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

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4424GEOMETRIÆ
Sit cylindricus, AE, ſectus a planis quomodocumque per latera.
Dico per eadem diuidi in cylindricos; ſint autem ſecantia plana, quę
in
cylindrico, AE, producant parallelogramma, AE, ME.
Quia
igitur
, AE, eſt parallelogrammum, ſi in ipſo ducantur rectæ lineæ
ipſi
, AD, HE, parallelæ, &
in, AH, DE, terminatæ, erunt ei-
ſdem
, AD, HE, æquales, &
ſubinde erunt æquales, & parallelæ
11Ex def. 3. regulælateris cylindrici, AE, vnde erit, AE, ſuperficies cylindra-
cea
deſcripta latere, AD, ſiue latere cylindrici, AE, ergo ſolidum,
ARXE
, erit cylindricus.
Eodem pacto oſtendemus ſolida, AM
HDVE
, MZHVIE, eſſe cylindricos, talibus igitur planis cy-
lindricus
, AE, ſemper diuiditur in cylindricos, quæ eſt prior pars
huius
Theorematis.
Secetur nunc duobus planis vtcumque inter ſe parallelis coinci-
dentibus
cum omnibus ciuſdem lateribus, quæ in cylindrico, AE,
producant
figuras, BNGK, COFL.
Dico ſolidum compræhen-
18[Figure 18] ſum inter has figuras, &
ijs incluſam ſuperficiem
cylindraceam
, eſſe cylindricum.
Sintadhuc pla-
na
per latera cylindrici, AE, vtcumque ducta, A
E
, ME, quæ ſecent figuras, BNGK, COFL,
in
rectis, BG, CF, NG, OF, igitur eiuſdem pla
ni
, &
ipſarum, BNGK, COFL, communes ſe-
ctiones
erunt parallelæ, quę ſint, BG, CF, ſicut
etiam
ipſæ, NG, OF, ſunt autem parallelę etiam
ipſæ
, BC, NO, GF, ergo, BF, NF, erunt pa-
rallelogramma
, &
latera eorumdem, BC, GF,
NO
, inter ſe æqualia, &
æquidiſtantia; ſi igitur
eorum
quoduis, vt, GF, ſtatuatur pro regula lateris ylindrici, ſu-
perficies
incluſa duabus figuris, BNGK, COFL, erit deſcripta
vno
laterum, BC, vel, NO, properante per circuitum figuræ, C
OFL
, ſemper ipſi, GF, æquidiſtante, donec redeat vnde diſceſſit,
igitur
hæc erit ſuperſicies cylindracea, cuius oppoſitæ baſes ipſæ fi-
guræ
, BNGK, COFL, &
ſolidum eiſdem incluſum erit cylindri-
cus
, quod erat poſterior pars huius Theorematis à nobis demon-
22Def. 3.ſtranda.
THEOREMA VIII. PROPOS. XI.
CViuſuis cylindrici oppoſitæ baſes ſunt fimiles, æquales,
&
ſimiliter poſitæ.

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