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

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3212 dratum D C ad quadratum B G, quare quadratum B F maius eſt quadra-
to B G; ideoque punctum F cadit extra ſectionem, vt & quodcunque
aliud punctum rectæ A C F, præter C. Erit ergo recta A C F Parabolen
contingens in in C. Quod erat faciendum.

MONITVM.

PRopoſitio 34. primi conic. licet ab Apollonio negatiuè ſit demon-
ſtrata, facilè tamen ad affirmatiuam reducitur, ſi ex ip-
ſa in principio demantur ea verba. _Si enim fieri po-_
_teſt, ſecet vt E C F_, ad finem verò. _Quod fieri non_
_poteſt_; nam ibi linea H G oſtenditur minor G F, vnde punctum F
cadet extra ſectionem, & ſic quodcunque aliud punctum rectæ E C H
præter C, quare ipſa E C H ſectionem continget in C: ſed vt clariùs
idem pateat, en afferemus noſtram directè concluſam demonſtrationem,
de qua in præcedenti Monito, præmiſſo tantùm (vice propoſitionis 169.
ſeptimi Pappi, qua indiget Apolloniana propoſitio) ſequenti Lemmate, in
quo interim duæ ſimul circuli proprietates detegentur haud iniucundæ.

LEMMAI. PROP. III.

Si circuli diameter A B inæqualiter ſecetur in C, & ad mino-
rem partem C B producatur, ita vt ſit A D ad D B, vt A C ad
C B, & ex C erigatur perpendicularis C E, iungaturque D E.
Dico quadratum ipſius D E æquari rectangulo A D B.

Si verò in recto angulo D C E, quælibet alia ſubtenſa F G
applicetur ipſi D E æquidiſtans, productam diametri partem
ſecans in F, aut infra D, aut ſupra, & perpendicularem C E in
G. Dico ampliùs quadratum applicatæ F G ſemper excedere
rectangulum A F B.

QVò ad primum, ſit circuli centrum H, & iungatur H E.

Iam cum ſit A D ad D B, vt A C ad C B, erit componendo A D
cum D B ad D B, vt A B ad B C, & ſumptis antecedentium
ſubduplis, erit H D ad D B, vt H B ad B C, & perlconuerſionem rationis
D H ad H B, vt B H ad H C, vel vt D H ad H E (ipſi H B æqualis) ita
H E ad H C: quare triangula D H E, E H C, cum habeant circa com-
munem angulnm H latera proportionalia, ſimilia erunt, vnde angulus
D E H æquabitur angulo E C H, ſiue rectus erit, ideoque D E circulum
continget, hoc eſt quadratum D E æquabitur rectangulo A D B. Quod
primò, & c.

Ampliùs iungantur E B, E A, quas, recta F G producta ſecet, in I &

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