553533LIBER VII.
COROLLARIVM.
_E_x proximè dictis manifeſtum eſſe poteſt quadratum ſolidum cuiuſ-
cumq; figuræ circa diametrum, regula baſi, quadruplum eſſe
quadrati ſolidi cuiuſuis eiuſdem portionum, quæ ab ipſa diametro ſe-
parantur.
cumq; figuræ circa diametrum, regula baſi, quadruplum eſſe
quadrati ſolidi cuiuſuis eiuſdem portionum, quæ ab ipſa diametro ſe-
parantur.
ANNOTATIO.
POſterior pars prop.
1.
lib.
3.
oſtendetur vt ibi dicta nominum
tantum mutatione facta cum Cor. ſicut etiam prop. 2.
tantum mutatione facta cum Cor. ſicut etiam prop. 2.
THEOREMA XXI. PROPOS. XXI.
ASſumpto ex ſchemate prop.
3.
ſemicirculo, vel ſemiel-
lipſi, ASFD, circa diametrum, AD, ſimul cum appli-
catis, RF, MS, quarum altera ſit regula, & parallelogram-
mo, NR, oſtendemus in illius figura, quadratum ſolidum
FB, ad quadratum ſolidum portionis, ICFS, eſſe vt rectan-
gulum, DRA, ad rectangulom ſub, DR, & ſub compoſita
ex {1/2}. RM, & ex, MA, vna cum rectangulo ſub, RM, & ſub
compoſita ex {1/6}. RM, & {1/2}. MA.
lipſi, ASFD, circa diametrum, AD, ſimul cum appli-
catis, RF, MS, quarum altera ſit regula, & parallelogram-
mo, NR, oſtendemus in illius figura, quadratum ſolidum
FB, ad quadratum ſolidum portionis, ICFS, eſſe vt rectan-
gulum, DRA, ad rectangulom ſub, DR, & ſub compoſita
ex {1/2}. RM, & ex, MA, vna cum rectangulo ſub, RM, & ſub
compoſita ex {1/6}. RM, & {1/2}. MA.
Producantur enim indefinitè ipſæ applicatæ, SM, FR, verſus,
MR, à quibus abſcindantur, CR, HM, ſingillatim ipſi, DA, æqua-
363[Figure 363]
les, vt etiam, GQ, HK, ſin-
gillatim pariter æquales ipſi,
DR, & , YR, LM, æquales
ipſi, MA, & iungantur, HG,
kQ, LY, QL, & ſit, TX, quę
cumq; inter, RF, MS, diame-
tro, AD, ſimiliter applicata,
quæ indefinitè hincinde ex-
tendatur ſecans, NF, in, V, A
D, in, T, LY, in, I, LQ, in, O, KQ, in, Z, & , HG, in, P. Erunt
ergo, HQ, KY, LR, parallelogramma, & rectangulum, GQR, æ-
quabitur rectangulo, DRA, cum autem, GQ, ſit æqualis, DR, & ,
YR, ipſi, MA, erit, QY, æqualis, RM, hoc eſt ipſi, YL, eſt autem,
QY, ad, YL, vt, OI, ad, L, ergo, OI, æquatur, IL, ideſt, TM, & , Z
I, ipſi, RM, ergo, ZO, æquatur, RT, ergo rectangulum, POT,
MR, à quibus abſcindantur, CR, HM, ſingillatim ipſi, DA, æqua-
gillatim pariter æquales ipſi,
DR, & , YR, LM, æquales
ipſi, MA, & iungantur, HG,
kQ, LY, QL, & ſit, TX, quę
cumq; inter, RF, MS, diame-
tro, AD, ſimiliter applicata,
quæ indefinitè hincinde ex-
tendatur ſecans, NF, in, V, A
D, in, T, LY, in, I, LQ, in, O, KQ, in, Z, & , HG, in, P. Erunt
ergo, HQ, KY, LR, parallelogramma, & rectangulum, GQR, æ-
quabitur rectangulo, DRA, cum autem, GQ, ſit æqualis, DR, & ,
YR, ipſi, MA, erit, QY, æqualis, RM, hoc eſt ipſi, YL, eſt autem,
QY, ad, YL, vt, OI, ad, L, ergo, OI, æquatur, IL, ideſt, TM, & , Z
I, ipſi, RM, ergo, ZO, æquatur, RT, ergo rectangulum, POT,