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

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4323LIBER I. vel, DC, planum per, EM, XR, ductum æquidiſtabit plauo, AC,
eſt
autem planum, quod tangit cylindricum in, O, æquidiſtans pla-
no
, AC, &
tranſit per idem punctum, O, per quod tranſit planum
per
, EM, XR, ductum, ergo illa duo plana fiunt vnum planum,
iacet
autem, EM, in plano per, EM, XR, ducto, ergo iacet etiam
in
plano ęquidiſtante ipſi, AC, &
cylindricum, AC, tangente, igi-
tur
tangit cylindricum in recta, EM.
Eodem pacto ſi in alio pun-
cto
extra, EM, in ſuperficie cylindracea ſumpto tangeret cylindri-
cum
, AC, oſtenderemus tangere ipſum in latere, quod per tale pun-
ctum
tranſiret;
in quo caſu tangeret cylindricum in lateribus vno
pluribus
, vt contingere poteſt.
Tangat autem ſecundò ipſum in
plano
, igitur in eo plano ſnmpto vtcumque puncto, tanget cylindri-
cum
in latere tranſeunte per tale punctum, igitur planum contactus
tale
eſt, vt in eo omnes ductæ rectæ lineæ æquidiſtantes ipſi, EM,
ſint
latera cylindrici, AC, &
ſubinde eidem, EM, æqualia, vnde
ſuperſicies
, in qua iacent erit parallelogrammum, igitur planum
contactus
in hoc caſu erit parallelogrammum, &
erit æquiangulum
parallelogrammo
, AC, nam eius latera ſunt parallela lateribus pa-
rallelogrammi
, AC, &
ideò continent angulos æquales contentis
à
lateribus parallelogrammi, AC, vnde talia parallelogramma ſunt
æquiangula
, igitur contactus plani æquidiſtantis plano per latera
cylindrici
ducto, vel fit in latere, aut lateribus contacti cylindrici,
vel
in parallelogrammo, ſiue parallelogrammis, in eiuſdem ſuper-
ficie
iacentibus, &
æquiangulis ei, quod fit a plano per latera ducto,
quod
oſtendendum erat.
THEOREMA VII. PROPOS. X.
SI cylindricus quomodocumque ſecetur per latera, diui-
ditur
in cylindricos à ſecantibus planis, ſi autem ſecetur
planis
omnibus eiuſdem lateribus coincidentibus inter ſe
parallelis
;
ſolidum compræhenſum conceptis in cylindrico
figuris
, &
incluſa ſuperficie cylindracea, erit cylindricus.

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