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

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9979LIBERI. ZX. Dico hanc tangere dicta ſolida, ſi enim non tangit ſecet, ve-
luti, D2N, in puncto, N, igitur punctus. n. erit extra planum figu-
ræper axem, nam ipſa, D2N, eſt parallela ipſi, ZX, quæ eſt ad
rectos angulos figuræ per axem tranſeunti, & ideò etiam, D2N,
118. Vndec.
Elem. eſt illi ad rectos angulos, occurrit autem illi in puncto, 2, ergo non
occurret illi in alio puncto, ergo, N, eſt extra planum figuræ per a-
xem, ducatur per, N, planum æquidiſtans plano, PXRZ, circuli,
2234. huius. quod producat circulum, BNFC, & ſit, BF, communisſectio ip-
ſius circuli, & figuræ per axem, quæ erit ipſius circuli diameter, & ,
33Corol. 34
huius. N, non erit aliquis punctorum, BF, ergo ſi ab, N, duxerimus ipſi,
ZX, parallelam, vt, NC, cum etiam, BF, ſit parallela ipſi, PR,
continebunt angulos æquales, ſed, ZX, ſecat perpendiculariter, P
4410. Vnd.
Elem. R, ergo, NC, ſecabit perpendiculariter, BF, ducta non ab extre-
mitate diametri, ergo intra circulum, BCFN, erit, & bifariam ſe-
cabitur ab ipſa, BF, ergo non tranſibit per circuitum figuræ per a-
xem ductæ, & per ipſum tranſit, D2N, ergo, NC, N2D, ſunt
duæ rectæ lineæ eidem, ZX, parallelę, ergo etiam inter ſe erunt pa-
rallelæ, quod eſt abſurdum, cum tranſeant per idem punctum, N,
ergo ducta per punctum ambitus figuræ per axem parallela ipſi, ZX,
tanget dicta ſolida: Sit nobis nunc punctus, N, pro puncto vtcung;
in ſuperficie ambiente ſumpto extra circuitum figuræ per axem, à
quo ducta ipſi, ZX, parallela, occurrat producta ſuperficiei ambienti
in puncto, C, oſtendemus ergo eodem modo ſupra adhibito (poſt-
quam duxerimus per, N, planum circulo, PXRZ, æquidiſtans,
quod in ſolido producat circulum, BNFC,) ipſam, NC, intra cir-
culum, BNFC, cadere, & bifariam diuidi à recta, BF, ſiue à figu@a
per axem ducta (nam eſt, NC, perpendicularis ipſi, BF,) quod o-
ſtendere opus erat.

THEOREMA XXXIV. PROPOS. XXXVII.

SI ſolidum rotundum, vel conus ſcalenus, ſecentur plano
per axem, & deinde alio plano ſecentur, cuius, & vnius
planorum rectè axem ſecantium communis ſectio ſit recta li-
nea perpendicularis communi ſectioni eiuſdem, & plani per
axem; figura à ſecundo ſecante plano in ſolido producta erit
circa axem, in cono ſcaleno autem erit circa axem, vel dia-
metrum, & axis, vel diameter erit communis ſectio per dicta
ſecantia plana productarum figurarum.

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