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# Title: LBr. 57, 1 · Bernoulli, Joh.
# Folio 160v, page id DE-611-HS-958803:0322
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Joh. Pernonllii problema. Act. Jun. 1695
invenire Luoao ABC ju naturae propretatis, ut durcta ducta a
puncto dato d recta de utcunque DEC, silidum sab
uno jegmento de, et quadrato ulterique DE sit.
plineam
vel fig. 3, ut fe s ) F ut qoadratum
sales R solidum sub GB et quadrato FB sit
estanti dato aequale; prius ex his considerare placet
bi
sit 
DE dx+v
e
ntot
FE y. 1) ((B) (C)
e
Xx+gu(XX)(yy) L a
Waec Hoc problema non est nimium,
seu non continet proprietatem pura
hopau, nec revocari potest ab tan-
gentium inversam. Nega enim  tam
ate accipitur, ut punctum unun cuivis alteri possit applicari
video posse assami vam quameunque arcumcurvae
HL, dende posse centro
s seu DC describi arcum irculi LM, et in ea
sun electo puncto L. junctaque recta IL, applicetur
curva assumta HL, ad L, sic D iptam et deguat
deinde sumta junctu DE secante arcum HEL i puncto
quocunque E quaeratur EC, quae erit a: DE.
eita etsi HC sit recta, tamen invenietur  curva LC
si D= tangat C lineam HL, jam melius coibunt in anum HL, et LC
et potent inveniri aequatio ipsis communis, ducendo ectum aequationes in
se invicem sed si una cuva aequalio selisfaciat utrique, s quae
sit in deprimibilis habebitur desideratum
Aequationi gu, (XX gy) = a auedet hoec
nae
2da) X: y = (X). (7) sisit et (tio y∞ V. et fiet
i sit y - x. F et ex 2 fiet X = (X) quod non procedit, nam fierent
X et H) aequales. () Si y - bx + X fiet e, (y)- 
L
b(X) testis
Ergo ex 2 per 3 etq fiet: (1t t ib. . C =b(  (ugnaL
quod si jam tottatur ex jam ex t ex cg. 1. tollantur y et (y) per
aeqque. 8 et 4, prodibit aequalio inter X et () coincidens aequatione 5.
Hinc poraeq pro aeq. 5 3 assumi posset formula generilis
Ex pn 2 fieril potest (X x+ gy, : gy = (XX + (yy),: (9y)
bo
ce
Forte utile erit adhiteri et tungentes. Nam ex aequ. 1.
fiet ( X X + gy. (xx  (y ly (: ( (X) A (yy)) + V( X) H(y) 2 Xdx + 2yly = 0
et cursq a ydx - xdy, : yy = (y) dx) = (X )dy), : (y))
sed itac non poterimus magnopere lucrari
Ex aequ. 1. frit E bX + Cx + X, Sb(X) = CX) + (X = a
o)
et tollendo urationalem fiet aequatio (10) quae concrdere
debet cum aeq. h, id est ducta aequatio in arbchanam
debet pnduce d a equ. 1 coinie t seu saltem
debent habere divisorem commmunem, vel potius sumponendo
ag. 5 esse indivisibilem debet aeq. 5 ducta in artitiariam
1gto)
Generatissime )y: X = X: a pro ag. 3. et (y). (X) = (X): a
produlere aequ. 10.
fiet. Erqu t X: a = (X): a Hini s si subessent cretronalie
sublatis illis et dividendo per X- (X) fieret aeq. ubi X et () se habent
Ex aeq. 1. fiet
eodem modo ida. Ergu jac x et guim si
(6b) a3: XL X X. (X (B = 1+ X: a, 1+ (Y): a Ergo fie
(7mo a. et : () =B A  + X: a. seus q  : XYa
et pro X. a ponendo vieraxa
sutor seu fll. XXG a eX: au B Vit Fa a 3. v B rvia
al. 2