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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7739Conicor. Lib. V.

PROPOSITIO LIV. LV.

ITaque oſtenſum eſt, vti memorauimus, quod ex concurſu
duarum breuiſſimarum ad coniſectionem non egrediatur alia
11a breuiſecans præter illas duas, & quod reliqui rami ex eorum
concurſu educti ad ſectionem habent proprietates ſuperiùs ex-
poſitas.

PROPOSITIO LVI.

In ellipſi ramorum, ſecantium vtrumque axim, à concurſu vl-
tra centrum poſito egredientium, vnius tantum portio, inter
axim maiorem, & ſectionem intercepta, erit linea breuiſsima,
22a ſiue menſura ipſam comparatam, nec non perpendicularis ipſam
trutinam ſuperet, æquet, vel ab ea deficiat.

SIt ſectio ellipſis A C B, & axis maior tranſuerſus A B perpendicularis
E F, centrum D, & ponamus D G ad G F, vt proportio figuræ, & ſi-
33b militer E H ad H F, & producamus per H rectam I H K parallelam ipſi A B,
& per G rectã I G L ipſi

53[Figure 53] E F, quæ ſibi occurrant
in I, & ducamus per
44c punctum E ſectionem
554. lib. 2 hyperbolen E M C cir-
ca duas eius continen-
tes L I, I K, quæ oc-
curret ſectioni A C B
ellipticæ, quia I L, I K
ſunt duæ cõtinentes ſe-
ctionem E M C, & pro-
portio E H ad H F po-
ſita eſt, vt D G ad G F;
66d ergo E H prima proportionalium in H I, nempe G F quartam, æquale
eſt D G ſecundæ in I G, nempe F H tertiam; ergo punctum M eſt in il-
lius diametro, & propterea ſectio hyperbole E M C tranſit per centrum
ſectionis ellipſis A C B; quare duæ ſectiones ſe inuicem ſecant, ſitque
concurſus in C, & producamus per E, C lineam occurrentem duabus con-
77e tinentibus ſectionem in L, K, & producamus duas perpendiculares C N,
K O ſuper A B. Et quia K C, L E ſunt æquales (16. exſecundo) erit G F
888. lib. 2. æqualis O N; quare F O æqualis eſt ipſi G N; atque E H ad H F, nempe
99f E K ad K P, ſeu F O (quæ eſt æqualis ipſi G N) ad O P eandem propor-
tionem habet, quàm D G ad G F, quę eſt ęqualis ipſi O N, & ideo G N
ad O P eſt, vt D G ad O N, & comparando homologum differentias D N
1010Lem. 3.

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