# Full text: Cavalieri, Bonaventura: Geometria indivisibilibvs continvorvm

## 424.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.

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,
productam, ſi opus ſit, occur-
rentes illi in punctis, X, C. Dico ergo parabolam, HNB,
ad ſegmentum, HN. A, eſſe vt
HNBV, eſſe vt cubum, BH,
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,
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,
culam, ASB, habet rationem compoſitam ex ratione trianguli,
HNB, ad triangulum, HAB, . i. ex ratione rectanguli, HOB,