Full text: Viviani, Vincenzio: De maximis et minimis, geometrica divinatio

diametro DE deſcribatur Parabole ADG, cuius AE ſit eius ſemi-applicata: dico primum Parabolen ADG, etiam ſi in infinitum producatur, totam ca-
dere intra ABC, & ſi ex A ducatur quæcunque ANMH, ipſam à Parabola
ADG ſecari in O, in eadem ratione, ac AC ſecatur in G, & AB in D.

Ducta enim ex A recta AP contingente Parabolen ABC, erit FB æqualis
BP; ideoque ED æqualis DR, vnde AR continget Parabolen ADG, & AH
ſecabit ipſam in O.

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Iam ductis ex H, O, ſemi - applicatis HI, OL, erit, ob Parabolen, FB ad
BI, vt quadratum AF ad HI, vel vt quadratum FM ad quadratum MI; qua-
re per Lemma præcedens, erit FB ad BM, vt BM ad BI, & per Coroll. eiuſ-
dem, in vtraque figura, erit FM ad MI, vt FB ad BM; eadem penitus ratio-
ne oſtendetur eſſe EN ad NL, vt ED ad DN, ſed eſt FB ad BM, vt ED ad
DN, quare & FM ad MI erit vt EN ad NL, ſed FM ad MI, eſt vt AM ad MH,
& EN ad NL, vt AN ad NO, quare AM ad MH erit vt AN ad NO, & in pri-
ma figura conuertendo, componendo, & permutando, HA ad AO, vt MA
ad AN; in ſecunda verò per conuerſionem rationis, conuertendo, & per-
mutando HA ad AO erit vt MA ad AN. Eſt igitur in vtraque figura HA ad
AO, vt MA ad AN, vel vt BA ad AD, ſed eſt BA maior AD ex conſtru-
ctione, quare & HA erit maior AO, ſed HA tota eſt intra Parabolen ABC,
vnde punctum O, quod eſt in Parabola ADG erit intra Parabolen ABC, & ſic de quocunque alio puncto Parabolæ ADG, etiam ſi ducta AH cadat in-
fra AC; quare ipſa cadit tota intra ABC: & cum ſit HA ad AO, vt BA ad
AD, vel vt FA ad AE, vel ſumptis duplis, vt CA ad AG, erit diuidendo
HO ad OA, vt CG ad GA. Quod erat, & c.

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