Full text: Cavalieri, Bonaventura: Geometria indivisibilibvs continvorvm

Sint intra curuam parabolicam, BAC, duæ vtcunquæ ductæ in
eandem terminatæ, DF, MC, quarum, DF, rectè, altera, MC,
obliquè ſecet axem, AP, ſit autem deſcripta linea, HR, vt ſit con-
ſtituta, HRC, figura diſtantiarum portionis, MFC, & ab eodem
vertice, H, à quo ducitur linea, HR, ducatur, HQ, parallela
axi, AP, & ſint diametri, AZ, HO, parabolarum, DAF, M
HC, inter ſeæquales. Dico ergo omnia quadrata parabolæ, DA
F, regula, DF, eſſe æqualia rectangulis ſub parabola, MHC, re-
gula, MC, & ſub, HRC, figura diſtantiarum eiuſdem parabolæ,
MHC. Iungantur ergo, DA, AF, MH, HC, & à puncto, M,
ducatur, MX, axi, AP, æquidiſtans, à puncto verò, C, perpendi-
cularis axi, AP, producta vſq; in, B, tandem à puncto, H, ipſa, H
I, perpendicularis ipſi, MC: Omnia ergo quadrata, DAF, para-
bolæ, regula, DF, adrectangula ſub parabola, MHC, regula, M
C, & ſub trilineo, HRC, habent rationem compoſitam ex ea,
quam habent omnia quadrata parabolæ, DAF, regula, DF, ad
omnia quadrata parabolæ, MHC, regula, MC, & ex ea, quam
habent omnia quadrata parabolę, MHC, regula, MC, adrectan-
gula ſub parabola, MHC, & ſub trilineo, HRC, regula eadem,
MC: Omnia verò quadrata para-
bolæ, DMF, regula, DF, ad om-
nia quadrata parabolæ, MHC, re-
gula, MC, ſunt vt omnia quadrata
trianguli, DAF, regula, DF, ad
omnia quadrata trianguli, MHC,
regula, MC, nam omnia quadrata
parabolarum ſunt ſexquialtera om-
nium quadratorum triangulorum in
eiſdem baſibus, & circa eoſdem axes cum ipſis conſtitutorum, regu-
lis baſibus: Omnia inſuper quadrata trianguli, DAF, regula, DF,
ad omnia quadrata trianguli, MHC, regula, MC, habent ratio-
nem compoſitam ex ratione altitudinum, & quadratorum baſium
. i. ex ratione, quam habet, AZ, ad, HI, & ex rationẽ, quam ha-
bet quadratum, DF, ad quadratum, MC, vel quadratum, ZF,
ad quadratum, OC, eſt autem, AZ, æqualis ipſi, HO, ex hypo-
teſi, & , ZF, ipſi, QC, ergo omnia quadrata trianguli, DAF, ad
omnia quadrata trianguli, MHC, regulis iam dictis, habebunt ra-
tionem compoſitam ex ea, quam habet, OH, ad HI, & ex ea,
quam habet quadratum, QC, ad qu adratum, CO, quia verò trian-
guli, HIO, OQC, ſunt æquianguli, ideò, OH, ad, HI, erit vt,
OC, ad, CQ, ergo illa habebunt rationem compoſitam ex ea, quam
habet, OC, ad, CQ, & quadratum QC, ad quadratum, CO, eſt

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