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

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166142 tranſire per H, adſcriptum eſſe Ellipſi ABCO, & Ellipſeos contingentem
EF circulum quoque contingere, ſed huius verticem G, cadere vltra Elli-
pſeos verticem B, cum ſit DE, vel DG maior DB, quare circulus ex 1188. h. erit Ellipſi ABCO circumſcriptus. Quod erat vltimò demonſtrandum.

2261. h.

THEOR. XLVIII. PROP. XCIII.

Si Parabolen, vel Hyperbolen, aut Ellipſim circa maiorem axim
quotcunque rectæ lineæ ad eaſdem axis partes, præter verticem
contingant, quibus à tactibus ductæ ſint perpendiculares axi occur-
rentes: ipſæ, quò magis contactuum puncta à maioris axis vertice
diſtabunt eò maiores erunt.

E contra: ſi Ellipſis fuerit circa minorem axim, huiuſmodi per-
pendiculares ſemper decreſcent, quò magis earum contactus à mi-
noris axis vertice remouentur.

SIt AB Parabole, vt in prima figura, vel Hyperbole, vt in ſecunda, aut
Ellipſis circa maiorem axim BR, vt in tertia, vel circa minorem, vt in
quarta, quas ſectiones duæ rectæ AE, DF ad eaſdem axis partes, & in Elli-
pſi in eodem quadrante BLM ad duo quælibet puncta contingant, præter
verticem B, quibus erectæ ſint perpendiculares AC, DG axi occurrentes
in C, G. Dico primùm in Parabola, & Hyperbola, ac in Ellipſi tertiæ figu-
ræ interceptam perpendicularem AC ex puncto A, remotiori à vertice,
maiorem eſſe perpendiculari DG ex puncto D propinquiori.

1. Nam, in ſingulis figuris, cum anguli CAE, GDF ſint duo recti, & con-
tingentes AE, DF cadant extra ſectionem, & ſi concipiatur iungi recta AD,
ipſa cadat tota intra ſectionem, anguli,
quos eadem A D conficiet cum perpen-
dicularibus A C, D G, minores erunt

131[Figure 131] duobus rectis, quare ipſæ conuenient
ſimul ad partem axis, vel vltra, vel inter
contactus, & axim.

2. Iam, ducta D I parallela ad AH, ſiue
axi perpendiculari, cum in Parabola pri-
mæ figuræ ſit CH æqualis G I, 3335. pri-
mi conic. enim eſt dimidium recti lateris, & 4490. h. maior D I, erunt quadrata ſimul C H,
AH, ſiue quadratum AC, maius qua-
dratis ſimul G I, D I, ſiue quadrato DG,
hoc eſt linea AC maior DG.

3. In Hyperbola verò ſecundæ figuræ ſumpto eius centro L: cum L H
ad H C, itemque L I ad I G, vt tranſuerſum latus ad rectum, erit L H 553. Co-
roll. 90. h. H C, vt L I ad I G, & permutando L H ad L I, vt H C ad I G, ſed eſt L H
maior L I, ergo, & H C, maior I G, eſtque H A maior I D, quare

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