359339LIBER IV.
G, ad parallelepipedum ſub, BG, &
quadrato, HF;
item omnia
quadrata, AF, ad omnia quadrata, AG, ſunt vt quadratum, FH,
ad quadratum, HG, . ſ. ſumpta, BG, communi altitudine, vt pa-
rallelepipedum ſub, BG, & quadrato, FH, ad parallelepipedum
ſub, BG, & quadrato, HG: _Tandem omnia quadrata,_ AG, dupla
ſunt omnium quadratorum ſemiparabolæ, BHG, ergo, ex æquali,
omnia quadrata figuræ, CBHF, demptis omnibus quadratis trili-
nei, BCE, ad omnia quadrata ſemiparabolæ, BHG, erunt vt pa-
rallelepipedum ſub, BG, & his ſpatijs . ſ. quadrato, FG, {1/2}. quadra-
ti, GH, & rectangulo ſub, FG, & ſexquitertia, GH, ab eodem
dempto {1/6}. parallelepipedi ſub, CE, & quadrato, FG, ad dimidium
parallelepipedi ſub, BG, & quadrato, GH, quod erat demonſtran-
dum.
quadrata, AF, ad omnia quadrata, AG, ſunt vt quadratum, FH,
ad quadratum, HG, . ſ. ſumpta, BG, communi altitudine, vt pa-
rallelepipedum ſub, BG, & quadrato, FH, ad parallelepipedum
ſub, BG, & quadrato, HG: _Tandem omnia quadrata,_ AG, dupla
ſunt omnium quadratorum ſemiparabolæ, BHG, ergo, ex æquali,
omnia quadrata figuræ, CBHF, demptis omnibus quadratis trili-
nei, BCE, ad omnia quadrata ſemiparabolæ, BHG, erunt vt pa-
rallelepipedum ſub, BG, & his ſpatijs . ſ. quadrato, FG, {1/2}. quadra-
ti, GH, & rectangulo ſub, FG, & ſexquitertia, GH, ab eodem
dempto {1/6}. parallelepipedi ſub, CE, & quadrato, FG, ad dimidium
parallelepipedi ſub, BG, & quadrato, GH, quod erat demonſtran-
dum.
THEOREMA XLIV. PROP. XLVI.
IN parabola ducta axi, vel diametro æquidiſtanter rect@
linea, ſi deinde fiat parallelogrammum ſub eadem du-
cta, & ſub baſi, angulum habens æqualem angulo i
tionis eiuſdem ductæ ad baſim, regula ſumpta baſi. Re
gula ſub parallelogram, in quæ dictum parallelogram-
mum diuiditur à ducta linea, ſunt dupla rectangulorum
ſub portionibus fruſti parabolæ, dicto parallelogrammo in-
cluſæ, per eandem ductam conſtituris.
linea, ſi deinde fiat parallelogrammum ſub eadem du-
cta, & ſub baſi, angulum habens æqualem angulo i
tionis eiuſdem ductæ ad baſim, regula ſumpta baſi. Re
gula ſub parallelogram, in quæ dictum parallelogram-
mum diuiditur à ducta linea, ſunt dupla rectangulorum
ſub portionibus fruſti parabolæ, dicto parallelogrammo in-
cluſæ, per eandem ductam conſtituris.
Sit parabola, AZG, in baſi, ZG, circa axim, vel diametrum,
243[Figure 243]
AQ, cui parallela ducatur vt-
cumque recta, DP, fiat autem
parallelogrammum ſub, ZQ,
DP, angulum habens æqualẽ
angulo inclinationis, DP, ad
ZG, . i. angulo, qui ſit, DPG,
vtcunque exduobus, DPG,
DPZ, ſit autem hoc paralle.
logrammum, HG, regula ve.
ro, HG. Dico ergo, rectãgula
ſub, HP, PE, dupla eſſe rectãgulorũ ſub portionibus, BDPZ, DGP.
Sumpto ergo vtcunq; in, DP, puncto, T, per, T, ducatur, RF, ipſi,
ZG, æquidiſtans ſecanſq; curuam parabolæ in, SI, & , AQ, in, O.
Rectangulum ergo, ZPQ, ad rectangulum, STI, habet
cumque recta, DP, fiat autem
parallelogrammum ſub, ZQ,
DP, angulum habens æqualẽ
angulo inclinationis, DP, ad
ZG, . i. angulo, qui ſit, DPG,
vtcunque exduobus, DPG,
DPZ, ſit autem hoc paralle.
logrammum, HG, regula ve.
ro, HG. Dico ergo, rectãgula
ſub, HP, PE, dupla eſſe rectãgulorũ ſub portionibus, BDPZ, DGP.
Sumpto ergo vtcunq; in, DP, puncto, T, per, T, ducatur, RF, ipſi,
ZG, æquidiſtans ſecanſq; curuam parabolæ in, SI, & , AQ, in, O.
Rectangulum ergo, ZPQ, ad rectangulum, STI, habet
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