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

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THEOR. XXXIX. PROP. LXXXIII.

Si binarum Ellipſium ſimul adſcriptarum altera alteri fuerit in-
ſcripta, & per terminos communis applicatæ ſe mutuò contingant;
quælibet alia Ellipſis datis adſcripta, cum eadem applicata, &
cum æquali tranſuerſo latere, inſcriptam Ellipſim omnino ſecabit.

SInt duæ Ellipſes ABCD, AECF ſimul adſcriptæ, circa communem ap-
plicatam AC, ſitque ipſarum altera, nempe AECF, alteri inſcripta, ita
vt in extremis tantùm A, C, ſe mutuò contingant: dico, ſi his alia adſcriba-
tur Ellipſis ALCI, circa eandem applicatam AC, & cum tranſuerſo LI,
quod æquale ſit ipſo BD tranſuerſo circumſcriptæ, ipſam ALCI omnino ſe-
care inſcriptam AECF.

Nam ſi alterum extremum tranſuerſæ

120[Figure 120] diametri LI, quale eſt punctum L, cadat
intra inſcriptam, vt inter E, & O, tunc aliud
extremum I neceſſariò cadet extra, infra F,
cum ſit LI, ſiue BD maior EF, ex quo ma-
nifeſtè patet, Ellipſim ALCI ſecare inſcri-
ptam AECF.

Si verò vtrunque extremum L, I, cadit
extra inſcriptam, vti exhibetur ab hac figu-
ra; tunc ducta AG, quæ circumſcriptam
ABCD contingat in A, ipſa, vtrinque pro-
ducta, ad alteram partium ſecabit 1181. h. no ALCI; vnde, quæ ex A cõtingit ALCI,
diuerſa erit ab AG, & ſit ipſa AM.

Iam ſi ALCI non ſecat inſcriptã AECF;
contingat in A, C, ſi poſſibile eſt. Cum ergo recta AM contingat Ellipſim
ALCI, atque hæc contingat inſcriptam Ellipſim AECF, eadem recta AM,
in A quoque continget AECF, ſed etiam AG eandem AECF contingit in
A: quare ex eodem puncto A ductæ erunt binæ rectę lineę eandem Ellipſim
contingentes; quod eſt impoſſibile. Non igitur Ellipſis ALCI 22ex 32.
pr. conic. inſcriptam AECF, quapropter in occurſibus A, C neceſſariò eam ſecabit,
Quod oſtendere propoſitum fuit. Sed hoc idem

ALITER affirmatiuè.

CVm recta AG contingat ad A circumſcriptam Ellipſim ABCD, atque
hæc ad idem A contingat inſcriptam AECF, ipſa AG omninò con-
tinget ad A inſcriptam AECF; ſed MA (quę vt ſupra oſtendimus, diuerſa eſt
à GA) hanc ſecat in A; quare MA producta, ad alteram partium omninò
ſecabit inſcriptam AECF, & eò magis Ellipſis ALCI, quam contingit ad A
recta MA, ad eandem partem ſecabit inſcriptam AECF. Quod erat, & c.

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114	90
115	91
116	92
117	93
118	94
119	95
120	96
121	97
122	98
123	99
124	100
125	101
126	102
127	103
128	104
129	105
130	106
131	107
132	108
133	109
134	110
135	111
136	112
137	113
138	114
139	115
140	116

Scan	Original
111	87
112	88
113	89
114	90
115	91
116	92
117	93
118	94
119	95
120	96
121	97
122	98
123	99
124	100
125	101
126	102
127	103
128	104
129	105
130	106
131	107
132	108
133	109
134	110
135	111
136	112
137	113
138	114
139	115
140	116

Scan	Original
111	87
112	88
113	89
114	90
115	91
116	92
117	93
118	94
119	95
120	96
121	97
122	98
123	99
124	100
125	101
126	102
127	103
128	104
129	105
130	106
131	107
132	108
133	109
134	110
135	111
136	112
137	113
138	114
139	115
140	116