Full text: Cavalieri, Bonaventura: Geometria indivisibilibvs continvorvm

GEOMETRI Æ libus figuris, in antecedentibus prop. conſideratis. Appendix au-
tem Cor 6 reſtauratione minimè indigere manifeſtum eſt. Et hæc
circa prop. lib. 4. adnotaſſe ſufficiat, reliquum eſt, vt ad lib. 5. exami-
nandum nos conferamus.

757. THEOREMA XXVII. PROPOS. XXVII.

IN Schemate prop. 1. lib. 5. regula eadem retenta, oſten-
demus quadratum ſolidum parallelogrammi, AF, ad
quadratum ſolidum hyperbolæ, DBF, eſſe vt, OE, ad com-
poſitam ex, NB, & . {1/3}. BE.

Aſſumatur . n. ex eo paral-
lelogrammum, CE, cum ſe-
mihyperbola, BEF, & recta,
OE, necnon, MG, quæcum-
q; ex ordinatim applicatis ad
diametrum, BE, extendantur
autem, CB, FE, & fiant BD,
EQ, ſingulæ æquales ipſi, E
O, necnon, RE, AB, ſingulæ
æquales ipſi, EB, & iungan-
tur, DQ, AR, AE, quas, GM,
indefinitè quoq; producta ſecet in punctis, P, S, T. Erunt ergo, D
R, DE, AE, parallelogramma. Quoniam verò quad. EF, ad quad. MH, eſt vt rectang. OEB, ad, OMB, hoc eſt vt rectangulum, QER,
ad rectangulum, PTS, permutando quadratum, FE, ad rectangu-
lum, QER, erit vt quadratum, HM, ad rectangulum, PTS, quod
& in cæteris oſten demus, ergo quadratum ſolidum, BEF, & re-
ctangulum ſolidum ſub trapezio, DQEA, & triangulo, AER, erunt
proportionaliter analoga, ac in proportione quadrati, FE, & re-
ctanguli, QER, Conſimili modo probabimus quadratum ſolidum,
CE, eſſe æqualiter analogum rectangulo ſolido ſub, OB, BR, & ad
ipſum pariter eſſe in proportione quadrati, EF, ad rectangulum, Q
ER, ergo dicta ſolida proportionalia erunt, & permutãdo quadratũ
ſolidumCE, ad quad, ſolidũ, BEF, erit vt rectangulum ſolidum ſub,
QB, BR, ad rectangulum ſolidum ſub, DQEA, ARE, hoc eſt vt, QE,
ad cõpoſitam ex {1/2}. QR, & {1/3}. RE, hoc eſt vt, OE; ad compoſitãex, N
B, (quæ eſt dimidia, BO,) & {1/3}. BE, igitur, viſo ſchemate dictę prop. 1 quadratum ſolidum, AF, ad quadratum ſolidum, DBF, erit vt, O
E, ad compoſitam ex, NB, & {1/3}. BE, quod demonſtrare oportebat.

757.1.

0562-01
39, & ſch.
40. l. 1.
17. huius.
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