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

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4222

Nam ſit quæcunque recta DBE ſectionem contingens in B: patet per 3.
ſec. conic. ipſam DE cum vtraque aſymptoto conuenire, & ad tactum B ſe-
cari bifariam, & quadratum vtriuſque portionis DB, BE æquale eſſe quarte
parti figuræ, quæ ad diametrum CB per tactum ducta conſtituitur; quare ſi
fiat CA æqualis CB, appliceturque quælibet GIH ipſi DB æquidiſtans,
aſymptoton, ſectionem, ac diametrum ſecans in G, I, H, & per I ducatur
IP parallela ad CD, ſecans diametrum in P infra C (nam punctum I eſt intra
angulum GCH) erit vt in præcedenti oſtenſum fuit rectangulum AHB ad
quadratum HI vt quadratum CB ad quadratum BD, vel vt quadratum PH
ad quadratum HI; vnde rectangulum AHB æquale erit quadrato HP, ſiue
recta HP erit media proportionalis inter AH & HB; hoc eſt punctum P ca-
det inter C & B; quare IP, quæ ipſi GC æquidiſtat contingentem BD ſeca-
bit in Q, eritque BD maior DQ, ſiue maior intercepta GI.

18[Figure 18]

Iam applicata infra G qualibet alia RN diametro occurrent in O, ex N du-
cta ſit NS parallela ad RC, quæ contingentem BD, ac diametrum ſecabit vt
ſupra in T & S. Cumque rectangulum AHB ſit æquale quadrato HP, vt mo-
dò oſtendimus, ſitque in directum ipſi AH addita quædam HO, erit, per
præcedens Lemma, rectangulum AOB maius quadrato OP, ſed rectangu-
lum AOB eadem ratione, vt ſupra, oſtenditur æquale quadrato OS; quare
quadratum OS maius eſt quadrato OP, hoc eſt punctum S cadit inter C, &
P, ſiue CP eſt maior CS, vel DQ maior DT, hoc eſt GI maior RN. Quare
aſymptoton CD, & ſectio BIN quæ in infinitum productæ, nunquam ſimul
conueniunt, ad ſe propiùs accedunt; idemque de aſymptoto CE. Quod erat
primò & c.

Præterea dico ipſas ad interuallum peruenire minus dato interuallo M.

Sumatur DT ex cõtingente BD, quę ſit minor interuallo M, & per T aga-
tur STN parallela ad CD diametro occurrens in S, ſeceturq; SV æqualis

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