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# Piece: https://leibnizlegible.com/api/records/5785/text
# Catalogue record: 5785 — an
# Date: 1690-09
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=5785&katnr=42058
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 305r (from the shelfmark LH 35, 13, 1 Bl. 305r° (1 Bl. in 8°)): 1 page, canvas 605, page ids 00068196:0605 to 00068196:0605
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=605
# Lines: 185 recognised on 1 of 1 pages, mean confidence 0.78
# Catalogue: Leibniz-Katalog / Arbeitskatalog der Leibniz-Edition (https://leibniz-katalog.bbaw.de/), ed. Arbeitsstelle Potsdam I der Berlin-Brandenburgischen Akademie der Wissenschaften, CC BY 4.0.
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# After the first blank line, page by page: a "## Folio <label> — <page id>" line ("## Canvas <n>" where no folio label is recorded), "# " lines naming its source image and model, then its recognised lines in reading order, one per line.

## Folio 305r — 00068196:0605
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000605.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 185 lines, status machine
Quod est nirum, cum enim F seu dyi dx detur
pe X. y,
hinc sequitur aidu:  dani per X. y
ergo dari per X. y: a
illx
datur aeddus dus per d
posito dxdx. =  X, a ddx de generaliter
ratur a ddy: dus per Xa et dx: dy
a
dx: dus datur per X, a, ergo d
denooo tequatio essavint
oper
2
ed
iublata. sequatio est  a adu: dus dx: dy addx .dy. 5)a
valorper
L
drem
en addy: dx C ddu: dx addx: dx X, a
a
Ergo si ddxa = . dabitur
into insius a ddy: dx per dy:d B, X, a
Exm ricissim addus:d datur per dy: dx X, a
sed g: dix datur ptrXa. Ergo a ddy: d
dabitur
ergx, a seu per rationem X ad a  pro
jam pro X substituatar iveram dax.c. Ergo habetur
ergo t daturidus datum ex dx per quoda
 aui
arp et
dis ea
Quid sit sic aequano u disferentia disstrentialis in qua
absit du. restet _ siet addus: du = actax  per addx. dx
deo hinc exitumus ulao
palio differentialis
a, 9.
praa
de dudi. dusvalori per addx: dy, : dy, R: a, G a d
Barpteris
s
Tertia  cilia adue
2Tenia assumbilia
erfa
seco
s
f: a in et Tertia assumeatia
addus: dus = y: a
Htiae si p12+ LV
de Hine si per 4g. si cotcs ex ag. 2F d
dx: dys et ex aeq. Z per eg.3 tollas via, rest abit postim
aequato adg: ay
addx: dy , X: a
sinipu aeque sumtusi sadd
An sic sit a proposita:
1. a= lyd
xDarec
 ay: dx evalori per X: A, Hisa a
Ned  reluno inde differentio differenti alis in qua Cextent
X
 Hinc
aaX
Hinc dhy: dax dalori per dy: aldx, Ax. adex, X: a, :a
sumatur 9ad addc:d valori per v
quae explicitur reapse, exaague da po d et Y
u
9. a
coclotaxia
2 et 5, sabeliae oe a, et rurius per .4. 1 et 2 hlabelur ae4 sine a
10  d= dx d 3 = y day dx = dy dax
e rbi
a
X E
alegans excuptum alititatis cum ay: du
sumitur a. ita sointa
tg.  
dxx + aDx y + bX, +CH 0 A maxima
possumus semper efficere ut aliqua litera
extet sola, ea scilicet cujus dd 20. ut sit
ddx = 0. firl
2dx dx X + asdxdy + ax dydys + a Xy ddytbaxdys
+ b X ddy + 2CGdy dx + = 0 tollaturX
ope aeq. 1 ex ag. 3. siet aeu (4) in qua solu
lt du e  y +. yadys + y
Ax= Sdy + im ady  est aequatio
himogeneatalis legem servens tran cenentem
u dy: gadque nitu est ratio nil mulans
in homo genertate itaque dx et d homo-
genea sunt (ay +sy ddus = ydy
quay
si ddx =0 eo ipso habetur certa relatio
inter dy et ddy
inter
ea igitur
1y
et dy et ddy et nempe
quaerendo differentialem quationis 4
inter 3I
y.dx=
.
deys
_ day ax- sdxay
 du dx ditus: dx - +ax dy : dx
dy: dx =, X+y: X_ X
L ponaum ddx-0
et dx- 1(4
siat adque= xrdus X - dy XFg, 2 47
seua tolla si ex aeg. +nt. 4 licare aliquid
du
XX
ulile exculpere sed dubilo
s. um assumi, posstst relata quaecumque inter
 X. dx ddx A unitur lulis ut
ejus ope et aeq. 1) et 4 ex
1B0a + H10Y F120yy + (01 dy =9 fiat y = 10a + n1V, 20atu
fiet 11DD. 20. 20 a43. 100. 20. 2 ( N. 100. 21. 21V
2 110. 10. 20aa +1. 110. 10.21) a v +110. II. 2IYV
H. 2et
9= 7 +120. 10.1044 + L. 120. 10. 11. av + 120. 11. 11. N 4 101. 11. 20 adv
+ (0I. a1o 2oaa H. 101 d1L. 20 aB
+ 1. 101 d10. 21 av + 101. di, l vB
= (01. 1042049 - 1. 101. 10d2 av
(2101. 1 2. VaV
- 101. 18. 4 A dV
=1. 101. 11 d20a v - 101. 11d2IVV
(- 101. 11. 21 vd
assumatur MB +n dy b+C =0 posita e constante differentiando
C
it
M
+dx +n drdr + n dur =
+an
et faciendi adr = dx V, fiet
o0 adm +wm. D. d0d v +adn dv = 0
a
a
fit n = 101 (11. 20 - 10. 21) = 100. 21. 21 4 110. 11. 21
120. H1. 11 +101 C11d 21 - 21d11)
posita 101 constante. quod semper licet.
2.120.1011 21.10
odu
oIudio
(10121
C- 1ed2o
Hni jam pro absoluto termino, ubi est dm descenderetur ao
dd, nisi facianus n =  11.20- 10.21 = constanti ooisra
efint n constans, et 00n 101a (11. 26 -, 10. 21) et
M = 2. 100. 20. 21 + 110510. 21. Ergo tandem fiet
2. 120. 10. 11
2HI. 20
110d217 1, (11d20)
(10. 21 +11021d105 Zrodus
2. d100. 20. 21 + 2. 100(1 d212 + 21265 10dy J. Fl110G 1 20
Ludo)
2)100. 20.20 + 101. 20 101 (, 20 dio p. positis 101, et n ostantibus
2 d120. 10. 11
Lud2o)
ope tium aeq. 12, 3, debet solvi quaestro Ex 2 est 7
120. 10. 10
=10d20
seu
20 = 00 n: 101 + . 10. 214, : 11  00. 00f + 10. 21a, : 11
20
posito f = 00 n: 10). Rursus ex aequ. l. est 20 = 101. 10. 21 +
+ 100, 25. 21 -+ 110. I1.2)
120. II. HI 101 (11 des = 2du)
: 11. 101. Ejq duos ipsius 20
valores aecqvando fit
H0. (1. 2
Ct) cam  E (101. 10. 2(a) +100. 31. 2) 41oi. u d 2
120. H. 11 = 101.21d4
00. 00. 101 f (+101. 10. 2+a) = 120 M _ tumta quacunque ru e
su 00n ubi utiliter evanescit 10
inter II et 2) evanescit du e d2, et habetur I1 per aequationem ordi-
nariam. Quodsi efficere possemus, ut in aeg. 3 evanesceret d10
non evanescente 10 haberetur solutio. prosit autem in aeque3e. b
ipsuis dio, (posito 21 = 11P) ; 1IBIIB(: 11) +2. 12c 11 F10. 11P + 110. 11B
101. 10f+19). 10p =B, et coanescet 11, sirant t not nen aseu
 4, habemus H, P, H ut opesey
a
 +1
vL pdO: 100. Et ita ue aequ
i volones substiluantur in a
habals d. ta  aliuil ao, quo t emi ti aotiri o el bunt et 6
1 +10) p
it  tio