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

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142118 circulum dico in punctis D, E, quæſitum ſoluere: nempe AE, & AD eſſe
quæſitas extremas. Nam cum ſit BD æqualis BE, erit data AB media ari-
thmetica inter inuentas EA, AD. Cumque ſit BF radius circuli EFD, & an-
gulus BFA rectus, erit FA ipſi circulo contingens, quare rectangulum EAD
æquabitur quadrato AF, ſiue quadrato AC, vnde data AC erit media geo-
metrica inter eaſdem inuentas EA, AD. Quare ignotæ extremæ, ſunt in-
uentæ, vti quærebantur. Quod, & c.

PROBL. XXX. PROP. LXXV.

Datæ Parabolæ, per punctum intra ipſam datum, MAXIMAM
Ellipſim inſcribere, cuius latera datam habeant rationem: & è
contra.

Datæ Ellipſi, per punctum extra ipſam datum, MINIMAM
Parabolen circumſcribere.

ESto data Parabole ABC, & datum intra ipſam punctum ſit E; oportet per
E _MAXIMAM_ Ellipſim inſcribere, cuius rectum latus ad tranſuerſum
rationem habeat R ad S.

Ducatur ex E Parabolę diameter BED,

110[Figure 110]& applicetur EF, & ſumpta V media pro-
portionali inter S, & R; fiat vt R ad V, ita
FE ad ED, iunctaque FD, quæ producta
ſectioni occurrat in G, ex quo 1127. pri-
mi conic. GHI, circa tranſuerſum latus EH, & ter-
minos applicatæ AC deſcribatur 22Coroll.
57. h. AECH. Hanc dico eſſe quæſitam.

Cum enim in Parabola ſint diametri 33Coroll.
1. 13. h. gmenta BH, BD, BE proportionalia, ſint-
que quadrata applicatarum IH, AD, FE
in eadem ratione ipſorum 4420. pri-
mi conic. erunt quoq; ipſæ applicatæ continuæ pro-
portionales, quapropter rectangulum ſub
IH, vel ſub HG, & FE æquabitur quadrato AD, ac proinde quadratum
AD, ad rectangulum HDE, erit vt rectangulum ſub GH, EF, ad idem re-
ctangulum HDE, ſed rectangulum ſub GH, EF, ad ſibi ſimile rectangulum
HDE, (habent enim circa rectos angulos latera proportionalia, cum ſit GH
ad HD, vt FE ad ED, & permutando GH ad FE, vt HD ad DE) eſt vt qua-
dratum FE ad ED (vtraque enim proportio, duplicata eſt proportionis linee
FE ad ED) quo circa, & quadratum AD ad rectangulum HDE, hoc eſt in
Ellipſi, @@ rectum latus ad tranſuerſum, erit vt quadratum FE ad ED, 5522. pri-
mi conic. vt quadratum R ad V, vel vt data linea R ad S. Deſcripta eſt ergo Ellipſis
AECH, cuius latera habent datam rationem R ad S. & eſt datæ 66Schol.
62. h. ABC inſcripta. Amplius dico, ipſam eſſe _MAXIMAM_ Ellipſium quarum
latera ſint in ratione R ad S, ſiue eſſe _MAXIMAM_ ſibi ſimilium: nam, quæ
cum minoribus lateribus datæ Parabolæ per E adſcribitur ad partes H,

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