Full text: Cavalieri, Bonaventura: Geometria indivisibilibvs continvorvm

LIBER V. integræ hyperbolæ, ſiat autem parallelogram mum ſub di-
cta baſi, & axi, vel diametio, in angulo ab eiſdon comen-
to, ſumpta baſi pro regula: Omnia quadrata dicti paralle. logrammi ad omnia quadrata trilinei extia hypeibolam
conſtituti, erunt vt idem parallelogran n@un ad ſuiieli-
quum ab eodem dempta ſemil yperbola, vna cum exceſſu,
quo dicta ſemihyperbola ſuperat. dicti parallel@ gram-
mi, cum {1/6}. parallelogrammi ſub targente hyperbolan, & axis, vel diametri hyperbolæ ea poitione, ad quam teli-
qua ſit, vt integra axis, vel dian eter ad eiuſdem latus
tranſuerſum.

Sit ergo axis, vel diameter hyperbolę, BE, cuius
dimidia, BED, latus tranſuerſum, AB, & in angu-
lo, BED, ſub, BE, ED, conſticutum parallelogrã-
mum, GE, ſit autem, vt, EB, ad, BA, ita, EH, ad, H
B, & per, H, ducta, HM, parallela ipſi, ED, quę ſu-
matur, pro regula, ita vt ſit conſtitutum parallelo-
grammum ſub, HB, & ſub, BG, quæ erit tangens
hyperbolam in puncto, B. Dico gitur omnia qua-
drata, BD, ad omnia quadrata trilinei, BGD, eſſe
ut, BD, ad ſui reliquum, dempto ab eodem ſemihyperbola, BE
D, vna cum exceſſu, quo ipſa ſuperat {1/3}. dicti paralle ogran mi, B
D, cum {1/6}. B M. Nam omnia quadrata, BD, ad rectangula ſub, B
D, & ſemihyperbola, BED, ſunt vt, BD, adiplam, BED, rectan-
gula verò ſub, BD,, & BED, æquantur rectangulis ſub, BOD, B
ED, ſimul cum omnibus quadratis, BED, ergo omnia quadrata,
BD, ad rectangula ſub, BGD, BED, cum ommbus quadratis, BE
D, erunt vt, BD, ad, BED; ſunt autem omnia quadrata, BD, ad
omnia quadrata, BED, vt, AE, ad compoſitam ex {1/2}. AB, & {1/3}. B
E, . i. vt, BE, ad compoſitam ex {1/2}. BH, & {1/3}. HE, quia, AE, BE,
proportionaliter diuiduntur in punctis, B, H, . i. vt parallelogram-
mum, BD, ad compofitum ex {1/2}. BM, & {1/3}. HD, . i. vt, BD, ad
compoſitum ex {1/3}. BD, & {1/6}. BM, ergo omnia quadrata, BD, ad
rectangula ſub, BGD, BED, erunt vt, BD, ad exceſſum, quo ſe-
mihyperbola ſuperat {1/3}. BD, cum {1/6}. BM, erant autem omnia qua-
drata, BD, ad rectangula ſub, IGD, BED, vna cum ommb. qua-
dratis, BED, vt, BD, ad, BED, ergo omnia quadrata, BD, ad
rectangula bis ſub, BGD, BED, vna cum omnibus quadratis, BE

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