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

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10884

THEOR. XXVI. PROP. XXXXVIII.

Similes Hyperbolæ per diuerſos vertices ſimul adſcriptæ habent
aſymptotos parallelas, & quando centrum interioris cadat vltra
centrum exterioris, tunc huius aſymptotos interiorem Hyperbolen
ſecabit, ac ipſæ Hyperbolæ neceſſariò ſe mutuò ſecabunt. Cum
verò centrum interioris idem ſit cum centro exterioris, tunc vnius
aſymptotos erit aſymptotos alterius; & ſectiones erunt ſimul nun-
quam coeuntes. Et ſi interioris centrum cadat infra centrum ex-
terioris, tunc eædem ſectiones erunt inter ſe nunquam coeuntes; &
aſymptotos inſcriptæ ſecabit Hyperbolen circumſcriptam.

SInt, vt in vtraque figura huius propoſitionis, duæ ſimiles Hyperbolæ
ABC, DEF per diuerſos vertices B, E ſimul adſcriptæ, quarum centra
ſint G, H, & ſectionis ABC aſymptoti ſint GI, GO; ſectionis verò DEF ſint
HM, HR; Dico has aſymptotos eſſe inter ſe æquidiſtantes.

Nam in ſimilibus Hyperbolis ABC, DEF, anguli IGE, MHE, ab earum
aſymptotis, & diametris ad homologas partes facti ſunt æquales, 11Coroll.
40. huius. alterni, quare ipſæ aſymptoti inter ſe æquidiſtabunt. Quod primò, & c.

Iam in hac prima figura, (in

75[Figure 75] qua centrum H interioris DEF
remotius eſt à verticibus B, E,
quàm ſit centrum G exterioris
Hyperbolæ ABC) cum ſint HM,
HR aſymptoti Hyperbolæ DEF,
& in loco ab eis, & ſectione ter-
minato ducta ſit GI alteri aſym-
ptoton HM æquidiſtans, ipſa
omnino ſectionem DEF ſecabit.
Quod ſecundò, & c.

Sed ipſa GI, cum ſit aſympto-
tos ſectionis ABC, tota cadit ex-
tra ipſam BA, quare occurſus
prædictæ aſymptoton GI cum ſe-
ctione ED, erit extra ſectionem
BA, vnde ipſa interior ſectio ED
neceſſariò ſecabit priùs exterio-
rem BA. Quod tertiò, & c.

Ad pleniorem autem doctrinam, ſi quæratur, quo nam in puncto huiuſ-
modi Hyperbolæ ſe mutuò ſecent, ita id conſequetur.

Sumpta enim GS æquali GB, erit tota BS tranſuerſum exterioris ABC;
item ſumpta HT æquali HE, erit tota TE tranſuerſum interioris DEF.

Iam, vel H centrum interioris cadit in ipſo puncto S; vel ſupra inter S, &
T, vel infra inter G, & S.

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Scan	Original
91	67
92	68
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95	71
96	72
97	73
98	74
99	75
100	76
101	77
102	78
103	79
104	80
105	81
106	82
107	83
108	84
109	85
110	86
111	87
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120	96