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

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22947

THEOR. XXII. PROP. XXXVII.

Proportionalia Acuminata, quorum baſes eorum altitudini-
bus ſint reciprocè proportionales, ſunt inter ſe æqualia.

SInt duo proportionalia Acuminata A B C, D E F, quorum diametri
ſint B G, E H, altitudines verò B I, E L, quæ inter ſe reciprocam
habeant rationem baſium A C, D F; ſiue ſit vt A C ad D F, ita E L ad
B I. Dico huiuſmodi Acuminata inter ſe æqualia eſſe.

Si enim poſſibile eſt,

192[Figure 192] ſit alterum ipſorum,
nempe A B C reliquo D
E F minus, & per con-
tinuam diametri B G
biſectionem, iuxta vul-
gatam methodum, cir-
cumſcribatur ipſi A B
C, figura exparallelo-
grammis conſtans æ-
qualium altitudinum A
L, M N, & c. quorum
altitudines I T, T V, & c.
æquales erunt (cum
altitudo B I in tot æ-
quales partes diuidatur
ab æquidiſtantibus parallelogrammorum baſibus A C, M O, & c. in quot
partes diameter B G ſecta fuit) huiuſmodi autem circumſcripta figura ex
parallelogrammis, acuminatum A B C ſuperet minori exceſſu, quò acu-
minatum D E F ponitur excedere idem acuminatum A B C, adeo vt ipſa
circumſcripta A B N L C ſit adhuc minor acuminato D E F, cui circum-
ſcribatur item figura D E R P F ex totidem parallelogrammis D P, Q R & c.
æqualium altitudinum K X, X Y, & c.

Iam, cum ſit baſis A C ad D F, vt altitudo E K ad B I, vel vt ſubmul-
tiplex K X ad æque-ſubmultiplicem I T, erit parallelogrammum A L, æ-
quale parallelogrammo D P. Et cum, ex conſtructione, ſit G B ad B Z,
vt H E ad E 3, erit, ex definitione proportionalium acuminatorum, A C
ad D F, vt M O ad Q S, ſed A C ad D F eſt vt E K ad B I, ergo, & M
O ad Q S erit vt E K ad B I, vel vt ſubmultiplex X Y ad æque-ſubmul-
tiplicem T V: parallelogrammum igitur M N æquatur parallelogrammo
Q R; & ſic de reliquis, ſingula ſingulis: ergo vniuerſa figura A B N L C
æqualis erit vniuerſæ D E R P F, ſed figura A B N L C facta eſt minor
acuminato D E F, quare figura D E R P F erit quoque minor eodem ſibi
inſcripto acuminato D E F: totum parte, quod eſt abſurdum. Nullum
ergo horum acuminatorum eſt reliquo minus, quapropter æqualia eſſe
inter ſe neceſſe eſt. Quod erat demonſtrandum.

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Scan	Original
111	87
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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
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131	107
132	108
133	109
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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
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133	109
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135	111
136	112
137	113
138	114
139	115
140	116