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

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10379

THEOR. XXIV. PROP. XXXXV.

Similes Hyperbolæ per diuerſos vertices ſimul adſcriptæ, &
quarum eadem ſit regula, ſunt inter ſe nunquam coeuntes, & in in-
finitum productæ ad ſe propiùs accedentes, ſed ad interuallum
nunquam perueniunt æquale cuidam dato interuallo.

SInt duæ Hyperbolæ ABC, DEF per diuerſos vertices B, E ſimul adſcri-
ptæ, quarum eadem ſit regula GH (ſic enim ſimiles erunt, 116. ſecúd.
defin. ductis contingentibus BI, EL; eſt tranſuerſum GB ad rectum BI, vt tranſuer-
ſum GE ad rectum EL.) Dico primùm, has in infinitum productas, nun-
quam ſimul conuenire.

71[Figure 71]

Protracta enim contingente LE, ſectionem ABC ſecane in M, N, quæ
erit ipſi ordinata, cum ſectiones ponantur ſimul adſcriptæ; patet ſectionem
DEF totam cadere infra contingentem MN. Iam ſumpto in ſectione DEF
quolibet puncto D, per ipſum ordinatim applicetur ADOH alteram ſectio-
nem ſecans in A, regulam verò in H: erit quadratum AO, ad quadratum
DO, vt rectangulum BOH ad rectangulum EOH (ob ęqualitatem) vel 22Coroll.
1. huius. altitudo BO ad altitudinem EO, ſed eſt BO maior EO, quare quadratum
AO maius erit quadrato DO, ex quo punctum D cadit intra Hyperbolen
ABC; & ſic de quibuſcunque alijs punctis Hyperbolę DEF: quare huiuſmo-
di ſectiones inter ſe nunquam conueniunt. Quod primò, & c.

Iam ſi datæ Hyperbolæ, per verticem E, adſcribatur Hyperbole P E Q
cuius latera ER, ES æqualia ſint lateribus BG, BI, vtrunque vtrique, ipſæ
Hyperbolæ ABC, PEQ, congruentes erunt, eritque, (ob 331. Co-
roll. 19. h. RE ad ES, vt GB ad BI, vel vt GE ad EL, quare Hyperbolæ DEF,

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130	106

Scan	Original
101	77
102	78
103	79
104	80
105	81
106	82
107	83
108	84
109	85
110	86
111	87
112	88
113	89
114	90
115	91
116	92
117	93
118	94
119	95
120	96
121	97
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125	101
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127	103
128	104
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130	106