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

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147123 _MI_ circuli, erunt, vt quadrata eorumdem numerorum diſparium ab vnita
te. Quod erat demonſtrandum.

PROBL. XXXI. PROP. LXXVIII.

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

Datæ Parabolæ, per punctum extra ipſam datum cum dato ſe-
mi- tranſuerſo latere MINIMAM Hyperbolen circumſcribere.

SIt data Hyperbole ABC, cuius centrum E, & punctum intra ipſam da-
tum ſit G. Oportet per G _MAXIMAM_ Parabolen inſcribere.

Iungatur EG ſecans Hyperbolen in B, & concipiatur EG eſſe mediam
arithmeticam, EB verò mediam geometricam inter eaſdem ignotas extre-
mas, quæ reperiantur, & ſint EH, EF, & per F applicetur AFC, & 1174. h. diametrum GF adſcribatur ipſi Hyperbolæ ABC, Parabole 2257. h. quarum communis applicata ſit AC. Dico ipſam Parabolen eſſe quæſitam.

Cum enim ſit FE ad EB, vt EB ad EH, erit

113[Figure 113] rectangulum FEH æquale quadrato EB, qua-
re AH Hyperbolen continget. Cumque 33conuerſ.
37. primi
conic.
Comand. EG media arithmetica inter FE, EH, erunt
ipſarum diſferentiæ FG, GH æquales, vnde
442. h. eadem AH Parabolen quoque continget: 5561. h. quare Parabole D G M Hyperbolæ ABC erit inſcripta. Quod autem ſit _MAXIMA_,
patet; cum quælibet alia per G adſcripta cum
recto minori, minor eſt AGC, quę verò cum
maiori, eſt quidem maior, ſed omninò ſecat
Hyperbolen ABC, cum ſectio Parabole in in-
finitum abeat, & ſuperficies ABCGA vndi-
que ſit clauſa. Quod primò, & c.

IAM ſit data Parabole AGC, & datum extra ipſam pũctum ſit B, per quod
oporteat, cum dato quolibet ſemi-tranſuerſo D, _MINIMAM_ Hyperbo-
len circumſcribere.

Ducatur per B diameter Parabolæ DGF, quæ vltra B producatur, ſuma-
turque BE ipſi D æqualis, & facta EB media geometrica, & EG media ari-
thmetica inter eaſdem ignotas extremas, reperiantur ipſæ extremæ, 6674. h. ſint EH, EF; & per F applicetur in Parabola AFC; & ſuper AC ad diame-
tri ſegmentum BF, cum ſemi- tranſuerſo BE deſcribatur Hyperbole ABC. 7757. h. Dico ipſam eſſe _MINIMAM_ quæſitam.

Si quidem iuncta HA, ijſdem omnino argumentis, ac ſupra, demonſtra-
bitur ipſam HA, & Parabolen, & Hyperbolen contingere, vnde ſectiones
ſe mutuò contingent, & Hyperbole ABC erit Parabolæ circumſcripta: 8861. h. eritque _MINIMA_; nam quælibet alia Hyperbole per B adſcripta cum eo-
dem tranſuerſo, ſed cum recto maiori, maior eſt ipſa ABC, quæ verò cum
minori, eſt quidem minor, ſed cum ipſi ABC ſit inſcripta, & ad partes

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