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

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4424 ſtans. Dico ipſam cum ſectione conuenire, eamque omnino ſecare.

Iungatur CS, quæ producta ſectioni occurret, per ſecundam partem 8.
huius, eritque ſectionis diameter: quare per Coroll. 2. præcedentis, ipſa STX
ſectioni occurret, vt in X.

Præterea, cum quæcunque contingenti æquidiſtans GI ſupra RX, inter
aſymptoton, & ſectionem intercepta, maior ſit ipſa RX, ſiue ipſa GZ, pun-
ctum Z cadet extra ſectionem, & ſic de quolibet alio puncto rectæ XTS. E
contra cum quælibet intercepta LY infra RX, parallela ad DB, minor ſit ip-
ſa RX, ſiue LF, punctum F cadet intra ſectionem, idemque de quolibet alio
puncto rectæ XF: vnde recta STX ab ipſo occurſu X cum ſectione, ad partes
verticis tota cadit extra, ad oppoſitas verò partes tota cadit intra ſectionem;
ideoque in vno tantum puncto X Hyperbolen ſecat. Quod erat propoſitum.

19[Figure 19]

COROLL.

HInc eſt, lineam alteri aſymptoton æquidiſtantem per punctum, quod
ſit, vel in ipſa ſectione, vel intra, pariter in vno tantùm puncto ſe-
ctioni occurrere, eamque ſecare.

Nam recta XS ex puncto X, quod cſt in Hyperbola, vel recta FS ex pun-
cto F, quod eſt intra, æquidiſtanter ducta aſymptoto CD, ſi ad partes cen-
tri C producatur, alteri aſymptoto CE omnino occurrit, (quoniam EC, ſe-
cans DC vnam parallelarum ſecat quoque alteram CE) vnde aliqua pars,
ipſius rectæ XS, vel FS cadit in loco ab aſymptotis, & ſectione terminato,
ac ideo ex his, quæ ſuperius oſtendimus, ipſa linea in vno tantùm puncto ſe-
ctioni occurret, ac Hyperbolen ſecabit.

Qua propter, quælibet linea alteri aſymptoton æquidiſtans, dummodo ſit
ducta ex puncto, quod ſit in angulo ab aſymptotis facto, in vno tantùm pun-
cto Hyperbolæ occurrit, atque eam ſecat.

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