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

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4323& fiat vt AV ad VS, ita AS ad SO, & per O ordinatim applicetur ONR ſe-
ctionem ſecans in N, rectam verò ST in X. Et cum ſit vt AS ad SO, ita AV ad
VS, erit componendo AO ad OS, vt AS ad SV, vel vt AS ad SB, & permu-
tando, & per conuerſionem rationis, vt AO ad OS, ita SO ad OB, ergo re-
ctangulum AOB æquatur quadrato OS: ſed rectangulum AOB ad quadra-
tum ſuæ ordinatim ductæ ON in Hyperbola ſemper eſt vt quadratum CB ad
BD (vt iam ſuperius oſtendimus) vel vt quadratum SO ad OX: quare permu-
tando rectangulum AOB ad quadratum SO, erit vt quadratum ON ad qua-
dratum OX, ſed eſt rectangulum AOB æquale quadrato SO, ergo & qua-
dratum ON quadrato OX æquale erit, quare puncta N, & X idem funt, ſed
eſt N in ſectione, quare recta TX conuenit cum ſectione in X, vel N, hoc eſt
RN & RX æquales erunt, ſed eſt RX æqualis ipſi DT, & DT minor M, vnde
RN, vel RX erit quoque minor M. Peruenit ergo aſymptoton CD cum ſe-
ctione ad interuallum RN minus dato interuallo M. Quod tandem erat de-
monſtrandum.

COROLL. I.

HInc eſt, quodlibet diametri ſegmentum inter quamcunque applicatam,
& rectam ex ipſius occurſu cum ſectione alteri aſymptoton æquidi-
ſtanter ductam, medium eſſe proportionale inter aggregatum ex tranſuer-
ſo latere cum prædicto diametri ſegmento, idemque ſegmentum. Demon-
ſtratum eſt enim HP eſſe mediam proportionalem inter AH, & HB; & OS
mediam inter AO, & OB.

COROLL. II.

PAtet etiam quamcunque rectam, ex puncto tranſuerſi lateris inter cen-
trum, & verticem ſumpto alteri aſymptoton ęquidiſtanter ductam ne-
ceſſariò ſectioni occurrere. Iam enim ſupra oſtendimus rectam STX, quæ
ex puncto S in tranſuerſo CB ducta eſt aſymptoton CD parallela, cum ſe-
ctione conuenire in N.

MONITVM.

HInc facilè erit oſtendere 13. ſecundi conicorum aliter, & affir-
matiuè, vt videre licet in ſequenti.

THEOR. IV. PROP. XI.

Si in loco aſymptotis, & ſectione terminato quædam recta linea
11Prop. 13.
ſec. conic. ducatur alteri aſymptoton æquidiſtans, in vno tantùm puncto cum
ſectione conueniet, eamque neceſſariò ſecabit.

SIt in præcedenti ſchemate in loco ab aſymptotis, & ſectione terminato
quodcunque punctum S, à quo ducta ſit STX aſymptoton CD

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