321301LIBER IV.
THEOREMA XII. PROPOS. XIII.
SIab extremo puncto baſis datæ parabolæ ducatur vſq;
ad
curuam parabolæ ſupra, vel infra baſim (indefinitè
producta ipſa curua) recta linea: Data parabola ad ſegmen-
ta ſub ductis lineis, & curua ab ijſdem abſciſſa comprehen-
ſa, ſingillatim ſumpta, erit vt cubus baſis ipſius datæpara-
bolæ ad cubum rectæ lineæ dicto puncto interceptæ, & alio
puncto eiuſdem baſis productæ, ſi opus ſit, in quod cadit
recta linea, quæ ducitur ab alio extremo puncto baſis re-
ſecti ſegmenti parallela axi, vel diametro ipſius datæ pa-
rabolæ.
curuam parabolæ ſupra, vel infra baſim (indefinitè
producta ipſa curua) recta linea: Data parabola ad ſegmen-
ta ſub ductis lineis, & curua ab ijſdem abſciſſa comprehen-
ſa, ſingillatim ſumpta, erit vt cubus baſis ipſius datæpara-
bolæ ad cubum rectæ lineæ dicto puncto interceptæ, & alio
puncto eiuſdem baſis productæ, ſi opus ſit, in quod cadit
recta linea, quæ ducitur ab alio extremo puncto baſis re-
ſecti ſegmenti parallela axi, vel diametro ipſius datæ pa-
rabolæ.
Sit ergo data parabola, HNB, inbaſi, HB, ſumpto autem vno
extremorum punctorum, H, B, ipſius baſis, H B, vtipſum, H, ab
eo ducatur vtcunq; recta linea, HA, ſupra baſim, HB, & indefi-
nitè producta curua, NAB, alia, HV, ſubterbàſim, vt ſint con-
ſtituta ſegmenta, ANH, VBNH, ſit autem axis, vel diameter,
NO, cui parallelæ ducantur per puncta, AV, verſus baſim, HB,
215[Figure 215]
productam, ſi opus ſit, occur-
rentes illi in punctis, X, C.
Dico ergo parabolam, HNB,
ad ſegmentum, HN. A, eſſe vt
cubus, HB, ad cubum, HC.
Eandem verò ad ſegmentum,
HNBV, eſſe vt cubum, BH,
ad cubum, HX, iungantur
puncta, B, A; B, N; N, H,
& ſit, CE, tertia proportiona-
lis duarum, quarum prima eſt
tripla, CH, ſecunda autem ipſa, BC. Quoniam ergo triangula,
11Coroll.1.
19.huius. NBH, BAH, ſunt in eadem baſi, BH, erunt inter ſe, vt altitu-
dines, vel vt lineæ, quæ a verticibus, NA, ad baſes ductæ cum
eiſdem æqualiter inclinantur . i. triangulum, HNB, ad triangu-
lum, HAB, erit vt, NO, ad, AC, . i. vt rectangulum, HOB,
ad rectangulum, HCB. Inſuper triangulum, HNB, ad portion-
22Defin.12.
l.1. culam, ASB, habet rationem compoſitam ex ratione trianguli,
HNB, ad triangulum, HAB, . i. ex ratione rectanguli, HOB,
ad rectangulum, HCB, & ex ratione trianguli, HAB, ad
extremorum punctorum, H, B, ipſius baſis, H B, vtipſum, H, ab
eo ducatur vtcunq; recta linea, HA, ſupra baſim, HB, & indefi-
nitè producta curua, NAB, alia, HV, ſubterbàſim, vt ſint con-
ſtituta ſegmenta, ANH, VBNH, ſit autem axis, vel diameter,
NO, cui parallelæ ducantur per puncta, AV, verſus baſim, HB,
rentes illi in punctis, X, C.
Dico ergo parabolam, HNB,
ad ſegmentum, HN. A, eſſe vt
cubus, HB, ad cubum, HC.
Eandem verò ad ſegmentum,
HNBV, eſſe vt cubum, BH,
ad cubum, HX, iungantur
puncta, B, A; B, N; N, H,
& ſit, CE, tertia proportiona-
lis duarum, quarum prima eſt
tripla, CH, ſecunda autem ipſa, BC. Quoniam ergo triangula,
11Coroll.1.
19.huius. NBH, BAH, ſunt in eadem baſi, BH, erunt inter ſe, vt altitu-
dines, vel vt lineæ, quæ a verticibus, NA, ad baſes ductæ cum
eiſdem æqualiter inclinantur . i. triangulum, HNB, ad triangu-
lum, HAB, erit vt, NO, ad, AC, . i. vt rectangulum, HOB,
ad rectangulum, HCB. Inſuper triangulum, HNB, ad portion-
22Defin.12.
l.1. culam, ASB, habet rationem compoſitam ex ratione trianguli,
HNB, ad triangulum, HAB, . i. ex ratione rectanguli, HOB,
ad rectangulum, HCB, & ex ratione trianguli, HAB, ad
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