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

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11591 Dico hanc eſſe _MINIMAM_ circumſcriptam quæſitam.

Cum ſint enim ipſæ Parabolæ congruentes, & per diuerſos vertices ad-
ſcriptæ, erunt inter ſe nunquam coeuntes quare ABC datæ GDH erit 1142. h.cumſcripta.

Præterea, quælibet alia Parabole per B adſcripta cum recto, quod exce-
dat BF, maior eſt ipſa ABC, quę verò cum recto BO, quod minus ſit ipſo BF,
qualis eſt PBQ, eſt quidem minor ipſa ABC, ſed omnino ſecat inſcriptam
GDH. Quoniam ſi fiat vt FO ad OB, ita BD ad DE, ac per E applicetur
EGP ſecans DG in G, & BP in P: cum ſit BD ad DE, vt FO ad OB, erit com-
ponendo BE ad ED, vt FB ad BO; vnde rectangulum ſub BE, & BO 221. Co-
roll. 1. h. quadratum applicatæ EP in Parabola PBQ æquale erit rectangulo ſub me-
dijs ED, & BF, ſiue DI, hoc eſt quadrato applicatę EG in Parabola GDH: 33ibidem. vnde EP, EG ſunt æquales. Occurrit ergo Parabole BP, ſibi adſcriptæ DG
per diuerſos vertices, in puncto P, quare in eodem occurſu, & ad alteram
partem ſe mutuò ſecant. Quapropter congruens Parabole ABC erit 4450. h. _NIMA_ circumſcripta quæſita.

PROBL. XVIII. PROP. LII.

Datæ Hyperbolę, per punctum intra ipſam datum MAXIMAM
Hyperbolen inſcribere, quarum eadem ſit regula.

ESto data Hyperbole ABC, cuius centrum D; & punctum intra ipſam da-
tum ſit E. Oportet per E Hyperbolen inſcribere, quæ ſit _MAXIMA_,
ſed tamen eius regula ſit quoque regula datæ ſectionis.

Iungatur ED ſecans datã ſectionem in

80[Figure 80] B, & producta ſumatur DF æqualis BD,
erit FB trãſuerſum ſectionis ABC, 5547. 1.
conic. vertex B, ſitque BG eius rectum latus, &
regula FG, quæ producatur, & per E ſit
ducta EH parallela ad BG, & per verticẽ
E, circa communem diametrum BE, da-
tę ſectioni ABC adſcribatur 667. huius. IEL, cuius latera ſint FE, EH, hoc eſt
eadem ſit regula FGH: patet ipſam IEL
datæ ABC eſſe inſcriptam, cum in infini-
tum productæ ſint inter ſe 7745. h.coeuntes.

Dico ampliùs ipſam IEL eſſe _MAXI-_
_MAM_. Quoniam quęlibet alia adſcripta
per verticem E, cum eodem tranſuerſo
FE, ſed cum recto, quod minus ſit recto
EH, minor eſt ipſa IEL; quæ verò cum recto EO, quod excedat EH, 882. Co-
roll. 19. h. lis eſt Hyperbole PEQ, eſt quidem maior ipſa IEL; ſed omnino ſecat 99ibidem. ABC. Nam ſi fiat vt OH ad HE, ita BE ad EM, & per M applicetur MPA
Hyperbolen PEQ ſecans in P, BA verò in A, & producta ſecet regulam
FH, in N, & iunctam regulam FO deſcriptæ Hyperbolæ PEQ in R.

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