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# Catalogue record: 60470 — [Aufz. zu Differentialgleichungen]
# Incipit: Sit aequatio data 50w + 60z + 51dw + 61dz + 560 = (0) = …
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=60470&katnr=57398
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 87 (from the shelfmark LH 35, 13, 1 Bl. 87): 2 pages, canvases 171–172, page ids 00068196:0171 to 00068196:0172
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=171
# Lines: 292 recognised on 2 of 2 pages, mean confidence 0.79
# 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.
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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 87r — 00068196:0171
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000171.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 144 lines, status machine
sint aequatun data 50d = 5ocd + 60Z + 51 deo + 6i dq + 5b0 ( =3t70co + 30 Z+ 21duo + g1 dZ + 780 = () -0
sit co = Dy + dv et Z - n y + H V.
5050c0 = 50Dy + 50Av
1560
1780=
60Z = 60py + 60HV
S1 deo = 51ABy + SIdA V + 51Ddy + SiA dv + sbC
61 dZ - 61 deny +6i d n v + 61 p dy +617 dv
70W = 7oby + 70A∞
80Z = 60piy + G0 r V
7Idco = nidby + 71 ds v + 71Bdy, 715 dv 478
e1dZ - B. dpiy + B1 dr V + 81 n ly r S1 r dv
Et ex aeggus. datis, interpretando cd et Z fiet
(3) 70 Hae novae aequationes componitae intelligantur in unum
ne
it
identificentur jam aequationes 5 et 6
per y fiet
715B. 81fM
fa. difa
ib
multiplicatione prioris facta per e, et posserioris per f.
paeterea Et videatur an aequatio composita possit reddi
tractabilis, si possit coincedentiari identificari osi
 huic C + K d0 + n multiplicatae per 1. posito
3 = gy + h D, +dO = dgy + dh V = gdy + h dv. Et fit
igy + ih v
ikdgis + ikdliv + ik gdys + ik h do + ik n 
e et perire
51 eB +b1 e , et  (51C)B + Sie AB +((51 2) n + 61 e de
615
Pa715
7Sat
670
1 SeteD +B1 C)
t de + E lic use 8 sp
I usa BiSM et s (o) _se A F BC LA + d(B C)X+ B1 edX + dix  ed + 6  )
S1 fdu
it
hoed +boee +5, edB +6i ede (L sig  ik dg =) d su e) B+ 5ue
71 fd N
0d+bIB ScK d
C12)
20fD 80f 1fdB 81 fde
-
AS dat
ubi evan in qua aequatione apte eva nescunt dd et dM, et habetur B: n =
Eodem modo per v habebitur A: H: ubi tantum cavendum est, ne sit A: N = B: n, sed hoc fiet
Nensee scecse, fiot 60 si has literas conservamus  uod si velimius destmere _ et M, (undetram destruentur A et I
rota incidemus in difficilem nimis inont inquisitionem ipsarum e et f
Itaque istis quoleunque comparatio machinas non licet efficere, ut aequatio ex trans datis transfor
Si in aeqqe 3 et 4 liceret facere, ut coefficientes duorum quorandam terminorum unius
matis composita reddatur tractabilis
aequationis haberent inter se eandem rationem quam habent coefficientes duorum respon–
dentium terminctum alterius aequationis, possent posset componi aequatio in qua illi
Si F Itaque si simul vellemus tollere dy et do fieret
51D +61 n, :, 51A + bI7 = 7IA +BIM, :, 71N + 8II. Unde multiplicando peo crucem, fit
termini deessent simul
(51. 71 AB) + 11. 71 X n + 51. 81DH (+ 61.81 (n 1 = (51. 7IAD) + 61. 61XB + 51. 81 A n (+61. (B1 n ri
unde fit 61. 27 (51. 81 - 61. 71) (2C = nD) = 0 und sed non licet facere An = 18, seu K B:  = p: H, ut
jam dictum est.
si simu vellemus tollere, D et de
50R + 60H + 51 dA + 6, d H
fieret
7on +807 + 7id + 81 de comperi non licere facere rationem
7I A + 81N
itaque ciet: + 507XX + G0. _ RH (+ SP. Y. Add + 51. BIrd X
5I A 46IN
+ 60. 8IHII + 60. YI NH = 80. 71 AdX - 61. 71 nd X
= 51. 70NA = 51. 80NH
=61. 8OMM - 61. 8O N H
-51. 70
+ 61. 8I rdu
= 61. 80
+ 61. 71 A dH
- 61. 81 Hdu - 51. 817 dH
G+ 8
constantem ipsarum 51A+6I, et 71 A + 8I H,
ita enim explicato H evanescet A
AA AN AX XA SL d T
co
N = B RN = 88
N. 6271
sed ita deo evanescit
- 61.71
i- deo b bedtes
+61. -1
sit, A = b et ( = cs (d fit dA = dco b et dn = des b + pdco b. Itaque fiet
ex aeq. 1. fier 30 + 318 + 32deo +,d. 2x aeq. 2 fit 40 + 4 G + 42 deo + 43 de
438deo
Ex caq. 1. fit 30p + 31c F 32 d ip +33d9 +32deo = 0 Ex aeg. 2 fit 40Y + 415 + +2 d ep + 43 d 9 + 42 deo
Muttiplic etur illa aequatie per 43, haec per 33, et 30. et a se invicem subtrahantur, fiet
Goded
43Gdeo
multiplicetur illa per e, haec per f, et coaddantur
 =0 d( l. V + mg) ~ di y + dm 8 + ld V + mde
3. 43G. t 3143 +32. 43d V + 33. 43 des 1 32
+ 30 Crp + 31C4. = 32 C d p + 33C des
+32 deo Y F 33 edco.
-40f2Y 4158 - 42 5d4 - 43 fdes.
∞32le Y - 433 de es
vd3ge 1 Gd 33fC 5
+424 d f  +438d54
+dg2hf 1d43f9
ABec 3ge9
arfgip +43058
datur huic aequationi
Ht3rgdet + 33)de G
+ge fers 13des
-424 dfy -43259
diffip 13df 8
+ 32eg +33914
fit l = 32C - 42f M = 33e - 43f
(et erit 30c = 40f a 2e +32 de d4 f +  df = 0
4259 1 - 4395
pdc
- 61. 715F 1pdg   inf + 51. 81 9 4 deo . 51. 81 Ve duo
Aliter A = b ( = 8b ds = Idco b + dip bo et dn = c)dco b + d9 bco
33
(2)
30 d + 31 n + 32 d a +33d n =0 40N 41  + q2 dR + 42 dpr = 0
61. 70
A 4.X + 40 + 31 dx + 41 dn =0
+33 deo
 elo
fin  1. Vi 1 8g +. ps . S. 9 plo F. 19 0
11
3adio e gr cof
31C 7- 414
37deoe -43 dco f
dise +3y de-dgrs + df -
cofficientium ipsins g. astat equatio actabiliis quae est (g (3 e)  + (d e)G
sed video u vel deo, ad idem reduci, seu atterurum ( d sore Aeg
altenntium non prodesse itaque potest poni 9-0
435.
jam aedendee sunt injiceum aega. una en  sibi scrpposilis, altera ex  sibi suppositis
divisa prius illa per 42, hac per 43, habebitur aequalio sinc ef letiam etiam tollitur f  non possumus efficere
, bis. Ergo habebitur des ter log. C. Et hoc videtur
necedere; sit sc. f= βC. firt da= 41 β3 + 1; 13 = 43βd4 4b2 (2 deo = . 3) + 4B + aZ  Bd 42 + 32 de - q2de, :, 32 qu
sed im prori de simpliciter. Et proinde dues ipsius deo ralores aequando evanescit. Ha patet ipso deo noss
f in 
sed ita inubiliter
evanescit deo
ta divelli potest in tres, una coefficientium ipsius  sibi suppositorum, altera similite

## Folio 87v — 00068196:0172
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000172.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 148 lines, status machine
EF in omni aeq. simplics duam differentiali duarum
alem
sit pamy et f nv
32dcomus + 3tdeonB
33dcopy + 33dco gV
32dm 1) +32dub +32 may + 32 nd
33 aps + 33 dgV +33 pdy + 334
Jomus
fiet 3on +,dce n +H 9+ 35dc) + qeduty dqu - tont 42 dcon + 919 + 43 des 9+ Fnt 13 
Ex his duabus aequt siuul tollamus V et de
30. 42 un (+3242 dconn + 3Ig2ng + 9f. 33. 4 2deb n9. + 32 qunlu 33 +undus
aequ.
32.40 nn (132. 42 dconn) 32. 41 ng
33. 4149. (+ 33. 43 deoq4ue) 3340 ng
32n F339
32. 43 deo n9 43. 439d932. 434dn
+. (32. 43 d0 ng F32. 42ng 152. 43 ndque
33edqu
qutmy
=d(iem 9 . dsiet
1328 mdy +33 C9d
4359)
33C p dy
43f p dy
4359dv
ex
42f
1354
endo et demendi
boli tot
ope ducrum ipsius deo valorum collamus de fit df
ita evanescit do, nec
oumF 3Iep-3Eam -3 Fdpe H32. em + 32 edmtz ndert
du2tmtaefum+
destmamus in hac aequatione tan ds,
quam fdf. ot af, fent
33. 42 egm 2 47 e
dexomiratores Y
(32. a2 emnds) + 23 42 e puds (42. g2. fmnat) + 4. 47 fnq edt
espetne
sed hoc nonlicet
peuaere et (et silitu Ls) Dum11 3 p - g 2n. + 439
3em + 33 ep +42Sn t 3sp- 3 eq 435 47 32e+425 = m2
43P
a
B3Pp+ 43fY
22t
32C+425
Faciemus 32 em +33Cp + 4u so + 43sp = 32 en + 33 eqtarfn + 43f4
ita fiit p= 4 sed ita Clial sieiet n = m
2
m
g2arem +33 V ep + 42 Lm + 43~. fp = 3raent 33 acq. + 4rafnt. 13 afq
ut 42. 438
R = 32 N e m + 2G ev ep + 42 v. fm + 43W sp - 33 aeq. 43af9
-42 43f9
hrac re g2af dui
seu n =  m + 37 e (qop a9) 1B (33 C + 435) C  p - a9). d estruntur spont m Cfi
42n 4 439 sit n l m  + 43pF
23ac - +2af
re 4et + 3re + 42f -0
eaemet  Lepe tes C3pdete negut cqne
e as2e Scde A1 sp 42 p= 43 pas-a sa massut aqa
Ar CN h Sep BCer dae ((3 d 3) P + (41 - d4) (33de + ~) p- 9
sed ita simul esd destruitur du et fiet
hoemgofm
20 en fB fes + d32em + 32nde a= 240fta d4fm
supersunt 40fn aa +d42 fn
et fit n = M
de
a43p93 fap +43p d
32em = 33C p + 42f m + 43 fp
32 en + 3309+4
q2m + 43p- 4397: 42
=32. 43 ep
Z243C
 faciamus potius denominatores in data ratione sit
etAm +4
et fit n. m +
+
my Esc lum affictis per dys  do- 0
a23 ex +33 Cdp +
v sabi super posilis destrecris
Bie
deo = - 40 5n- a15
532 42. en mdf) f, 3. 42 Cgmdf +42. 2. fn md5 + 42. 435m q. ed
quarum ope toltamus n, et p, nempet
et fiet +33 grep + 42 a3 fp
37. 42 19 = 42. 435)4
am -42Fd
lodem modo e
32. +3 emgts (137 42 e 9p d5) + 4 2.43 5mg df (+ 43. 43 fp9 ds
quicquam lucrimur
rent asmentu 
ded =  40
si licet
32. 43 en paf (+33. 43gpds 42. 43. fn pas (+43. 43fqpdf
a
qn +4Bq
Et de reliquo t frat ex y totleado ibi supersiositis destinctis
n do
qufdeotus arfdcono. arfm dus +42fndo
40fnv
grtn
Bt
dremt3s ex +Em p+3) y + d (32 en tcq +g Sn 14) sy) B
rant
et compositam fentemus compiet  (rem reprartmta3s
33dcd e q
32edn
n
1.
smassiplicasam addamu in 
33. 42 deong 43. 434 9) 337 42 gda
hinc per illam
Quod si aequationes eddnts tractabilem
gredomys +32dconv
+330dcopy
+2 edmys
33CdpG
+3legV
475P
teral eiferam anguiao; itaque
+30 + 31C5 =32 d( + 33dG -d et 404, f 418 +42dip + 13d9
+3 alco  133dco G
20 siniliter tedous etc
31. 4349 (+ 33. 43 deo49 36.43
30my + 3onv
31 epys
ortns
42co 43dco
si= mg+no et g - py + qV. fiet et dyp - dmgs + dn v. + msus Ando it d - dpst dqu + Pdy t9 do
abemut aequativnem
Et obtinai uaes novam quantitatem de
30mys = 30nv
1py +31qV
finc
tmd
42 n
(31 - d39) e + (41 - d43) f - 33de - 43df ) p-
(d32 - 30)C + (d42 - 40) f + 32 de + 42 df
3 a, p -9
quos binos ipsius m valores aequando
M = (_ (31 - d3.) C + (41 - d43) f = 33de = 43 df
d32 - 30) e + (d 42 - 40)f + 32 de + 42 d
(33 e + 435) (a p - a9 a, :, v - a
fit
23 aC - 42af