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# Piece: https://leibnizlegible.com/api/records/368/text
# Catalogue record: 368 — [Aufz. zu Differentialgleichungen]
# Incipit: 10m + 11n + 12 differentiando 10dm + 11dn + d12
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=368&katnr=57944
# Work: https://leibnizlegible.com/work/00068199
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folios: Bl. 93–94 (from the shelfmark LH 35, 13, 2c Bl. 93-94): 4 pages, canvases 185–188, page ids 00068199:0185 to 00068199:0188
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068199&page=185
# Lines: 380 recognised on 4 of 4 pages, mean confidence 0.67
# 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.
# Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German.
# Licence: CC BY 4.0, Leibniz Legible. Machine output: do not ingest it as verified text; keep this provenance with the text.
# Wording rule: say "the machine reads it as …", never "Leibniz wrote".
# 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 93r — 00068199:0185
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 94v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068199/jpgs/default/00000185.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 95 lines, status machine
d: c
na
+ rav  la 7 at fet  
 alm
e u a
 o1t 1 L 0
1.  + 1+.
sunt a e  pleo  eet
denam soda d 2 t po ue vn contaca p
6 d ma a’atre
 oico actal 2 ms  
=p nse ahoF +O d in XGAIC
et 0c -c et
1 te Alac. ext
ch A + stivC L. pd dr s. nC = p
at EF  ochm OC + sL. Or 25  Dck M CF e tab03, 05
ad
2x = (o 05 1 -
 ban
nm 
et m +
so me ct nerit co
= 4
C.) pnas  A E sM 1. A mnve d  simaiiana
1he D. c artib n al hele +BCA
or larcactit at igelitant  li a ato abet doti -9 a at t
ab  1X 3 oct d Sct B.
e. De = 1od
a. 12
BoA 2E 15 = 0ct ac 4 _t 10
S sit mbut d aa in ina
en doru n alove
aviria 1B 
neorelt
d a
deBE t d 0b. 2s
10.D it d 1E 0. 1 9 L. d 1
 tam leviadia nas osur monoje.
19 13
n
 + n 0.  + att t do unt
11
se L
1t
ut  A cex mtaore d reL. vo A  acitsine
2, i A0H y
xditare
hec
ab
1: octa 1 M d+
est
et et e 1
a0 B
dh ced. + a
il
ia
a rmm e    1 l
∞i Ae Ai f ert N E s + geboeS A ex + sthioEdot. L out 05
nLuc + getiahos + et haca ox + mb. A ed 05 + rel sot.. 2 8
atis. m: nem pd. e = 3 1 L. 1 5 = a d f 15. c, : X lip DF. B5 = A  1t: vd = 05: d6 a 1d. D5 = cF 13 7
at 40d 05 + ae metpaeL
0o toc. 65 - 0 F1 . BEt v, 1 ad = 1 3t 0= 15. BC: BCl. OE BB = 0 pl 05. 05 + 0C b0t 05 = 1
OFhe,. 60 = + 15. 39. DB = 10 ) as Cp. = ab 15. 08 c
r 1. 05. B5  p1. 1:. B5, OV. us _ f = 1 = L. 35 05 ∞5 - 0  LDE, DD ge L OR OA = 7 111 Op. BA 1 12
vola   lo res
bi1 = EC = OCV   t  18 9 
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d5

## Folio 93v — 00068199:0186
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 94r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068199/jpgs/default/00000186.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 75 lines, status machine
siom dx +, dua t att n dx + 13 an
10m + un + 12 diffeentiando 10dm + mdio + 11dn + n du. + d12
Esto aegus. 20dm + 21 mdx + 23 dn + 24 n dx + 25 dx = 8
e3 (tom + Hn) dx F sit g = rom + un
Solvi potest (ex rgyax + dy. sit y - tom + un
qux + P y dx + + dy
solvi potest fdx + gydx +hdy = 0 sit y = 10m + un, et fiil aequatiosruble e et fiet
Squalio solubilis fax +g.0m ax + ∞ un dx + iolam + utan = 0. haec comporretur dalae O
fiet 10 = 20:h et 1 = 23: h et f = 25
120 -2odh = l20- 20dh. et g = 2h = hdro = 20d 4 24h - hd 23 +23 dh
+Em dio n du1
21 = g. 10 + hd1o C seu g = 21 - hd10; 10
24 = g. 11 + hdu seu g - 24 - hdu: 1
du
et grodx + h lo = 21 b
Unde si calculus probus sequeretur rem succedere, seu haberi summabilitatem aequationis cqualis 
dio = hdro 2odh hd10 = h
tch 20:h
in quibus m et n nunquam habeat
Resumamus Dectasit aequatio differentialis trium literarum, m, n) solbi potest ( dx + qydx + hdy =0 sit y-   +60n
erit dy = 4in d40 + d
affectionem replicalam, ut 20 dm + 21 m dx +33 dx=0 dato
dn
31 n dx
31, 1ondbo +fix = 3.
g = 21dx - hd40, 31 dx - h d60
40 dx
om ax g6onx
60dx
 40h = 20 60h = 30
h d4o = sid (20: h) = ( hd20 - 20 dh = hd 2o  20dh, et g = 2 1hdX -hd 20 + 20dh  uh dx - hd2o + 20 d h
c
b
a 60 du
Exprodibit
q4n fiet f = 23
sed pari jure h dbo = hd(3o. h) = hd - 3odh et g = 31hdx = hdzo +3odh seu g = 31hdx hd3o+ 30dh, quos duos
t, (20: h) dx
e
d: h) dx
20dx
30dF sed ita evanescit h
BCha∞ +20. 31 hdx - B20h d30 ( 2030 dh =0:
quod non mirum. putem tamen succuri posse per arbitrias fiat  2odp + 2 pdx + E3 dx et fiat
30. 2
1. 30)
319dx
30hd20
o= c0m et dp = sodm + m d5o g = yon et aq = 7odn + n d70. Ergo in 1eq. C pro m et n, pe fudo p et 9, est
dq
explicando pper Som, et 4 per 70n, fint exaeg. C. 20. sodm + 20 mdgo + 30. 7odn + 30mdzo F 23 d
=O
on x no 50 et 30 substituantur
40h = 20. 50 60h = 30. 70 9= 20d50 + 2150dx - hdgo Y 21 3 
valores ∞ = goh   20dso = 20, 20godh p 20h d40 - 4oh dro - 40dh + hdg6 - 4 teda) goh d2o: 20) et 30 d 7 0 - bodla thd6o eh as
Ergo ex ducbus mabbilbus ipsus gnfiet aodAM 4. esu d 40 ) h duo + 21 L ex C m. 14) = ut A n 190). h ago + 31  ( hac
10
66
Licebitne in data aequatiane O poro m ponere  p+9 et pro B ponere +p f9
essi lergo ne sit quidem nam quaesita evanescunt
piastar
o= 4oh20dm +40h 3odn + 40 h rimdx + 4ot. 31 ndx =F 40t. 23 dx
h = 20. 40gdx :, 40. 21 _d X - 20d 40 - 20. 69 glx:, 40. 31 dx - 20460
0= 20. goh dm + 2obohan + 2olmdto + 2ohndbo F 20fdx
+ 20940m dX + 20gbonax
40. 40.31 dx - 40. 20 d60 600. 21 dx - 60. 20d40 60= 684. 30, 20 unde
40. 30 = 20.60 seu 30: 20 =60:40
a 60 = 20. 40d30 +20.30d40 = 40. 30d20, : 20. 20 20460 = 40 d30 +30d40-40. 30 d20
et fit 40. 40. 31dx = 40. 40 d30 - 40. 30d40. + 40. 40. 30 d29 = 4 V 40. 40. 30. 21 dx 40. 30d40
Unde omnibus divises per 40, fit 40 sc3idx
20
ille
corax
 sed tollendo utrobus - +0. 3od40, connescit 40
et fa 21 dx = d30  L30d 30 = 30riap

## Folio 94r — 00068199:0187
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 93v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068199/jpgs/default/00000187.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 113 lines, status machine
1ea + uy =0 differentiand sit diea + ydu + 11 dy L0. qram dividendo per H. prodibit. co+ y du + du
esta data diferentalis ejusdem formae, quae sit resalienda  am dx + nydx + aacy ifiet du.u  ndu
etoderc seu Sduiu = Sndxet Rio = StImax et Ga sumax : 1 = yC
M ma + ny =0. ubi pono m et En dari ex x differentiando fiet: adm + dy + yan + ndy-
vel divisis omnibus per n, fiet a dm. +y dn 4 dy =0. sit, jam data differentialis ejusdem formae, qua
netiam bC dantur per
resolvere oporteat:  dx +Cy ax + dy =0 (fiet Sdn: n = Sedx: aa et m = Sbaxiaa sbnax: aa
Ascendamus altius, et esto maxtanus + pyy =0. et diffeientiando aaam + aydn rgydp + andys + 2pydy = 0
Esta, differentialis data ejusdem formae aabdx + acyax + eygex fdy + agyay = 0. divis illa
aequationem per 2p, honc per ag, coincidentiemus, et fiet, an: 2p = af: 9 seu n: p = 25: g, et rursus
ap: 2p = e dx: ag. Et datur p. Ergo etiam datur n. sed fieret denud. a dn: 2p = cdx . g. jam p = gn: 2C
et fiet ca afdn: in = cdx: 9 seu dn: n = ccx: af.. Quod non procedi, nisi in casu speciali
ubi ea ost relatio inter data per quam duo ipsius n valores coincidere possint. Sumammus ergo
aliam differentialem resolvendam bax + C vdx + C vvdx + fdo + v do = c. Et esto v = y + F, et
prodibits + bdx +9 7
ex
+et ax Eyd
e Etax 28 Fydx + e yydx
+Fd+
+ + dt +ydt
F5dy)
+5dy
+ + dy + ydy
audm + a gdn tgy de tandy +ydy 20
2PS
Ex yydx fit do: p =  2 edx: aa. itaque datur P
Exdus fit an: = =F 2f P + 2pt, et a dn = 2f Ht) p + 2p (a s+ d)
sed exyaxsit adn = 2e px + 4 eptc +rpat. Quos duos ipsius
adn valores aequando fit 2aa fap p raat p fraavater aapatm
= 2a cpdx + 4 eptaxA 2a a pat) seu a
quia autem f habetur per X, ideo possumus facere ads = k ax, et aequando duos valores ipsius ap: p fiet
2ef + 2et, = ac + 2C+ = ak. Sed ita infeliciter evanescit t.
AAliter tentemus: adm + ydn + ndy. multiplicemus per aq+ py, proditat:
aatam + aty dn + pyyan + agndus +npydys
+ ap yam
de dr f. efdx- ad df
hanc dividamus per np, et coincidentiemus
ipsi. aequationi O
a+: p = f+t an: n = eax: ua. Ergo datur n.
Compendio in aeq. C terminus infimus sit Saxet peninfimus Ex fiet am = N np: aaque, et rursus an-
npp se dx = npg 7 dx = aagqudn, adeque fier
dn: n = p4 = dx = ApP SVAX,: aa qq = edx: ad seu p9) - pp5 = 9ge, seu
Inp dx = aagdn,: aap. seu
F Sea 43dx, (od 4. P = 5+ t. Ergo explicatione facta redit difficultas prior, secundum +
Assumamus maa + ang + p +  a5 dy: y =0 et differentiando prodit aaam +aydnt andys + dy: 7
eax op nec lucramur
ita frt aayam + agy an t dys  anydus dato simili comparanm
mbyax +cdu summ abile reddi pt multiplicando per in, nam fiet
bmydx +emdus comparandim huic dm yan+ ndus nam fiit n = cm et da =t
et du = Cam + mdc = bm. Eigo prodit dm: m = b - dc,: C. adeoque datur m et. n
Ergo summari  byax +Cdus, m, posito dm: m = b -dc; c entque summa Cmus
(5
fbys dx + Cclus) compendio vocetuo Si ergo summari potst potest srsi = emus
md N = d scmus) d(sm) = Nam t mds datur (m da = datur Et fiet,
et fit se = fbydx + cdys ,: m) ((d Cemus) am) = sbydx +scdu
msi fdim + emus seu N =  SNam + cy
dse = dCemy): m
potitus, am: m = b - dc: c
Sed haec methodus non procedit ad plura simul summanda, ut, si esset
bydx + edus + egydx+ fydy. non licet una eademque multiplicatione efficere, ut
ut ambae partes summabiles reddantur
ad 1042) +  Z db + Chy ryde  p det fdy tyds
in summabilia plura
sed videndum an non liceat divellere, C
2, hedb + eydc
+ f ydb +f yydc
37f bde 4bby df
cyde tegydf
+ ecdus
-
xase ydx, aam textang ax tpgnax t angulus
uit ia
fiea  4fdc ged5 +4d ef dx
= apec dx  apbF Ax
et ase db A = bde + qubeax = Am ec dx tcmbsax
ila habentu
+fc ydy
+bsdy tef ydus
+ g besx + gbsyex + gesqycx
si 2 =1 et 32=
 ita habitil
std aquatio
sin dan ecdxe an cfdx-q
bfax L qedx
gg = apecax a apb f ex- 4495 dc + uaqc d
tos +  et b + Cy
si 3=-E evanescet ydy.
+ e + 57d
Habemus ita quinue arbitrariis
b, c, e, f, g. Cum pim initro
adassent solum tres
fa db + cdy +y de + f ydf + fay + gdx = 0
a
Bovius sic
ab +cy
et fiar
1a db + aa ydc
 abydf
ad + egyax +cgy df
faacdu
+abfdys
+ ef 
sit jam data aequato differenlialis
aax +arydx tpyydx + aagdus + aaydus = 0
.Et datur
firit, df: f = pdx: aa et datur f. Rurtus aa + ab = 4
ab rabgdx: cf = mdx fit g = ncf. dx - a d b,: abdx
9 = a dc + aabdf - aC fndx,: acdx fiit mecfdx= aecab =
fint abac + aabbas- sbefndx abde t acdb
ir dugbus puosemantbus a et valorilysfit apbcedx. ap bb fax- aaqufbde + aaqbe ds = me fedx- a mbsf ax- aaq esab +aaqu fbde
13
49 = mecdx = amb filx- a ag ed b + aaqude
aa, f + t

## Folio 94v — 00068199:0188
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 93r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068199/jpgs/default/00000188.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 97 lines, status machine
d + 1 S.   Ein A   DL  Mau 1 = vd. 11 11L Y + 1 2
nta ir.s 1 vdo sa Q + Xtov A L at aer data tti ab. dudin de datum ua
P = 1:   a  F XC  Xm 1I A = .d 
vd  ed  + cel e F 1F A F : 2 oa al B.  e  e
Port A ux an e 4. D. a C, egil N (su, 2c 0 = vd ∞ Mbo v
BM. Xt 1d1 aror ntiet = 1N 1d BN: X 3 (  B aedi
et nb G u d 1
t P  a diois aehuccatia do
ie i
sed did namar at ari
ita
x 32 dmat s)ot V
at d
 9 di 4e d
+. ( B
X
1 e  2
e+2
deve
din  cnva d
2730 = et + xt x3+
R tp a nonm am a
1
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