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

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9167

THEOR. XIX. PROP. XXXVIII.

In Parabolis quibuslibet, vel in ſimilibus Hyperbolis, aut ſimi-
libus Ellipſbus, ſegmenta diametrorum ſectionum lateribus pro-
portionalia, ſuſcipiunt applicatasijſdem lateribus proportionales.

SInt, vt in prima figura, duæquælibet Parabolæ, velvt in ſecunda, duæ
ſimiles Hyperbolæ, vel vt in tertia, duæ ſimiles Ellipſes ABC, DEF,
quarum diametrorum ſegmenta BG, EH, rectis earum lateribus BI, EL, vel
tranſuerſis BM, EN ſint proportionalia, dico & applicatas GA, HD ipſis la-
teribus eſſe proportionales.

Nam in Parabolis pri-

61[Figure 61] mùm, cum ſit rectum BI
ad rectum EL, vt ſegmẽ-
tum BG ad EH, erit per-
mutando IB ad BG, vt
LE ad EH, vnde rectan-
gulum I B G ſimile erit
rectangulo LEH, quare
rectangulum IBG ad LE
H, erit vt quadratum la-
teris I B ad quadratum
homologilateris LE, ſed
rectãgulum IBG 111. huius. tur quadrato GA, & re-
ctãgulum LEH quadra-
to HD, vnde quadratum
GA ad HD, erit vt qua-
dratum IB ad LE, vel
applicata GA ad HD, vt
rectum IB ad rectum LE.
In Hyperbolis autem, &
Ellipſibus cum ſit vt BI
ad EL, vel ob ſectionum ſimilitudinem, vt MB ad NE, ita BG ad EH, erit
permutando MB ad BG, vt NE ad EH, & in Hyperbolis, componendo, in
Ellipſibus autem diuidendo, MG ád GB, vt NH ad HE, quare rectangulum
MGB ſimile erit rectangulo NHE, ſed rectangulum MGB ad quadratum
GA eſt, vt MB ad BI, vel vt NE ad EL, vel vtrectangulum NHE ad 2221. pri-
mi conic. dratum HD, quare permutando rectangulum MGB ad rectangulum NHE,
vel (ob ipſorum rectangulorum ſimilitudinem) quadratum BG ad quadra-
tum EH, vel quadratum BI ad quadratum EL, erit vt quadratum GA ad
quadratum HD, hoc eſt rectum BI ad rectum EL, vt applicata GA ad appli-
catam HD. Quod erat, & c.

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