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

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11490

PROBL. XVII. PROP. LI.

Datæ Parabolæ per punctum intra ipſam datum MAXIMAM
Parabolen inſcribere, & è contra.

Datæ Parabolæ per punctum extra ipſam datum MINIMAM
Parabolen circumſcribere.

SIt data Parabole ABC, & oporteat primò per punctum D intra ipſam da-
tum _MAXIMAM_ Parabolen inſcribere.

Ducatur diameter

79[Figure 79] BDE, cuius rectum
latus ſit BF, (quod
in poſterùm intelli-
gatur ſemper ex ver-
tice cõtingenter ap-
plicatum ſectioni,
prout in prę; cedenti-
bus factum eſt, & in
quinta primarũ defi-
nitionũ monuimus)
& per verticem D,
circa diametrũ D E
adſcribatur datæ Parabolæ ABC Parabole GDH, cuius rectum DI 115. huius. le ſit recto BF; nam ipſa erit congruens datæ Dico hanc eſſe _MAXIMAM_
in ſcriptam quæſitam.

Nam cum ipſæ ſint congruentes Parabolæ per diuerſos vertices ſimul ad-
ſcriptæ erunt inter ſe nunquam coeuntes: quare GDH datæ ABC erit 2242. h. datum punctum D inſcripta.

Ampliùs, quælibet alia Parabole per verticem D adſcripta cum recto,
quod minus ſit ipſo DI minor eſt Parabola GDH, quæ verò cum recto 332. Co-
roll. 19. h. quod excedat ipſum DI, qualis eſt MDN, eſt quidem maior GDH, 44ibidem. omnino ſecat circumſcriptam ABC. Nam ſi fiat vt LI ad ID, ita BD ad DE,
& per E applicetur EMA ſecans BA in A, & DM in M. Cum ſit BD ad DE,
vt Li ad ID, erit componendo BE ad ED, vt LD ad DI; vnde rectangulum
ſub extremis BE, & DI, ſiue BF, hoc eſt quadratum applicatæ AE in 55Coroll.
1. huius. rabola ABC, æquale erit rectangulo ſub medijs ED, DL ſiue quadrato 66ibidem. plicatę; ME in Parabola MDN, ac ideò AE, ME ſunt æquales, quapropter
Parabole DN occurrit ſibi adſcriptæ BA, per diuerſos vertices in puncto M,
& ob id in eodem occurſu, & ad alteram quoque partem ſe mutuò ſecabũt: 7750. h. Itaque congruens Parabole GDH erit _MAXIMA_ inſcripta quæſita. Quod
primò, & c.

IAM datæ Parabolæ GDH, oporteat per punctum B extra ipſam datam
_MINIMAM_ Parabolen circumſcribere.

Ducatur BDE diameter datę; GDH, cuius rectum ſit DI, & ei adſcribatur
per B, cum recto BF, quod æquet ipſum DI, congruens Parabole ABC:

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