544524GEOMETRIE
ra autem fuerit indiuiſa:
Solidum rectangulum ſub indi-
uiſa, & ſub diuiſa contentum, æquabitur ſolidis rectangu-
lis ſub eadem indiuiſa, & ſub partibus diuiſæ, regulis ijſ-
dem, contentis.
uiſa, & ſub diuiſa contentum, æquabitur ſolidis rectangu-
lis ſub eadem indiuiſa, & ſub partibus diuiſæ, regulis ijſ-
dem, contentis.
Sint duæ expoſitæ ſuperficies, AC, CH, ſolidum recrangulum,
FC, iuxca regulas, kC, CB, continentes, earum autem altera, vt,
AC, ſit diuila in quotcumq; partes, vt per lineam, DEC, ſecan-
tem quaſcumq; intra ſuperficiem, AC, ipſiregulæ, BC, parallelas,
357[Figure 357]
in duas partes, DEC, ADECB, ipſa
verò, HC, ſit indiuiſa. Dico ſolidum
contentum ſub indiuiſa, HC, & ſub
diuiſa, AC, ideſt, FC, æquari ſolidis
contentis ſub, DEC, CH, & ſub, DE
CBA, & ſub eadem, CH. Intelliga-
tur ergo quandam rectam lineam fer-
ri peripſam, CED, indefinitè produ
ctam, donec totam percurrerit, ac
ſemper moueri ipſi regulæ, KC, æqui-
diſtanter, deſcribet ergo ſuperficiem
cylindraceam, quæ ſit, KEH, & ab-
ſcindet à ſuperſiciebus, FK, AC, ſu-
perficies cylindraceas, HIK, DEC, & , HC, eſt cylindracea, & hoc
ſiue ſit in ambitu contenti ſolidi, ſiue non, alioquin non poſſent
latera, quæ per ſolidum, FC, ſecantia plana, ipſi, GC, æquidiſtã-
tia ſignantur in ipſa ſuperficie, HC, omnia vni regulæ, kC, æqui-
diſtare, ergo ſolidum, HIKCED, ſuperficiebus cylindraceis com-
prehenditur, quarum regulæ ſunt, kC, CB, inuicem perpendicula-
res ergo ſi ſolidum, HIKCED, ſecetur planis ipſi, kB, parallelis fiẽt
in ſolido parallelogramina ipſi, kB, æquiangula, hoc eſt rectangu-
la, & ideò dictum ſolidum erit ſolidum rectangulum contentum
ſub, HC, CED, ſuperficiebus: Eodem modo oſtendemus, HIKC
EDAG, eſſe ſolidum rectangulum contentum ſub ſuperficie, DIC,
hoe eſt, Dk, il i homologa iuxta planum, BK, ac ſub, DECBA, eſt
autem ſolidum, FC, æquale duobus ſolidis, HIC, CIHFB, ſimul
ſumptis, ergo ſolidum rectangulum contentum ſub indiuiſa ſuperö
ficie, HC, & ſub diuiſa, AC, æquale eſt ſolidis rectangulis conten-
tis ſub eadem indiuiſa, HC, & ſub partibus diuiſæ, DEC, DECB
A, regulis ſemper ijſdem, BC, CK, retentis, quod oſtendere opus
erat.
FC, iuxca regulas, kC, CB, continentes, earum autem altera, vt,
AC, ſit diuila in quotcumq; partes, vt per lineam, DEC, ſecan-
tem quaſcumq; intra ſuperficiem, AC, ipſiregulæ, BC, parallelas,
verò, HC, ſit indiuiſa. Dico ſolidum
contentum ſub indiuiſa, HC, & ſub
diuiſa, AC, ideſt, FC, æquari ſolidis
contentis ſub, DEC, CH, & ſub, DE
CBA, & ſub eadem, CH. Intelliga-
tur ergo quandam rectam lineam fer-
ri peripſam, CED, indefinitè produ
ctam, donec totam percurrerit, ac
ſemper moueri ipſi regulæ, KC, æqui-
diſtanter, deſcribet ergo ſuperficiem
cylindraceam, quæ ſit, KEH, & ab-
ſcindet à ſuperſiciebus, FK, AC, ſu-
perficies cylindraceas, HIK, DEC, & , HC, eſt cylindracea, & hoc
ſiue ſit in ambitu contenti ſolidi, ſiue non, alioquin non poſſent
latera, quæ per ſolidum, FC, ſecantia plana, ipſi, GC, æquidiſtã-
tia ſignantur in ipſa ſuperficie, HC, omnia vni regulæ, kC, æqui-
diſtare, ergo ſolidum, HIKCED, ſuperficiebus cylindraceis com-
prehenditur, quarum regulæ ſunt, kC, CB, inuicem perpendicula-
res ergo ſi ſolidum, HIKCED, ſecetur planis ipſi, kB, parallelis fiẽt
in ſolido parallelogramina ipſi, kB, æquiangula, hoc eſt rectangu-
la, & ideò dictum ſolidum erit ſolidum rectangulum contentum
ſub, HC, CED, ſuperficiebus: Eodem modo oſtendemus, HIKC
EDAG, eſſe ſolidum rectangulum contentum ſub ſuperficie, DIC,
hoe eſt, Dk, il i homologa iuxta planum, BK, ac ſub, DECBA, eſt
autem ſolidum, FC, æquale duobus ſolidis, HIC, CIHFB, ſimul
ſumptis, ergo ſolidum rectangulum contentum ſub indiuiſa ſuperö
ficie, HC, & ſub diuiſa, AC, æquale eſt ſolidis rectangulis conten-
tis ſub eadem indiuiſa, HC, & ſub partibus diuiſæ, DEC, DECB
A, regulis ſemper ijſdem, BC, CK, retentis, quod oſtendere opus
erat.