Viviani, Vincenzo - De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei

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9773 rallelæ, ex quò ſi iungatur MZ, & DY, ipſæ æquales, & parallelæ erunt, &
facta conſtructione vt ſupra, idem omninò demonſtrabitur, nempe interce-
ptam YX minorem adhuc eſſe ipſa DS. Huiuſmodi igitur Parabolæ con-
gruentes, quò magis à tangente EA remouentur ad ſe propiùs accedunt:
quod ſecundò, & c.

ALITER.

SEd hoc idem aliter in nouo hoc ſchemate, in quo item oſtendetur inter-
ceptam contingentem EA maiorem eſſe intercepta applicata DI, & DI
maiorem infra intercepta

66[Figure 66] ML, & hoc ſemper, ſi ſectio-
nes in infinitum producan-
tur. Ducta enim DN paral-
lela ad EB, eadem penitus
methodo, qua ſuperiùs vſi
ſumus, demonſtrabimus DN
ipſi EB æqualem eſſe, & pa-
rallelam, quare, & coniun-
ctæ BN, ED æquales erunt,
ac parallelæ: ſi ergo BN ſe-
cetur bifariam in O, duca-
turque POT diametro BE
æquidiſtans, patet ipſam
TOP eſſe vtriuſque 1146. pri-
mi conic. bolæ diametrum, & BN eſſe
vnam ei applicatarum in
Parabola ABC, vti etiam QDER ipſi NB æquidiſtantem: quapropter QP,
& PR æquales erunt, ſed eſt DP æqualis PE (ob parallelas, & quia NO
æquatur OB) quare reliquæ QD, ER æquales erunt, ideoque rectangulum
QDR æquabitur rectangulo QER. Ampliùs ducatur TV æquidiſtans ad
QR, vel ad NB: patet TV ſectionem contingere in T, & contingenti 2232. pri-
mi conic. occurrere in V, (nam hæc, cum ſecet in B alteram parallelarum BN, ſecat
quoque reliquam TV.) Cumque duæ rectæ TV, BV, ſectionem ABC con-
tingentes, in vnum conueniant, ſitque QR ipſi TV, atque IS, & AC ipſi BV
æquidiſtantes, ac ſe mutuò ſecantes in D, & E, erit rectangulum QDR ad
IDS, vt quadratum TV ad BV quadratum, itemque rectangulum QER 3317. tertil
conic. AEC, vt idem quadratum TV ad BV, quare vt rectãgulum QDR ad 44ibidem. ita rectangulum QER ad AEC, & permutando rectangulum QDR ad QER,
vt rectangulum IDS ad AEC, ſed QDR, QER ſunt ęqualia, vt modò oſten-
dimus, ergo & rectangulum IDS æquatur rectangulo AEC, ſiue quadrato
AE, quare vt SD ad EA, ita EA ad DI, ſed SD maior eſt EA, cum ſit 5532. h. maior CE ſiue EA, vnde AE quoque, maior erit DI. Eadem ratione oſten-
detur rectangulum LMX æquale quadrato AE: vnde rectangula IDS, LMY
inter ſe æqualia erunt, ſed eſt latus MY maius later@ DS, cum eius ſegmen-
tum ZY maius ſit huius ſegmento XS, & reliquum ſegmentum MZ 66ibidem. reliquo ſegmento DX, quare latus LM minus erit latere ID, & ſemper,

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