552532GEOMETRI Æ
idem probabimus circa alias quaſcumque applicatas.
In ellipſi
verò oſtendemus rectangula, ℟XB, QDC, eſſe vt quadrata, BP, C
N, ſicut rectangula, ℟XB, QZC, vt quadrata, BP, CM. Ergo ſi
intelligamus ſolidum rectangulum fieri ſub parallelogrammis, GX,
362[Figure 362]
XE, &
quadratum ſolidum, EP,
communi regula, BP, erunt hæc
ſolida inter ſe æqualiter, vel pro-
portionaliter, analoga, cum ſint
in eiſdem planis parallelis, nem-
pè tranſeuntibus per lineas, ℟P,
GH, & quæcunq; plana his pa-
rallela præfata ſolida ſecantia,
producant in ipſis æquales figu-
ras planas, vel ſaltem proportionales, ſicut patuit de rectangulo, Q
ZC, æquali quadrato, CM; vel ad idem exiſtente, vt rectangulum,
℟XB, ad quadratum, BP. Eadem ratione, quia probauimus re-
ctangulum, QDC, æquari quadrato, CN, vel ad idem eſſe vt re-
ctangulum, ℟XB, ad quadratum, BP, concludemus ſolidum rectã-
gulũ ſub trapezio, EG℟X, & triangulo, EXB, eſſe ęqualiter, vel pro-
portionaliter, analogum quadr. ſolido, EBP, iuxta communem re-
gulam, BP, igitur rectangulum ſolidum, ſub GX, XF, æquabitur qua-
drato ſolido, EP, & rectangulum ſolidum ſub, EG℟X, EXB, ęqua-
bitur quadrato ſolido, EBP, vel ſaltem erunt proportionalia in el-
lipſi, ergo quadratum ſolidum, EP, ad quadr. ſolidum, EBP, erit
111. huius.223. huius. vt rectangulum ſolidum ſub, GX, XE, ad rectangulum ſolidum
ſub, EG℟X, & , EXB, hoc eſt, vt, ℟X, ad compoſitam ex {1/2}. ℟X,
& {1/6}. XB, ideſt, vt, RB, ad compoſitam ex {1/2}. RB, & {1/6}. BE, ergo,
iterum conſpecta figura prop. 1. lib. 3. quadratum ſolidum portio-
3316. huius. nis, DEP, ad quadratum ſolidum, EP, erit vt compoſita ex {1/6}. BE,
& {1/2}. BR, ad ipſam, BR, cum enim ſemiportiones, DEB, BEP,
ſint homologę ſecundum regulam planum tranſiens per regulam,
BP, cuiæquidiſtant plana ſolida ſecantia, ſicut etiam, FB, BH, &
cum quadratum ſolidum figuræ, FP, diuiſæ per lineam, EB, æque-
44Cor. 3. 15.
huius. tur quadratis ſolidis, FB, BH, & duobus rectangulis ſolidis ſub, FB,
BH, ideſt quatuor quadratis ſolidis, BH, ideò quadratum ſolidum,
FP, quadruplum erit quadrati ſolidi, BH, ſicut etiam patebit qua-
dratum ſolidum portionis, DEP, quadruplum eſſe quadrati ſolidi
ſemiportionis, EBP, ergo, vt quadratum ſolidum, EBP, ad qua-
dratum ſolidum, BH, ita eſt quadratum ſolidum portionis, DEP, ad
quadratum ſolidum, DH, ideſt vt compoſita ex {1/6}. BE, & {1/2}. BR, ad
ipſam, BR, quod oſtendendum erat.
verò oſtendemus rectangula, ℟XB, QDC, eſſe vt quadrata, BP, C
N, ſicut rectangula, ℟XB, QZC, vt quadrata, BP, CM. Ergo ſi
intelligamus ſolidum rectangulum fieri ſub parallelogrammis, GX,
communi regula, BP, erunt hæc
ſolida inter ſe æqualiter, vel pro-
portionaliter, analoga, cum ſint
in eiſdem planis parallelis, nem-
pè tranſeuntibus per lineas, ℟P,
GH, & quæcunq; plana his pa-
rallela præfata ſolida ſecantia,
producant in ipſis æquales figu-
ras planas, vel ſaltem proportionales, ſicut patuit de rectangulo, Q
ZC, æquali quadrato, CM; vel ad idem exiſtente, vt rectangulum,
℟XB, ad quadratum, BP. Eadem ratione, quia probauimus re-
ctangulum, QDC, æquari quadrato, CN, vel ad idem eſſe vt re-
ctangulum, ℟XB, ad quadratum, BP, concludemus ſolidum rectã-
gulũ ſub trapezio, EG℟X, & triangulo, EXB, eſſe ęqualiter, vel pro-
portionaliter, analogum quadr. ſolido, EBP, iuxta communem re-
gulam, BP, igitur rectangulum ſolidum, ſub GX, XF, æquabitur qua-
drato ſolido, EP, & rectangulum ſolidum ſub, EG℟X, EXB, ęqua-
bitur quadrato ſolido, EBP, vel ſaltem erunt proportionalia in el-
lipſi, ergo quadratum ſolidum, EP, ad quadr. ſolidum, EBP, erit
111. huius.223. huius. vt rectangulum ſolidum ſub, GX, XE, ad rectangulum ſolidum
ſub, EG℟X, & , EXB, hoc eſt, vt, ℟X, ad compoſitam ex {1/2}. ℟X,
& {1/6}. XB, ideſt, vt, RB, ad compoſitam ex {1/2}. RB, & {1/6}. BE, ergo,
iterum conſpecta figura prop. 1. lib. 3. quadratum ſolidum portio-
3316. huius. nis, DEP, ad quadratum ſolidum, EP, erit vt compoſita ex {1/6}. BE,
& {1/2}. BR, ad ipſam, BR, cum enim ſemiportiones, DEB, BEP,
ſint homologę ſecundum regulam planum tranſiens per regulam,
BP, cuiæquidiſtant plana ſolida ſecantia, ſicut etiam, FB, BH, &
cum quadratum ſolidum figuræ, FP, diuiſæ per lineam, EB, æque-
44Cor. 3. 15.
huius. tur quadratis ſolidis, FB, BH, & duobus rectangulis ſolidis ſub, FB,
BH, ideſt quatuor quadratis ſolidis, BH, ideò quadratum ſolidum,
FP, quadruplum erit quadrati ſolidi, BH, ſicut etiam patebit qua-
dratum ſolidum portionis, DEP, quadruplum eſſe quadrati ſolidi
ſemiportionis, EBP, ergo, vt quadratum ſolidum, EBP, ad qua-
dratum ſolidum, BH, ita eſt quadratum ſolidum portionis, DEP, ad
quadratum ſolidum, DH, ideſt vt compoſita ex {1/6}. BE, & {1/2}. BR, ad
ipſam, BR, quod oſtendendum erat.
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