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

360340GEOMETRIÆ compoſitam ex ea, quam habet rectangulum, ZPG, ad rectangu-
lum, ZQG, ideſt ex ea, quam habet, DP, ad, AQ, & ex ra-
tione rectanguli, ZQG, ad rectangulum, SOI, vel quadra-
ti, QG, ad quadratum, OI, ideſt ex ea, quam habet, QA, ad,
113. huius. AO, & ex ratione rectanguli, SOI, ad rectangulum, STI, ideſt
ex ratione, AO, ad, DT, ergo rectangulum, ZPG, vel, RTF, ad
rectangulum, STI, erit vt, PD, ad DT, abſciſſam. Et quoniam,
223. huius. HG, eſt parallelogrammum in eadem baſi, & altitudine cum fruſto,
BZGD, & per punctum, T, vtcunq. ſumptum ducta, BP, regulæ
parallela, quę eſt baſis, ZG, inuentũ eſt rectangulũ BTP, ad rectan-
gulũ, STI, eſſe vt, PD, ad DT; quatuor ergo horum magnitudinum
33Yux. Cor.
3. 26.l. 2. ordinibus conſtructis, iuxta has quatuor magnitudines, quę inuentę
ſunt eſſe proportionales, & hoc modo ſolito, reperimus rectangula
ſub, HP, PE, ad rectangula ſub portionibus, BZPD, DGP, eſſe vt
maximę abſciſſarum, DP, ad omnes abſciſſas, DP, recti, vel eiuſdẽ
44Corol. 2.
2. obliqui tranſitus . i. eſſe eorum dupla, quod oſtendere opus erat.

THEOREMA XLV. PROP. XLVII.

IN anteced. figura oſtendemus, regula eadem, ZG, omnia
quadrata, DG, ad omnia quadrata, DPG, eſſe vt, ZP,
ad compoſitam ex {1/3}. ZP, & {1/2}. PCOmnia verò quadrata,
DC, ad omnia quadrata trilinei, DGE, eſſe vt, ZP, ad ſui
reliquum, demptis ab eadem {2/3}. ZP, cum {1/6}, PG.

Rectangula enim ſub, HP, PE, adrectangula ſub, HP, & por-
55Coroll. 1.
26.I. 2. tione, DPG, ſunt vt, EP, ad portionem, DPG, . ſ. vt, ZP, ad
compoſitam ex {1/4}. ZP, & {1/6}. PG; eadem autem rectangula ſub, H
P, PE, ſunt dupla rectangulorum ſub portionibus, DBZP, DP
663. huius. G, . i. ſunt ad illa, vt, ZP, ad {1/2}. ZP, ergo ad reſiduum rectangulo-
77Ex antec.

244[Figure 244] rum ſub, HP, & , DPG, dem-
ptis rectangulis ſub portioni-
bus, DBZP, DGP, ideſt
88Jux. A. 23.
l. 2. ad rectangula ſub trilineo, D
P G, & trilineo, BHZ, . i. tri-
lineo, DEG, erunt vt, ZP,
ad {1/6}. PG, . i. ſumpta, PG, cõ-
muni altitudine, vt rectangu-
lum, ZPG, ad rectangulum
ſub, PG, & {1/6}. PG, . i. ad {1/6}.
quadrati, PG, ſunt autem omnia quadrata, DG, ad rectangula

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