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

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22442

Iam ducatur per D quælibet alia F D G.

186[Figure 186] Et cum in triangulo D G C ſit externus
angulus D C L maior interno D G C, fiat
angulus D G H ipſi D C L, ſiue D A F æ-
qualis, eſtque angulus G D C æqualis an-
gulo A D F, & duo ſimul D A F, A D F
minores ſunt duobus rectis, ergo & duo
D G H, G D C erunt duobus rectis mino-
res, ſiue G H cum D C producta conue-
niet, vt in H, eritque reliquus angulus H
in triangulo D H G æqualis reliquo F in
triangulo D F A: quare huiuſmodi trian-
gula ſimilia erunt, & circùm æquales an-
gulos ad D habebunt latera proportio-
nalia, ſiue vt A D ad D F, ita G D ad D
H, vnde rectangulum A D H æquale erit
rectangulo F D G, ideoque rectangulum
A D C minus erit rectangulo F D G, & hoc ſemper vbicunque applicata
ſit per D, recta F D G præter A D C. Quare rectangulum ſub ſegmentis
A D, D C eſt _MINIMV M_ quæſitum. Quod erat faciendum.

PROBL. V. PROP. XXXIV.

A puncto intra coni-ſectionem dato rectam applicare, cuius
rectangulum ſegmentorum ſit MINIMVM. In Ellipſi verò, &
MAXIMVM rectangulum reperire.

ESto primùm A B C Parabole, vel Hyperbole, vt in prima figura, cu-
ius axis B D, & datum intra ipſam punctum ſit E. Oportet per E re-
ctam ſectioni applicare, ita vt rectangulum ſub eius ſegmentis ſit _MINI-_
_MVM_.

Applicetur per E recta A E D C axi ordinatim ducta. Dico hanc ip-
ſam quæſitum ſoluere: ſiue rectangulum A E C eſſe _MINIMVM_.

Nam applicata per E qualibet alia inclinata F E G: non abſimili mo-
do, ac in 26. ſecundi conicorum, demonſtrabitur applicatas A C, F G in-
tra ſectionem ſe mutuò ſecantes in E, in ipſo E nunquam bifariam ſimul
ſecari, ex quo ipſarum applicatarum diametri diſiunctæ erunt inter ſe,
ideoque B vertex portionis A B C non erit vertex portionis F H G: is er-
go ſit H; ducaturque ex B ſectionem contingens B I, ſiue applicatę A C
æquidiſtans; itemque ex H recta contingens H I, ſiue F G parallela, que
contingentes ſimul conuenient in I. Erit ergo rectangulum A E C, 1158. pri-
mih. rectangulum G E C, vt quadratum B I ad quadratum H I; ſed eſt 2216. tertij
conic. tingens B I, ad axis verticem, minor contingente H I, ergo & quadra- tum quadrato minus erit, ſiue rectangulum A E C minus rectangulo F E
3387. primi
huius. G, & hoc ſemper quæcunque ſit quæ per E applicatur diuerſa ab appli-
cata A C, ergo rectangulum A E C eſt _MINIMVM_ quæſitum.

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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