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

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5636

Iam applicata quacunque OPQR, tùm in Parabola AED, tùm in ſemi-
Parabola DHC; cum ſit quadratum AD ad OP vt linca GF ad FS, vel vt DH
ad HQ, vel vt quadratum DC ad QR, ſintque antecedentia AD, DC ęqua-
lia, erunt & conſequentia OP, QR æqualia, nempè applicata OP æqualis ap-
plicatæ QR, & ita de omnibus & c. quare integra Parabole AED æquatur
ſemi-Parabolæ DHC.

Amplius ducta quacunque TVX
parallela ad BD, erit BD ad TX, vt

32[Figure 32] rectangulum ADC ad AXC, vel vt
HD ad VX, & permutando, cum ſit
BD dupla DH, & TX erit dupla XV,
& ſic de omnibus interceptis, & æ-
quidiſtantibus in ſemi-Parabola DB
C, & in ſemi-Parabola DHC, vnde
tota ſemi-Parabole DBC dupla eſt
totius ſemi-Parabolæ DHC, & ſum-
ptis æqualibus; ſemi-Parabole ABD
dupla Parabolæ AFD, ſiue trilineum
ANBDFA, æquale erit Parabolæ
AFD.

Tandé, ſi ſit AE contingens ABC
in A, erit EB æqualis BD, & ducta
in trilineo AEBDFA quacunque IKZ parallela ad ED, erit IK æqualis 113. Co-
roll. 13. h.& ſic de omnibus alijs interceptis in trilineis AEBNA, & ANBDFA quare
totum trilineum AEBNA æquabitur toto trilineo ANBDFA, ſed hoc, modò
oſtenſum fuit æquale Parabolæ AFD, quapropter totum triangulum AED
erit ſeſquialterum ſemi-Parabolæ ABD, vel erit vt 6 ad 4, ſed ad triangulum
ABD eſt vt 6 ad 3; quare ſemi - Parabole ABD ad inſcriptum triangulum
ABD erit vt 4 ad 3, & duplum ad duplum, hoc eſt Parabole ABC ad trian-
gulum ABC, ſuper eadem baſi AC, & eiuſdem altitudinis cum Parabola,
erit vt 4, ad 3, nempe ſeſquitertium. Quod erat demonſtrandum.

Sed iam tempus eſt vt ſuſceptum opus aggrediamur, initio facto à defini-
tionibus.

Definitiones Secundæ.
I.

CONI SECTIONES ÆQVALITER INCLINAT Æ

33[Figure 33] vocentur illę, quarum ordinatim ductæ æquales inuicem
angulos cum earum diametris efficiunt.

Videlicet coni - ſectio ABC vocabitur æqualiter incli-
nata, vel eiuſdem inclinationis, ac ſectio conica DEF, cum
vtriuſque ordinatim ductæ AGC, DHF, earum diametros
BG, EH, ad æquales diuidunt angulos, hoc eſt cum an-
gulus AGB, angulo DHE, & qui ei deinceps CGB reli-
quo FHE æqualis fuerit.

II.

Coni-ſectio vel circulus, coni-ſectioni, vel circulo

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