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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 48v, page id 00068196:0096
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Hinc
intendum posset fortisse inveniti solvi problema;  differentiale si series
differentiarum primarum secundarum tertiarum et. esset exponentialis
vel alia tractabilis
sit dy = dx N(a + b X 4). 4a seu dydy = a + b: a) X, det
fic 2 dus Rds sit dx constans, fiet 2dy ddy = 4b: a) X dx 2 aa seu ddy = (4: 2) b: a) x du .
rursus  dydhy + ddy ddy = 2b: 2. 3. (6: a) X dxt: na sed ddy =  a )bb. aa) X dx:
et fiet du d’y + 2bb: aa) X dx = 2. 3. (b: a) xx dx dydy
seu eret:
By = 2. 3. (b: a) x  (a + b: 4) X2)  - 2 1bb: aa) X, dx:, a 1b. a) , atb X
seu d’y - 2.3 (b: a) a . + (2. 3 - 2) bb: aa) X, dx; a + (b: a) X, 1(a + b: a) X2)
in dy dhy = 2. 3 b: a) a Xx + (2. 3 - 2) (bb: 4a) X, dx
3d ydy ady) + dy) dy = 2.3 (b: a) a X + (2. 3 - 2. 3) (bb: aa) X dx
et fiel
A. a)X (a abia X) du et fiet
(2y = 2. 3sb. a) a N+  (2. 3 = 2. 3) bb aa) X - 2. 3 (b: a) da X i a - 1b. a X) 
(a + (b: a) X3) (Y (a +b: 4) p)
ax
Et ita pergendo, uidendum an haberi posset lex. quae si esset retro
continuabiis, posset haberi, y. Et si autem hoc in casu nonsit talis, posset ex cogi–
Et quia in potestate est assumere
relationem inter, X, dx, et dax proarbituo, hinc fortasse faciluis tates leges
tari casos, in quo continuabilis esset
ex cogitari possent.
sit aliqua langentiam inversa, seu differentialis primi gradis aequatio
veluli adue= axs X=+ eax. sumanus X et y ut constantes d et Z, nenoe primas
jam proxima. Erquacdys) = (X + dx + y+ du) (dx) et pro du ponendo valorem, fiet
aldys) = x(dx) + qd Xax) + y A X) +  X dx (d X) + y dx (dx)
X+ c
L(dx) +
seu a(dy) = (X + y) (dx) + a + x + Z dx AX) seu (dy) =
a (dy)) = (X + dx + (dy) + y + dy +(dy)) dx (dxD
a+xts
aa
y) = X (dX)) + d x pd X D) + (dX) pdX) + y fdx) + dy (dx) + fau  (dy) dX)
illeq
ppste
) roZ dx (dxI
Xa Z(dx) l  + d   Ax r  X)
dy) = ( p.. X+ 7 puX) = a + X +y (dx (dXD + (d X) (d X) + a + x 19 dx(.tX fl 7
dx = (1X) = (dx ) fint  dy=
= (X + Axaclx) tXD Fyrdus adus y) dx) (dy)-
ax  ydC +XHLy 1a d, p (d y) =
 L. X Y andx + atxFZ dx + xty apdx
2 a Z dx + ara y
+ dx, a
dyIX, a =
 +dx, a =
atpay
Unde tundem
 dx
dx +
(dy 1
1 d X dx
dx + 2 dxdx +
a
aa
XLy
X + y ex (d- ( 6B
esc
X + 7 dx +
atx tZ dedx
x +y dx + a+ x+7 dxdx +
aa
at 2 ax
axty
a
(y).y
Sed summa omnium d.y+ dx, a est
a+ x +y
etsumma valorum est 
Potuissemus et summare
aa
dx. _ 1+
atxty
1
a + x+y
2
dy
as ++
a
x) a (YL fxea
iaest autam numnus 14 dx X: a a + X +
eaaratim cdx. + idyct
et idede rant
1. dx) in
aal
1 dx + 3dx dx + 3 dx
vi tatio e)
sihed
in finitus qui
stuctus in infinite parvum, facit constantem
1 dx + a dxdx + 6. dx + 4 dx F 1d7
Hinc
ac
a +X + Z dxdx
aa
aa
XtZ dx + 3 a +XFy Axdx +
3
e
cujus valor aliunde haberi potest
Ubi notandum si aliter valorem iave
niremus pure seu sine (Y) ita ut ipsa
(y) haberetur, et poncietur esse
(d =  (1H dx: a) a fore
Si adus .  +ysr. idcoquia dato sus = pA ryB ) dx: C = (1 + (dX: a)). X. a seu fore
aa
 1= a log. a videtur negligi posse dx: a et fieri = 1. sed
y ∞W s dx 2. 10 potio B= Sod
ax lt hoc nonlicet, tam inde videretur absurdum nam toret
+ a) (AXX: a): pax) atia
 C 
x me
 u  intialte pavantordinn  aditamm
xligaetat ast etata  , a. at prt ex sud
um. Uinc st pncii infinite parvam ordinan additam
dy)
itE
+y
+ax, a -
+x+y
= ((d y)) + dx, a
A + X+ y
1xx +y
ait quim dx, 1r   X.: a