Viviani, Vincenzo - De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei

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8965 maior erit ipſa CE. Quò ergo in-

59[Figure 59] terceptæ PO magis remouentur à
vertice B, eò ſunt maiores: quare
huiuſmodi ſectiones inter ſe ſunt
ſemper recedentes. Quod ſecun-
dò, & c.

Præterea ſit BY ſectionem con-
tingens in B, & bifariam ſectis trãſ-
uerſis lateribus, nempe GB in V,
& HB in X: cum ſit tranſuerſum
GB maius BH, erit dimidium BV
maius dimidio BX: iam ex centro
V ducatur VY aſymptotos inſcri-
ptæ Hyperbolæ DBE, & ex cen-
tro X agatur XZ aſymptotos cir-
cumſcriptæ ABC, quæ aſymptoti
contingentem BI ſecent in Y, Z, &
per X agatur X & parallela ad BY contingentem ſecans in & .

Itaque quadratum BY ad BZ eſt, vt rectangulum GBF ad rectangulum
HBF (vtrumque enim quadratorum eſt quarta pars ſuæ figuræ) vel vt 118. huius. GB ad BH, vel ſumptis ſubduplis, vt VB ad BX, vel ob parallelas, vt YB ad
B& , quare BZ eſt media proportionalis inter BY, & B& : cum ergo inter pa-
rallelas VY, Z& recta ZX ſecet alteram parallelarum X& in X, ipſa produ-
cta ad partes Z ſecabit quoque alteram parallelam VY infra BY: vnde ha-
rum ſectionum aſymptoti infra contingentem ex vertice inter ſe conueniunt.

Amplius cum aſymptotos VY inſcriptæ occurrat diametro BG vltra cen-
trum X circumſcriptæ Hyperbolę ABC in puncto V, ipſaque aſymptotos
VY conueniat cum XZ aſymptoto circumſcriptæ ABC, vt modò oſtendi-
mus, ſi producatur, ſecabit quoque Hyperbolen ABC. Quare 2235. h. inſcriptæ ſecat Hyperbolen circumſcriptam.

Tandem cum harum ſectionum aſymptoti infra contingentem BY ſe mu-
tuò ſecent, & XZ aſymptotos circũſcriptæ BCP, cadat totas extra ipsã BCP,
harum aſymptoton occurſus erit extra eandem BCP, vt in 2: & cum VY 2
aſymptotos inſcriptæ, ſecet Hyperbolen BCP circumſcriptam, eſto earum
communis ſectio in 3, & recta 2 3, producatur ad inferiores partes 4, atque
ex 3 ducatur recta 3 5 parallela ad X 2 aſymptoton circumſcriptæ BC 3, quę
recta 3 5 nunquam conueniet cum ſectione 3 7 ad inferiores partes, 3334. h. etiam recta 3 4 nunquã conuenit cum ſectione BE 6 ad eaſdem partes (nam
eſt eius aſymptotos) & duæ rectæ 3 5, 3 4 ſunt ſemper ſimul recedentes,
& ad interuallum perueniunt maius quolibet dato interuallo; quare eò ma-
gis interuallum ſectionum BC7, BE6, datum quodcunque interuallum ex-
cedet. Quod erat vltimò demonſtrandum.

COROLL.

EX hac manifeſtum fit, quod Hyperbolarum per eundem verticem ſimul
adſcriptarum, & idem rectum latus habentium aſymptoti infra

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