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

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10076

THEOR. XXIII. PROP. XXXXIV.

Hyperbolæ congruentes, per diuerſos vertices ſimul adſcriptæ,
ſunt inter ſe nunquam coeuntes, & ſemper ſimul accedentes: ſed
ad interuallum nunquam perueniunt æquale cuidam dato inter-
uallo.

SInt duæ congruentes Hyperbolæ ABC, DEF per diuerlos vertices B, E
ſimul adſcriptæ, quarum recta latera ſint BG, EH (quæ inter ſe æqua-
lia erunt) & ipſarum tranſuerſa ſint BI, EL (quæ item æqualia erunt 111. Co-
roll. 19. h. ſectiones ponantur congruentes.) Dico primùm has ſectiones nunquam
inter ſe conuenire.

69[Figure 69]

Nam producta contingente HE donec ſectioni ABC occurrat in A, & C,
ipſa quoque erit ordinata in ſectione ABC (cum ſint ſectiones ſimul adſcri-
ptæ) & ſectio DEF tota cadet infra contingentem AEC; ſumptoque in ipſa
ED quocunque puncto D, applicetur SDO, quæ iunctis regulis IG, LH oc-
currat in K, R; & cum ſit triangulum IBG ſimile triangulo LEH (habent
enim circa æquales angulos B, E, æqualia latera, vtrunque vtrique) erit
angulus BIG æqualis angulo ELH, vnde regulæ IGK, LHR æquidiſtant,
ideoque IK cadit extra LR, cum ſit punctum I ſupra L, ergo OK maior eſt
OR, ſed eſt OB maior OE, igitur rectangulum BOK ſiue quadratum 22Coroll.
1. huius. maius eſt rectangulo, EOR ſiue quadrato DO; hoc eſt punctum D cadit 33ibidem. tra ſectionem ED, & ſic de quocunque alio puncto eiuſdem ſectionis infra
contingentem EA: quapropter huiuſmodi Hyperbolæ inter ſe nunquam
conueniunt. Quod primò, & c.

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