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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215177Conicor. Lib. VI.

PROPOSITIO XVI.

SI ſectiones A B, C D ſimiles inter ſe, quæ ſint prius para-
bolæ, tangant lineæ A E, C F terminatæ ad earum axes
E B, F D, & contineant cum illis angulos æquales E, F, &
in qualibet earum educantur ordinationes G H, I K ad diame-
tros L A M, N C O tranſeuntes per puncta contactus axibus

239[Figure 239] æquidiſtantes, & fuerit proportio ſuarum abſciſſarum A M, C
O ad lineas tangentes A E, C F eadem; vtique ordinationes
abſcindent ex ſectionibus ſimilia ſegmenta, & ſimiliter poſita, vt
G A H, I C K. Si verò ordinationes ſecuerint ſimilia ſegmen-
ta; vtique ſectiones ſimiles erunt, & abſciſſarum ad lineas tan-
gentes proportio erit eadem, atque lineæ tangentes continebunt
cum axibus angulos æquales.

Educamus enim duas B L, D N ſuper duos axes B E, F D perpendi-
culares, quæ tangent ſectiones in B, D: & ponamus A P ad duplam A
1132. lib. 1. E, vt R A aſſumpta ad A L ei ſimilem, nec non C Q ad duplam C F,
vt aſſumpta S C ad C N; igitur P A, Q C ſunt erecti duarum diametro-
rum L M, N O (52. ex 1.) ergo G M poteſt P A in A M, (12. ex 1.)
2249 lib. 1.& ſimiliter I O poteſt O C in C Q, (12. ex 1.) & propter æquidiſtan-
3311. lib. 1.
lbidem. tiam E B, L A, atque F D, C N ſunt ſimilia E R B, R L A, atque D
S F, S N C; & duo anguli E, F ſuppoſiti ſunt æquales; igitur angulus R
A L æqualis eſt S C N, & N, L ſunt recti; quare R A ad A L, nempe
P A ad duplam A E eſt, vt S C ad N C, nempe vt Q C ad duplam
C F, & M A ad A E ſuppoſita eſt, vt O C ad C F: ergo M A ad A P
eſt, vt O C ad C Q, & angulus O æqualis eſt M. Oſtendetur igitur (vt
44a

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