectangula I P, & K R
ſimilia ſunt duobus rectangulis G I, H K inter ſe ſimilibus; & ideo illa inter
ſe quoque ſimilia erunt, & habent latera homologa æqualia, illa nimirum, quæ
opponuntur æqualibus abciſsis A L, & D N, igitur rectangula P I, & R

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