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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199161Conicor. Lib. VI.

LEMMAV.

IN eiſdem figuris rurſus G B ad B D maiorem proportionem habeat,
qnàm K F ad F 1 : Dico quod minimè reperiri poſſunt axium ab-
ſcißæ erectis proportionales, quæ habeant eandem rationem ad contermi-
nas potentiales.

Secentur quælibet abſciſſæ, B C, F H ita vt C B ad B D ſit vt H F ad F I,
& ducantur ordinatim ad axes applicatæ A C, E H, quæ productæ ſecent, con-
iunctas G D, K I in P, L, atque fiat γ B ad B D vt K F ad F I, iungatur-
que γ D ſecans A P in M. Manifeſtum eſt rectam C M inæqualem eſſe C P,
(propterea quod γ B minor eſt, quàm G B, cum ad eandem B D minorem pro-
portionem habeat, quàm G B, ideoque punctum Y, & recta γ D cadent intra,
triangulum G B D, & punctum M intra ipſum cadet, aut extra G D pro-
ductam). Quoniam D B ad B γ eſt vt I F ad F K, & erat C B ad B D vt
H F ad F I ; ergo ex æquali C B ad B γ erit vt H F ad F K, & comparando
terminorum ſummas in hyperbola, & differentias in ellipſi ad antecedentes, γ C
ad C B erit vt K H ad H F; eſt verò M C ad C R vt L H ad H K (eoquod
triãgula M C R, & L H K ſimilia ſunt triangulis ſimilibus B D Y, I F K,) ergo
ex æquali M C ad C B erit vt L H ad H F, & rectangulum M C B ad quadra-
tum C B eandem proportionem habebit, quàrn rectangulum L H F ad quadra-
tũ H F; ſed rectangulũ M C B æquale nõ eſt rectangulo P C B (cum M C oſtenſa
ſit inæqualis P C); ergo rectangulum P C B, ſeu quadratum A C ad quadratum
1112. 13.
lib. 1. C B non eandem proportionem habet, quàm rectangulum L H F, ſeu quadratum
E H ad quadratum H F; & propterea A C ad C B non eandem proportionem
habebit quàm E H ad H F. Idem oſtendetur in reliquis omnibus abſciſſis ſimi-
liter poſitis. Quare patet propoſitum.

COROLLARIVM I.

MAnifeſtum eſt in coniſectionibus non ſimilibus duci poſſe duas ſeries appli-
catarum ad axes, itaut abſciſſæ ſimiles, ſeu proportionales inter ſe adcõ-
terminas potentiales non ſint in ĳſdem rationibus.

COROLLARIVM II.

Colligitur pariter conuertendo, quod in duabus ſectionibus eiuſdem nominis
ſi duæ ſeries abſciſſarum ſimilium in axibus poſitæ fuerint, & in vna ſe-
rie abſciſſæ ad conterminas potentiales maiorem proportionem habeant, quàm in
altera ſerie, fieri poteſt vt ſiguræ axium non ſint inter ſe ſimiles: Quod verifi-
catur ſaltem in caſu præcedentis propoſitionis.

His præmiſſis, quoniam paſſo in definitione poſita eſſentialiter conuenit defini-
to eſt impoſſibile, vt eidem ſubiecto definito competant duæ paſſiones diuerſæ, &
inter ſe oppoſitæ, exempli gratia, fieri non poteſt, vt in triangulis ſimilibus

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