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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folio 177r, page id 00068199:0353
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H: a = dv: deo et X: a = dx: deo
dy: dx = ddv: ddzue posito adao = d
sit jam aequalio data D dy: dx = XX +gy: aa fiet inde ddy: daqu dedZe t avdv: deo deo
A assumatur = (dz: deo; dv: deo) = ddv: ddzue. differentietur jam aequatio 2, et fiet
2dZ ddque + 2d0ddv = decy ddz d ex (dz. devi dv: deo) + deodeo ddv d(dZ. deo, d v: deo)
Ex aequ. 2 fiat deo (av; aZ) quam differentiando habebimus Ado  (dv; dZ) + da  (d d)
adeoque ddv: ddz = 5) A(dv; dZ): d (dvidz. Porro ex aequ. 1. per 4 fit ddo idaz = (6) durdv: (v, 12.
Et per 5 et 6 fiet dL de d7 dZ + dvdv,: qu dv; d2) =  (d0; d2): d (dv; d2) quae aequatio
dz et du, debet esse ideutica verum video remsic
allumto valore ipsius (0 per 3. amt et ly: dx, habebur ex ddo iddque statim initio
quae cum  solas continea
ergo t rursus per aequ. 1 et 4 ergo ag. dx dantur per a0: dx seu per X: y
non posse procedere sid pat
res reducta esset ad quadraturas etiam sine aqu. b. sed oportet tam
ere
verum video rem non procei
alb: ddzue datur ex dz: dv aequ. 5. sed fix Ergo dfdx  jam ado: ddzue = du: dx,
et dy: dZ = y: X. Ergo dy: dx daretur ex X: y. Sed jam sup initio datur
ita
C4. 7G esse identicam
ilares procedet
seu haberetur solutio sine quadraturi
y: aL avido, et _ x: a = d dz: de et j: x = dv: dz
dy: dx exx, y, a ergo dy: dx daretur bis,
jam la aq. data dy: dx = (5) XXF yy,iaa.
Cet ddp: adue = avdo + d3d2: deo deo fio jam Alg. (ddque dZ -(dg. deo; dv: deo)
Ret ergo sic potius
pro arbitrio et tint dazidu = (9) ddZ = (43. Ao, d video) tddo APdg ideo; dv: deod
et ddy dy: dx = (4) dev: ddZ.
per 6
(2
deo = sadZ: di + (Z; du) et diferentiando erit ∞ dd set ddque daque sade dz
+ ddzue d(azi dv) + dav  Cdis  0 = deo ddzi de +, d (dz; du) +  alr (d2: d0) et diel
deo d = (9) qu ddzu ddz qu d (dz: du) + ddD ddv qu a(d2. dv) = 2 dd 7dd v Adzidv) a (z: dv)auqc
dddo + dZd2, ddZ: ddv, quae est aequ. in qua solum do de d2. de dvi dz  et dav: de)
nam si daretur dr. dt Z ddv: adz ex dv: dz daretur
son licet ob jam dicta
et X: a = (ddz: dv
um liceret
et dx dy: dx = dv. d2; et res redi substituendo hos valores in aequatione
sic ergo potius si sit X y: a = Sdd.
yid X ex X: y o quo
data, per unam diffeviliationem tolletur una simmalitas, et res reducta
crit ad aequationem talem (dd Z: do = (E. ddr: d) ea  (12: du; ddv: da
Fial dav: ddz = (2) (dz: deo si)dv: deo) eritque fddZ: du = (dZ. dv; Cdq: deo; v: deo). Ergo
ddque dx - (4) ddZ d C.. ) + d ddo  (eut seu compendio a Sddz. de = (7). r: a
siet ddque. dv = (5) ddque d r: a + ddvar: a et ddv: ddz = (6) (1: d0) Fe- dr: a, r: a
= per 3 per 2 -(7 mo) (dZ: deo; de: dio) vel per compeindium fi
Et ex aequ.I. fiet sadzue ido = (dz: do; r: a
d (dZ. d0) d (dz. dv; r: a) + dria)  d (dz: dv; r: a)
dav: ddZ = r: a = 4) (dZ: lco sildp: deo)
sed d (dz: du d = do dd2 - dZddv: dudo et dr daque trc- ddv r. Ergo fiet
ex aeq. 8 fint per 9 et 10 fiat ddz: dv= (dv ddz - dz ad v,: d vdv) et (dz. dv, r.: a) -
et pro per aequ. 3. pro idz
fiet Y: do = doY - dZa,: dvdv) d (dz: dx; r: a)
et adze: du = (6) d  (dZ: d vsr: a)  (7
(ddZ dr + d d ddv Ar, : a) _ d(dz: du; r: a)
+  + Y dr = a ar : a) d (dz: dv; r: a, quae aequatio debet esse identica, seu Y
10
ubi nota d esse differentrationem secundum
ponendo Y, et pro ddo ponendo a,
42: do, et d esse differentiationem secundum r:a, et dr esse differculcatione
psius Y secundum dZ, et r esse differentiationem ipsius r secundum do, et pro
talis assumi debet, ut hoc succedat
Y ponendo (do ut autem relationes datas vel cognitas ab incognitis
et incognita
fiet
distinguamus, poterit fieri cognita f Y (dZ: db; r: a)
Hinc explicando aequ. 12.
v (dZ: deoj; )dv: deo) (2  du (dZ: deo si (dv.: deo) d (d Z. du; (dy. deo. )dv.: dev) ( (d Z. de t. av. deo)  a 2: tuosi hor
_ Y: a = (dZ: dco (5) dv: deo).
+ vy B (dZ. deo s;(dv: deo) d az: dco (; (dv. de0) d (dz: dv; (dz . deo 1; (dv: deo )
Don + vV d (dz: deosi)dv: deo) d (dz: dv; (dz: deo (3 (dv: deo)
et debet relatio sil assumta esse talis, ut aequatio 13 fiat identica
diqe dio