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

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9470

THEOR. XXI. PROP. XXXXI.

Similes Hyperbolæ per eundem verticem ſimul adſcriptæ, ſunt
inter ſe nunquam coeuntes, & ſemper magis recedentes, ſed ad in-
teruallum nunquam perueniunt æquale cuidam dato interuallo.

SInt duæ ſimiles Hyperbolæ ABC, DBE per eundem verticem B ſimul
adſcriptæ, & Hyperbolæ ABC maiora ſint latera, tranſuerſum nempe
FB, rectũ autem BG; & DBE minora ſint, tranſuerſum HB, rectum verò BI.
Patet primùm has ſectiones eſſe inter ſe nunquam coeuntes; cum enim DBE
alteri ABC ſit inſcripta, ipſæ, licet in infinitum producantur, 115. Co-
roll. 19. h.conuenient.

Dico ampliùs, eaſdem in-

64[Figure 64] ter ſe longiùs ſemper recede-
re. Applicatis enim in alte-
ra ſectionum, nempe in in-
ſcripta, duabus vbicunque
rectis MN, LD, fiat vt BH ad
BF, ita BM ad BQ, & BL ad
BR, & per Q, R, applicentur
in circumſcripta Hyperbola
rectæ QS, RA; & cum dia-
metrorum ſegmẽta BM, BQ,
BL, BR ſint tranſuerſis BH,
B F proportionalia, 2238. h. quoque applicatæ MN, QS;
LD, RA ijſdẽ lateribus pro-
portionales, quare M N ad
QS erit vt LD ad RA; cum-
que ſit vt tota BL ad totam BR, ita pars BM ad partem BQ, erit & reliqua
ML ad reliquam QR, vt tota BL ad totam BR, vel vt BH ad BF, vel vt MN
ad QS, & vt LD ad RA, vt nuper oſtendimus; quare iunctis rectis DN, AS
in menſalibus DM, AQ, erunt ipsæ DN, AS inter ſe parallelæ. Iam 3339. h. ductis MN, LD vſque ad circumſcriptam ſectionem ABC, in P, & O, ſi iun-
gatur PO, hæc omninò ſecabit iunctam AS, vel intra ipſam ſectionem; (ſi
nempe vnius iunctarum ſectioni occurſus, alterius occurſibus contineatur)
vel extra (ſi nullius occurſus, alterius occurſibus amplectatur) ſed AS 4425. ſe-
cundi co-
nic ducta ad partes verticis tota cadit extra ſectionem in SK, & punctum P eſt in ipſa ſectione, quare punctum P eſt inter parallelas lineas ASK, DN, ſed
5510. pri-
mi. conic. producta PO conuenit cum altera parallelarum AS, vt modò monitum fuit;
taliſq; occurſus eſt omnino ad partes O infra applicatam PN, cum punctum
S cadat infra P (nam ex ipſa conſtructione applicata QS eſt infra applicatam
MP) quare eadem OP producta conueniet quoque cum altera æquidiſtan-
tium DN, ad oppoſitas tamen partes, vtputa ſupra ipſam applicatam PN,
vnde intercepta applicata OD, maior erit intercepta PN vertici B

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