Full text: Pergaeus, Apollonius: Apollonii Pergaei Conicorvm Lib. V. VI. VII. paraphraste Abalphato Asphahanensi

Conicor. Lib. VI. æqualem eße O V, eo quod in perallelogrammis Q I, & S K latera oppoſita ſunt
æqualia, & ipſæ ordinatæ E K O I; nec non M N, Q R æquales oſtenſæ ſunt: Deinde producantur, B E, O I ad ſectionem in C, P; Et quia differentia qua-
dratorum B Z, L X, ſeu T Z, ideſt rectangulum B T C æquale eſt differentiæ
rectangulorum Z G F, & X G F ſeu rectangulo ſub abſciſſarum differentia X Z,
& latere recto G F. Simili modo rectangulum O V P æquale erit rectangulo ſub
abſciſſarum differentia R I, & latere recto G F: ſuntque rectangula contenta
ſub X Z, G F, & ſub R I, G F æqualia, propterea quod later a X Z, R I æqua-
lia oſtenſa ſunt, & latus rectum G F eſt commune; igitur rectangula B T C, & O V P æqualia ſunt; ideoque vt T C ad V P, ita reciprocè erit O V ad B T. Et primò quia diametri G Z, H K coincidunt, & parabolæ H D compræhendi-
tur ab A G: erit G Z maior quàm H K, ſeu quàm G I, & B Z maior quàm
E K, & L X quàm M N. Si verò B E, L M parallelæ ſunt alicui rectæ lineæ
H Y diuidenti angulum G H K; ergo Y Z, ſeu ei æqualis H K, vel G I minor
erit, quàm G Z. Eadem ratione G X maior erit, quàm G R; quare ordinatim
applicata B Z maior erit, quàm O I, & Z C maior, quàm I P; pariterque L
X, ſeu T Z maior erit, quàm Q R, ſeu V I; ideoque T C maior erit, quàm
V P: erat autem O V ad B T reciprocè, vt T C ad V P; ergo O V, ſeu ei æqua-
lis S M maior erit, quàm B T: ij verò addantur æquales L S, T E, quæ in
parallelogrammo S T ſunt latera oppoſita, igitur L M, maior erit quàm B E.

234.1.

0252-02
ex 10.
ex 21.
huius.
ex II.
lib. I.

Deinde quando diametri G I, H K ſibi mutuo congruunt ſit b minor qualibet
data recta linea, & à vertice H ducatur H d cuius quadratũ æquale ſit rectangulo
H G F, & fiat vt b ad H d, ita H d ad aliam rectam lineam æqualem C E; atq; vt H d ad ſemiſſem sũmæ C E, & b potentia, ita fiat longitudine H G ad G K,
ducaturque B K C ordinatim applicata ad diametrum G I. Quoniam quadra-
tum E K æquale eſt parallelogrammo H K, G F (propterea quod parabolæ ſunt
æquales, & diametri ſimiles) & ijs adduntur inter ſe æqualia quadratum d H,
& rectangulum H G F, erunt duo quadrata E K, & d H ſimul ſumpta æqualia
rectãgulo K G F, ſeu quadrato B Z; quare differentia quadratorũ B K, & E K,
ideſt rectanguli B E C æqualis erit quadrato d H; & propterea d H media pro-
portionalis eſt inter C E, B E, ſed facta fuit media proportionalis inter C E,
& b; Ergo B E æqualis eſt b; ideoque R E minor @@ qu@libet recta linea data. Quando verò diametri G Z, H K ſunt æquidiſtantes, ijsdem poſitis ducatur O
n parallela diametris ſecans B E in n. Quia n Z eſt æqualis O I. & erat E K
æqualis O I, ergo n Z, & E K æquales ſunt, & addita, vel ablata comm@ni Z
E erit n E æqualis Z K; & propterea quælibet intercepta B E @@ior erit in
ſecundo caſu, & minor in tertio, quàm n E, ſeu Z K à diametris compræben-
ſa.

234.1.

II. lib. I.

Tertio quando B E, L M parallelæ ſunt alicui rectæ G a diuidenti angulum
H G I, erit K a, ſeu ei æqualis G Z minor, quàm H K, ſeu quàm G I, atq; vt
prius rectangula B T C, & O V P æqualia erunt, & eorum latera reciprocè
proportionalia, eſtque S M æqualis minori O V, ergo S M minor erit quàm B
T; & additis æqualibus L S, & T E, erit L M minor quàm B E.

Tandem ſint interceptæ B E, L M parallelæ G V, H C portionibus interce-
ptarum diametrorum non congruentium, & à terminis B, E, L, M, ducan-
tur ad diametros ordinatim applicatæ, eas ſecantes in Z, K, I, N, O, S, & ſectiones in P, & R; & cadat B E inter duas diametros. Quoniam punctum

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