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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198160Apollonij Pergæi

LEMMA IV.

SI G B ad B D maiorem proportionem habuerit, quàm K F ad F
I: Dico in ſingulis ſectionibus reperiri non poſſe binas axium ab-
ſciſſas inter ſe proportionales, quæ ad conterminas potentiales ſint in eiſ-
dem rationibus.

Si enim fieri poteſt, ſit A C ad

216[Figure 216] C B, vt E H ad H F, & Q R ad
R B ſit, vt T V ad V F, atque C
B ad B R ſit vt H F ad F V; con-
iungantur rectæ G D, K I quæ ſecẽt
ordinatas in S, P, X, L; & ſecen-
tur C a æqualis R S, & H b æqualis
V X, ſuntq; æquidiſtantes; ergo co-
niungentes S a, R C æquales ſunt,
& parallelæ, & ſic etiam coniun-
gentes X b, & V H, quare quadratum A C, ſeu rectangulum P C B ad qua-
dratum C B eandem proportionem habet, quàm quadratum E H, ſeu rectangu-
1112. 13.
lib. 1. lum L H F ad quadratum H F; ideoque P C ad C B eandem proportionem ha-
bet, quàm L H ad H F; eſt verò C B ad B R, vt H F ad F V, & per conuerſio-
nem rationis C B ad C R eſt vt H F ad H V, ergo ex æquali C P ad C R eſt
vt L H ad H V: Eodem modo oſtendetur, quod S R, ſeu a C ad R C eſt, vt
X V, ſeu b H ad V H; erat autem P C ad C R vt L H ad H V; ergo a P dif-
ferentia ipſarum S R, P C ad G R, ſeu ad S a eſt vt b L differentia ipſarum
X V, L H ad H V, ſeu ad X b; eſtque D B ad B G vt P a ad S a (propter pa-
rallelas a S, C G, & parallelas a P, & B D) pariterque I F ad F K eſt vt L
b ad b X, ergo D B ad B G eandem proportionem habet, quàm I F ad F K;
quod eſt contra hypotheſim, non ergo binæ axium abſciſſæ inter ſe proportionales
reperiri poſſunt in ſectionibus A B, & E F, quæ ad conterminas potentiales ſint
in eiſdem rationibus; quod erat oſtendendum.

COROLLARIVM.

HInc conſtat in duabus ſectionibus eiuſdem nominis ſi axium figuræ G B D,
& K F I non ſuerint ſimiles, neque ſectiones A B, & E F, ſimiles eſſe.
Nam eſt impoſſibile, vt omnes, ideſt infinitæ axium abſciſſæ inter ſe proportio-
nales ad conterminas potentiales ſint in eiſdem rationibus, cum neque bine in
ſingulis reperiri poſſint ex hac propoſitione.

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