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.
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.
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.
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.