Full text: Cavalieri, Bonaventura: Geometria indivisibilibvs continvorvm

451. THEOREMA XXVI. PROPOS. XXVIII.

SI intra curuam parabolicam duæ vtcunque ductæ fue-
rint rectæ lineæ in eandem terminantes, quarum vna
rectè, altera obliquè ſecet axim; omnia quadrata conſtitu-
tæ parabolæ per eam, quæ axim rectè ſecat, regula eadem,
ad rectangula ſub parabola conſtituta per obliquè ſecantem
axem, regula huius baſi, & ſub ſigura diſtantiarum eiuſ-
dem parabolæ, erunt vt quadratum axis primò dictæ para. bolæ ad quadratum diametriſecundò dictæ parabolæ.

Sintigitur intra curuam parabolicam, ADH, duæ ductæ rectæ
lineæ in eadem terminantes, quarum vna rectè, altera obliquè ſecet
axim, ſi ergo conſtitutarum ab ijſdem parabolarum diametri ſunt
æquales, pater veritas Propoſitionis ex antecedenti Theor. non ſint
autem conſtitutarum parabolarum diametri æquales, ſint autem
duæ parabolas conſtituentes, AH, rectè ſecans axem, DO, & C
G, obliquè ipſum diuidens, exiſtatq; axis, DO, maior diametro parabo-
læ, CEG, quæ ſit, EM, & ſit du-
cta linea, ER, & conſtituta, ER
G, figura diſtantiarum parabolæ, C
EG. Dico ergo omnia quadrata
parabolæ, ADH, regula, AH,
ad rectangula ſub parabola, CEG,
& trilineo, ERG regula, CG,
eſſe vt quadratum, DO, ad quadratum, EM, abſcindatur ergo
ab, OD, DN, æqualis ipſi, EM, & per, N, ducatur ipſi, AH,
parallela, BF. Omnia ergo quadrata parabolæ, ADH, ad omnia
quadrata parabolæ, BDF, regula communi, AH, vel, BF, ſunt
vt qúadratum, OD, ad quadratum, DN, vel ad quadratum, E
M, ſedomnia quadrata parabolæ, BDF, regula, BF, ſunt æqua-
lia rectangulis ſub parabola, CEG, & trilineo, ERG, regula, C
G, ergo omnia quadrata parabolæ, ADH, regula, AH ad re-
ctangula ſub parabola, CEG, & trilineo, ERG, regula, CG,
erunt vt quadratum, OD, ad quadratum, EM, quod erat oſten-
dendum.

451.1.

0340-01
22. huius.
Ex antec.

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