# Full text: Viviani, Vincenzio: De maximis et minimis, geometrica divinatio

## 287.THEOR. XXIX. PROP. XLVIII.

MAXIMA portionum eiuſdem anguli rectilinei, vel Hyperbo-
le, & quarum diametri ſint æquales, eſt ea, cuius diameter ſit axis
dati anguli, vel Hyperbolæ.

ESto primùm, in prima figura, A B C angulus rectilineus, circa axim B
D, cui applicata ſit perpendiculariter quæcunque A E C, eum ſecans
in E. Dico portionum, ſiue triangulorum ex dato angulo abſciſſorum, & quorum diametri ſint æquales ipſi B E, _MAXIMVM_ eſſe A B C.

Nam cum B E ſit perpendicu-
laris ad A C, facto centro B in-
teruallo B D, ac circulo deſcri-
pto, eius peripheria continget re-
ctam A C in D, anguli latera ſe-
cans in F, K; quare diametri æ-
quales abſciſſorum triangulorum
ad peripheriam F E K pertingẽt: ſumpto igitur in ipſa quocunque
puncto G, iungatur B G, & du-
catur per G recta L G M ipſi A C
æquidiſtans, axim ſecans in N,
& erit L N æqualis N M, vnde
L G minor G M; ſecetur ergo G
O ipſi L G ęqualis, & agatur O I
I G, & producatur, quæ cum O I
ſecet in I, alteram quoque paral-
lelam B A ſecabit in H, eritque I G æqualis G H, ſed anguli ad verticem
I G O, H G L ſunt æquales, ergo, & triangulum I G O triangulo H G L æ-
quale erit, & communi addito trapetio B L G I, erit quadrilaterum B L O I
æquale triangulo H B I, ſed triangulum A B C maius eſt quadrilatero B L
O I, totum ſua parte, quare triangulum A B C erit quoque maius triangulo
H B I, cuius diameter B G æqualis eſt axi B E trianguli A B C, & hoc ſem-
per de quolibet alio triangulo circa diametrum ipſi B E ęqualem; quare
triangulum A B C eſt _MAXIMVM_. Quod erat primò, & c.

### 287.1.

Sit præterea, in ſecunda figura, Hyperbole A B C, cuius centrum D,
axis D B E, ex quo dempta ſit B E, eique per E applicata A E C, & ſit
quælibet alia diameter D F G, ex qua ſumatur F G ipſi B E æqualis, appli-
ceturque H G I. Dico portionem A B C portione H F I maiorem eſſe.

Nam cum ſit ſemi-axis D B ſemi-diametrorum _MINIMA_, hæc erit ma- ior D F, eſtque B E æqualis F G, quare D B ad B E minorem habebit ra-
tionem quàm D F ad F G: fiat ergo D F ad F L, vt D B ad B E, & habe-
bit D F ad F L minorem rationem quàm D F ad F G, ideoque F L maior
erit F G, ſi ergo per L applicetur M L N, quæ ipſi H G I æquidiſtet, erit

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