Cavalieri, Buonaventura - Geometria indivisibilibvs continvorvm : noua quadam ratione promota

SI duæ rectæ lineæ ducantur, quarum altera parabolam
tangat, altera verò axi, vel diametro parabolæ æqui-
diſtanter ducta eandem ſecet, in idem punctum concurren-
tes: Omnia quadrata parallelogrammi, regula tangente,
erunt ſexcupla omnium quadratorum trilinei ſub dictis tan-
gente, & ſecante, & curua parabolæ ab ĳſdem incluſa,
comprehenſi.

Sit ſemiparabola, ACB, quam tangat lineæ, AD, & à pun-
cto, A, ducta, AB, axis, vel diameter, integræ parabolæ, dein-
de agatur vtcunque, DC, parallela axi, vel diametro, AB, ſecans
curuam parabolæ in, C, & occurrens tangenti, AD, in, D, duca-
tur tandem à puncto, C, ipſi, AD, æquidiſtans, CB, ſecans, AB,
in, B, regula autem ſit, AD. Dico ergo omnia quadrata parallelo-

230[Figure 230] grammi, AC, eſſe omnium quadratorum
trilinei, ADC, ſexcupla: Omnia enim
quadrata, AC, ad rectangula ſub, AC,
11Coroll. 1.
26. l. 2.& ſemiparabola, ABC, ſunt vt, AC, ad
ſemiparabolam, ABC, . i. ſexquialtera
. i. vt 6. ad 4. omnia autem quadrata, AC,
2221. huius. ad omnia quadrata ſemiparabolę ABC,
ſunt dupla . i. vt 6. ad 3. ergo omnia qua-
33Coroll.
23. l. 2. drata, AC, ad reſiduum demptis omni-
bus quadratis ſemiparabolæ, ABC, à re-
ctangulis ſub, AC, & ſemiparabola, ABC, . i. ad rectangula ſub
ſemiparabola, ABC, & trilineo, ADC, erunt in ratione ſex-
cuplaideſt vt 6. ad 1. ad eadem verò bis ſumpta, vt 6. ad 2. quo-
niam verò omnia quadrata, AC, ad omnia quadrata ſemipara-
bolæ, ABC, ſum vt 6. ad 3. vt dictum eſt, ideò omnia quadrata,
AC, ad omnia quadrata ſemiparabolæ, ABC, cum rectangulis
bis ſub ſemiparabola, ABC, & trilineo, ADC, erunt vt 6. ad 5.
ergo ad reliquum ſcilicet ad omnia quadrata trilinei, ADC, omnia
44D. 23. l. 2. quadrata, AC, erunt vt 6. ad 1. ideſt erunt eorundem ſexcupla,
quod oſtendere oportebat.

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