323303LIBER IV.
ſit ab illis tribus æquale eſt cubo mediæ ideſt parallelepipedum ſub
1145.1.2. altitudine, CE, & ſub baſi rectangulo ipſius, BC, ductæ in tri-
plam, CH, æquabitur cubo, BC, remanet adhuc parallelepipe-
dum ſub altitudine, HB, baſi t@ibus rectangulis, BCH, quod (al-
titudinem, BH, diuidentes in duas ſilicet in, BC, CH,) diuidi-
mus in parallelepipedum ſub altitudine, HC, baſirectangulo, H
2235.1.2. CB, ter ſumpto ideſt in parallelepipedum ſub altitudine, BC, baſi
quadrato, CH, ter ſumpto, & in parallelepipedum ſub altitudine,
3336.1.2. BC, baſi rectangulo, BCH, terſumpto ideſt in parallelep pedum
ſub altitudine, HC, baſi quadrato, BC, ter ſumpto; parallepipe-
4436.1.2. dum ergo ſub altitudine compoſita ex, HB, CE, baſi rec@angu-
lo, HCB, ter ſumpto, æquatur parallelepipedis ter ſub, BC, &
quadrato, CH, terſub, HC, & quadrato, CB, cum cubo, CB,
ad hæc ergo ſimul ſumpta cubus, HB, erit vt parabola, HNB,
ad trilineum, HASB; quia verò parallelepipedum ter ſub, BC,
5538.1.2.& quadrato, CH, cum parallelepipedo ter ſub, HC, & quadrato,
CB, cum cubo, CB, deficiunt à cubo, BH, quantitate cubi, HC,
ideo, per conuerſionem rationis, parabola, HNB, ad ſegmentum,
HNA, erit vt cubus, BH, ad cubum, HC.
1145.1.2. altitudine, CE, & ſub baſi rectangulo ipſius, BC, ductæ in tri-
plam, CH, æquabitur cubo, BC, remanet adhuc parallelepipe-
dum ſub altitudine, HB, baſi t@ibus rectangulis, BCH, quod (al-
titudinem, BH, diuidentes in duas ſilicet in, BC, CH,) diuidi-
mus in parallelepipedum ſub altitudine, HC, baſirectangulo, H
2235.1.2. CB, ter ſumpto ideſt in parallelepipedum ſub altitudine, BC, baſi
quadrato, CH, ter ſumpto, & in parallelepipedum ſub altitudine,
3336.1.2. BC, baſi rectangulo, BCH, terſumpto ideſt in parallelep pedum
ſub altitudine, HC, baſi quadrato, BC, ter ſumpto; parallepipe-
4436.1.2. dum ergo ſub altitudine compoſita ex, HB, CE, baſi rec@angu-
lo, HCB, ter ſumpto, æquatur parallelepipedis ter ſub, BC, &
quadrato, CH, terſub, HC, & quadrato, CB, cum cubo, CB,
ad hæc ergo ſimul ſumpta cubus, HB, erit vt parabola, HNB,
ad trilineum, HASB; quia verò parallelepipedum ter ſub, BC,
5538.1.2.& quadrato, CH, cum parallelepipedo ter ſub, HC, & quadrato,
CB, cum cubo, CB, deficiunt à cubo, BH, quantitate cubi, HC,
ideo, per conuerſionem rationis, parabola, HNB, ad ſegmentum,
HNA, erit vt cubus, BH, ad cubum, HC.
Nuncdico parabolam, HNB, ad ſegmentum, HNBV, eſſe
vt cubum, BH, ad cubum, HX; ducatur per, V, ipſi, BH, pa-
rallela, VZ, ſecans curuam parabolæ productam in, Z, & à
66@. huius. puncto, H, ipſi, NO, vel, XV, demittatur parallela, HI, oc-
currens ipſi, VZ, in, I, eſt ergo parabola, BNH, ad parabo-
lam, VBNHZ, vt cubus, BH, ad cubum, VZ, item parabo-
la, VBNHZ, ad ſegmentum, VBNH, (quia, VH, eſt ſupra
baſim, VZ,) eſt vt cubus, ZV, ad cubum, VI, vel, XH; æqua-
lis, VI, quia, XI, eſt parallelogrammum; ergo, ex æquali, pa-
rabola, HNB, ad ſegmentum, HNBV, conſtitutum per lineam
ductam à puncto extremo, H, baſis, BH, properantem infra
eandem baſim, BH, erit vt cubus, BH, ad cubum, HX, quæ o-
ſtendenda erant.
vt cubum, BH, ad cubum, HX; ducatur per, V, ipſi, BH, pa-
rallela, VZ, ſecans curuam parabolæ productam in, Z, & à
66@. huius. puncto, H, ipſi, NO, vel, XV, demittatur parallela, HI, oc-
currens ipſi, VZ, in, I, eſt ergo parabola, BNH, ad parabo-
lam, VBNHZ, vt cubus, BH, ad cubum, VZ, item parabo-
la, VBNHZ, ad ſegmentum, VBNH, (quia, VH, eſt ſupra
baſim, VZ,) eſt vt cubus, ZV, ad cubum, VI, vel, XH; æqua-
lis, VI, quia, XI, eſt parallelogrammum; ergo, ex æquali, pa-
rabola, HNB, ad ſegmentum, HNBV, conſtitutum per lineam
ductam à puncto extremo, H, baſis, BH, properantem infra
eandem baſim, BH, erit vt cubus, BH, ad cubum, HX, quæ o-
ſtendenda erant.
THEOREMA XIII. PROPOS. XIV.
SIintra curuam parabolæ ducantur vtcunque duæ rectæ
lineæ in eandem curuam terminantes, parabola ab vna
ductarum conſtituta ad parabolam ab alia conſtitutam erit,
vt cubus primò ductæ ad cubum rectæ lineæ, quæ,
lineæ in eandem curuam terminantes, parabola ab vna
ductarum conſtituta ad parabolam ab alia conſtitutam erit,
vt cubus primò ductæ ad cubum rectæ lineæ, quæ,
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