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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 295v, page id 00068196:0586
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exemplum primum et Mirum summationis inter duo nullam
plane habentia connexionem, ut sciti detur
log. X deo
quocu generuli firmula quacunque demum sit relatio inter X et co
ut el datur (a b 
quia reducendo b ad a, potest x et y reducu ad unam ++ y.
porro qua ex X = V sit eo dx: X + sdx: x deo = dv: v
co
seu ce cdiX dy: deo + Sdx: x = de: do: v hinc differentiando
fiet: Co: X de ddx: deo - dx: des ddeo + d x. deo lx. d - dx. du dan
=V ddv: ded = dy: deo ddco - dv. deo dv. VY
1
+ dx: deo dco: X - codx: XX L dx: X
= e ddv: deo - dv ddco: deo : v B- dv: deo dv: vv
et multiplicalis omnibus perWV . Ret
COXVV deo ddx =, Coxvv axdda + ex x vrda - Wcodxdeo
+ X VV dco = XX VIC ddv = xx V du dd - XX dudodco
itaque shaec aequalio differentio differentialis ordinaria solvi potest
per transcendentem X = v
 Quoniam
autem aequatio 6 tam late patet, ut  una sit
in ea indeterm ut novam aequationem issumere liceat
hinc videtur sic explicari posse ut comprehendat aequa
tionem differentio- differentialem datiamm Meci alerimn
dinarias ve ordiparie transpendentes
ledue
ad lanc foimoni
isaesedum cum aluude semper diffeontio disperen di ales quandam arbitrariam assceolictat
curvas Nos
d
elur sequi ongnes
X, = 
eo W
quin et si elset X y
V, vel eliam
V = V fieret _ log. X + log y = log. V
a e
 sed si esset X
i
seu ot co log X = E log. y = log. V et ideo tatequa
a summatoriorvinculo liberemur  descendendum est ad differen
tialem 3tii gradus. Et adeo rursus amplissium habetur
supponendi campus cum duae sint indeterminaet arbitrario
Atque ita lac methodo semper licet resfoere curvas
invenire perfinitas equares. Illud videndum an
semper hiac Mhethado inveniri possintaequationes ioes
quando pollibile est. Si sit 
x
a =
sit e AX = y7 fei d6 = 8. sit g = xy - y
et siet X Y = X, 97
.
s
e
seu
l
sei X
Xy
Bx
3 yy
= y
IX
seu Xy = q2V)
quae aequatio est ad parabolam, etsi
minime appareat. nam 1 log X = 3gy- bx log. y nec deteri
potest et ut delegatur descendendum, erit ad differentio differentralem
forte lame et alia diee aagnoscendi ratiro haberi pt nempe
sit 
 X. y = X. y. Posito esse aequationem relationes
(1)
interasrationales. fiet pr. 
4
es
(2)
L2 C5)
6
(6)
fiet = Xy: X Y = X. g. 
x.
 4
 p2.7. sit G = Xy. X3
jam ex agiuitit y= F relaoe quacunque siet g = s. g: f =
et s
4
et fiet  Xy. X
X. y 
X.  (X 4) = F. y. A. h. 9.  ~ (4.9
et fiet denique
pels
o
X=
1
gy- x. x
et ponendo X - yy
fit X
P= y
ugus. gu
seu X = 
seu X = y2 ut aute
Ergo si talis aequatio est
reducibilis ad ordinariam,
alo semper exporen.
tam qaxandet constant
qua ulitati