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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 363r, page id 00068196:0721
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dy: dx = e
Var
aite
sit  le dyfeax le Et ay =axe + e. firt ady = a xde taedx + de de
Ergo fit a e=  ax de: dx + ae, + de de: dx. unde destinc
Unde destructis destruendis fit ax + de =0 unde ex aeq. 2 et 4 habetur
t
quaesitum absolute, quod est memorabile tuo res adhuc generalior reddi potest.
Sit it ay = aXe + e fit ady = ax de de  a ede + de de
fit a e = aX d e derx + a C + det do ixund geentuus.
seu de: Ax = + aC D- a C : a x de + det se elt exy- ry
Seu dx: de = X. ade
t
+ 4u t
aeca +acl a
ry+ Xe  + sxdus
uturum exemplo sit Cy= axet aec f posito dy: dx e fiet dy = Xde+ edx traede
jam dy = eax ex 2. Ergo sit (e dx = Xde F edx) +2ae de seu x= iac u unde tollendo
e’ex aeq fier y- naec  y =LXX: 2a + (aXx: 4a seu y = ax:qa. Videamus
jam au succedat ag. 1. rempe dus: ax = x2a seu eZ BX: 2a et ee = G2xO: taa
et fit  a3 ex , 2 per 7 et 9 -exxiaa = _X AX: 2a + saXX: ral seu sit
3XX = 6XY + 4XX seu  XX = - 2XX + XX quod succedet
s alculus ergo mug et
y = xdy: dx + a tus. du fit dy: du + _dq +  = 5 + X seu dy: dx = = 4y +X C
si jam sit ae = a + e dulu
4
(
t 
 de + e X ax + X de de
Pix =q sit dy. dx = Z ex dato problemate Ex 1. fit Xdy - ytx : xx = dq, seu dy. X - 2 dx x = d3
seu dy: AxX = Z: X = dZ: dx seu Z: X - Z. X = dz: ax seu dX: x = a Z : 3Z Z. Hinc
si dy: dx A - Z = Z. X: a posito dy. dx = y: X = Z, seu si dy: dx =   a 1.X +y.
Seo si differentia inter dy:dx et inter y: X sit producta ex duabus quantitatibus quam unisit facta
7X = Z fit asdx + xous = dque seu Xydx + Xxdy = xdZ seu Z + AX dy: dx = x aqu. dx
ex Fia altera ex y: x
grx = Z du Y emade 7X:, yAx = Z y+ k yax + Xdus = yX dx - 9X dus, 1 y x d
si sit XX dy: dx = _ y. d
sb seu gu ax + XXay, :, ( yex = dZi
: a = du. du _ sit i _ y: a= dv: dco fit sit ddeo =0 fit dy: dx = dd videz
et f sit deo sit az. deo ddv: ddue
X . y y: x = dv: du
sed non licet explicare constantem deo, seu ipsam lad dus vel aliam, quia, duae sunt solum
quantilites lg. du: di et ddv: ddZue Ergo non possunt tres X: a, et y: 4 et dy: dx per
idia eblum habert nam danentur in inter se ex ipsis duae, u Hypothesim
Aater co, non sit constans faciamus: dx: a  = deotz - dZ adeo,: ded et dy: a - deo ddv- dx ddco: 
fit dy: dx = dco ddo - dr adco: d eoddze- dz du
g: a =  dv: a dlio x = d X x: a = ax: addeo dy: dx = du ddx: du daque
du
()
dy: dx = 1: Zv: 1: 22 (2 z. v
setq: p - - a. 2) y: a = s, a dx: XY
sit aequdata dy: dx(Z XX + y: aa fiet: Z: v = aa  + da sA2. 2B   aa seu J. v - 1 + sdY. 2B, D
X: a -
d
 EZ  ex du = d. v. 2 =  Zdv - vdzo: 2
X: a = (Z10 - vdZ, : Z7 = v. Z (y: a = (Z dv - vdz: ZV dy: dx = 1: ZV: 1: 7Z = Z. v
seu
(Kut - vlzu - du
sed hoc absurdum nam daretur relabio inter X: a et dy: dx
Xta  fZ X: a = sidv: v
y: a = (dv: X fit dy : dx = v: X sin 2- d
dge sit dy: dx = x + 9: a fiet v: X = 5dv: D + Sav: ∞ seu _2dv. vdZ = dv + A
linc si haberetur si sit tynax  C Hinc patet si X+y detur per ayiax et constantes
utcunque haberi solutionem per quadraturas. seu si detur X+y, ia per dg: dx uscunque
1
sit x+yC = G
fiit dxtius = dque et ag dg= dq. dx Ergosi ziadetur per dq. Ax = 11
Sed ita res jam tunc patet
Xia Soto. Cy. a = Z. bf dg. dx - t vtu -
X: a = A dZ - Zdv,: vz
y a = fudze 240.   flo3
N: a = dp. 1X g: a - R. f. dy. ex