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

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9571 quiori, & hoc ſemper, & c. Quapropter huiuſmodi Hyperbolæ ſunt ſemper
ſimul recedentes. Quod ſecundò. & c.

Præterea ſit TX aſymptotos inſcriptæ DBE, & VZ aſymptotos circum-
ſcriptæ, quæ contingentem GB productam ſecent in X, Z; & cum huiuſmo-
di Hyperbole ſint ſimiles, ſintque earum aſymptoti VZ, TX ad partes ęqua-
lium inclinationum ductæ, erit angulus ZVB æqualis angulo XTB, 11Coroll.
40. h. TX æquidiſtat VZ, ſed eſt VZ. Aſymptotos circumſcriptæ, vnde TX pro-
ducta ſecabit circumſcriptam Hyperbolen ABC; ſecet ergo eam in 2, & 2211. h. per 2 applicetur 3 2 4 5 alteram aſymptoton, inſcriptam ſectionem, ac
diametrum ſecans in 3,4,5 dico huiuſmodi Hyperbolas, licet ſemper inter
ſe magis recedant, nnnquam tamen ad interuallum peruenire æquale inter-
uallo 3 2, quod inter æquidiſtantes aſymptotos intercedit, & iuxta ordina-
tim ductas metitur.

Nam cum in ſimilibus Hyperbolis ABC, DBE, ex æqualibus, immo ex
eodem diametri ſegmento B 5, ducta ſit quædam applicata 5 4 2 3 ſimi-
limium Hyperbolarum aſymptotos ſecans in 2, 3; erit intercepta 3340. h. applicatæ portio 3 2 in Hyperbola maiorum laterum, maior intercepta
portione 2 4, in Hyperbola minorum. Ampliùs applicata infra 3 2 4 5,
qualibet alia 6 7 8 9; erit ob eandem rationem, & portio 6 7 maior por-
tione 8 9, quare addita communi 7 8; erit 6 8 ſiue 3 2 maior 7 9, &
hoc ſemper, vbicunque ſit intercepta 8 9 infra 2 4 licet ipſae; interceptæ
continuè augeantur. Vnde ſimiles Hyperbolæ per eundem verticem ſimul
adſcriptę, quamuis ſint ſemper magis recedentes ad interuallum tamen non
perueniunt æquale cuidam dato interuallo. Quod erat vltimò, & c.

COROLL.

HInc eſt, quod ſimiles Hyperbolæ per eundem verticem ſimul adſcriptę
habent aſymptotos parallelas, & aſymptotos inſcriptæ ſecat Hyper-
bolen circumſcriptam: nam vltimò loco oſtẽdimus TX ęquidiſtare ipſi VZ,
& ſecare inſcriptam in 2.

THEOR. XXII. PROP. XXXXII.

Parabolæ congruentes, per diuerſos vertices ſimul adſcriptæ,
ſunt inter ſe nunquam coeuntes, & in infinitum productæ ad ſe pro-
pius accedunt, & ad interuallum perueniunt minus quolibet dato
interuallo.

SInt duæ congruentes Parabolæ ABC, DEF per diuerſos vertices B, E,
ſimul adſcriptæ, quarum recta latera ſint BG, EH (quæ inter ſe æqua-
lia erunt, cum ſectiones ponantur congruentes.) Dico primùm has in 441. Co-
roll. 19. h. finitum productas nunquam inter ſe conuenire.

Nam producta contingente HE ſectioni ABC occurrent in A, & C, hæc
erit quoque ordinatim ducta in ſectione ABC (cum ſint ſectiones ſimul ad-
ſcriptæ) & Parabole DEF tota cadet infra contingentem AEC,

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