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# Catalogue record: 60533 — [Aufzeichnung betr. Differentialgleichungen. Formale Reihenrechnung]
# Incipit: Inserviunt series ad Tangentium inversas et per eas reducitur problema
# Date: 1693-10-01
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=60533&katnr=57129
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 371–372 (from the shelfmark LH 35, 13, 1 Bl. 371-372): 4 pages, canvases 737–740, page ids 00068196:0737 to 00068196:0740
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=737
# Lines: 382 recognised on 4 of 4 pages, mean confidence 0.81
# 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 371r — 00068196:0737
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 372v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000737.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 108 lines, status machine
21 sestetbil: 1693.
nserviunt series ad Tangentium inversas d peras redu
et per eas reducitur problema ad aliud faciliis. Nam cum seriei infinitae
aequatione composi Nam quando y habet pro valore seriem infinilam per X, et ita
repraesentantur ejus affectiones vel per se vel in ipsis e affectiones ductae, componitur inde
series infinita; et si jam addatur terminus qui constat ex solis affectionibus
aliu
is lac
explcata
aliquois non habet valorem infinitum, nec procude in seriem infinitam compositam influit, ut
V. g. si esset ax dx, negligendo de potest intelligi secundo gradui serici infinilae compositae ascubi
ax. I seu seniem re licebit  sed ut nos ax evilemus, summatim seriem tantum consideratimus
post  X, inciprendo ab X3, et ita quaeremus coefficientitis opeserici incipientis ab Xh quasi
nihil antecederet. Quod si jam series haec a composita ab X tendens in infinitum est
summabilis, utique linea quaesita este stdinaria. Furmmabilis autem est si omissa quantilile
a X dx aequao assumta ad curvam esset dabilis finite ita enim Y hoc series. Itaque et
A
pus
vi
tunc erit summabilis liaet accedat ax
pro aydx ponam bydu et pro
esaus et distrnctus uorem
sit ergo data aequalio differentialis aadus + Xydy +xy dx taq dx = axdx
singamus aadus ehydus, Zydx + aa d F a faciamus aa do + xVdo +Xrdx + ax dx = o
gaes
 ypes
schedicusmat dicrsse resolaesi resolui queat
a
sed si jam quaeramus X per _ v, sequitu sequi videtur hanc aequationem posse sumiliter
sed hinc 
Porro similithr suitur
resolod, ponendo, XV av +XoAX + avdX = 0 seu ZdV + ZdZ + a d3 = 0. unde sit
v = Z_. alog. Z. sit ergo Z = 10B + 11V + 12 v + 13 v3
(et adz -= a, 1 +, 12V +3BV + 4. 14v + 515V +6. 16 V ex)
7d0 = 10 + 11 + 12 + 13 + 14. + 15. ex
7dZ =. 21 72. 2 + 3.23 +4. 24 +5. 25 +. b. 26 et
(6
v =c
12= 20 +2IV + 2IV + 23V et
per con pend pro 10r utoripec
Erga f V = Z alogs Z t si sit. Erco ex terminis v, v, V, etc. v, et, V, V, etc. quaeratur
series exprimens valorem ipsius Z per V, seu valor ipsam 11, 12, 13 etc. nam 10 manet arbitiaria
porro sed ponamus haec jam ut inventa.
Et ja pergamus ad, aequ. 2, quae et fiet
etfier X et fal    30 +31V + 32 VV + 33V + 24 V et c. et jur quod poo
tiet + apde-  A . V + 2. 32V, 333V + 4. 34V 4 5. 35Y
29(30 +31 +32 +33 +34
X
vx = + B) (41 +2 p. 13 4 14p 15 45
+VYdo=
(8)
= 0
2 Xdx - 21964g X r41
Ansc
ZX- Gut bosex +St VISiPst t C
itaque ope aeg. 8 quaerendo sed quod morum videtur haec aequatio esse impossibilis
aado = ee tu
qua quoscunque terminos assignes ipsi D, nit reperdes quod ipsi a aequetur
caellies teminiscascrbas
Vi si quater
A = 29 + tube sde + 30 +31V. fort nam ita Vxdo dat h
X = 29 + 2. 29(d0 e 2.29. 30 + 2. 29. 31 +
 itaque quia ha haereo nec possum X exprimere
sed puto non scadere quia
fit _ ax - 13 24 - 3. 12, 4. 3+ er Chepdx = a1 242 d 1 eu
fiet: +a Vdx = a 7 X - 131 - 2 12 - 3. 12 - 4. 34 ere
pe
X var = (5 3.1) 31 + 32 + 33 + 24 +35 et
+yxdx =Z,
(Le) . 242 4 3.97 - 4. 41
=a dv - -aa
1 30 2 0  40=0 a1 -
fiet ad Z = a X 49 H  1 3t - 25+ - 3. 5) eu
Zv- 40r5s,  2 + st + 5 +54 ey
Idq  (. 1b v. G 4 -161 - 2 5r 2 1. 87
Sa hac est infinita
ad
sit 238 4 51t 51 +53 ei  00+ e p x
st 
ab = a, X X - 1 - 251 - 3. s 
2d- (59+ st + 3 +  +, et et
Z  (61)- 2.61 9.69 er
nonatur 5o= Bliq et 61 = 0 et 60 2 non iitspai
sit Z. 490+ 501 516152 etc 20- 59049v +60451 vr6ib aeu
Hmo patet omassit temminis uti
vi
0 se
su
 ac ficret aquat i possibilis
40= 30 4 = 2.303, 42 - 2.3032 +31. 31 437 - 2. 30. 33 + 2. 31. 32
44 = 2. 30. 34 + 2. 31. 33 + 32.32 et ila porio
52 +2.50.51
62 = 2.49,43 +2. 50.52 +51. 51 63 = 2. 49, 54 +2, 50. 53 + 2. 51. 52
64- 2. 49. 55. +2. 50. 54 + 2. 51.53 452. 52
sed indeo tamen nec sie rem posse u ferri
-
ostenres
seu cntrobius)eodem modo. inveniri
hunc sinterdum
coefcientes. Et hoc est artificiuqusvo naa
 tar erti e
non esset
Jam posset eli expumus
54V + 9
uo
it = et
et 3) determina
ne curva

## Folio 371v — 00068196:0738
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 372r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000738.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 91 lines, status machine
An ergo aliter tentemus verssumendo primam aequo pagiuae scedentis:
aa du +x ydy + xydx + aydx = axdx quaeratur X per V, et sit
XC 10+ Wy + 1y+ 1393+14 y4 15y+ 16y ex et X = 10 + 210119  2ronus et fre 
Xydy = 10Y + 11y + 12 y3 + 13 y + 14 15 +15 y
aydx =a, 1.1, +2.12 +3.13 +414 +5.15 4616
xdxte pion ru asuiva et. eu ert
Eiri
Bxip -i 12. 10u1 25 0  3( 2 . 1
aa = - aa
20.
si fiat Zydu taydx-axdx +yxdx radys-
opt; s qe qua differet d valore
ipsius X nisi quod 10-0, videamus an
aliqud hinc duci possit
Nempe si eq aequ. 4. suferas + 10y
=a, 1, 2. 10. 11 2. 2. 18. 12 y P3. 2. 10. 13)14. 2. 10. 1415 ey
XY
. 2. 101ry +2. 2 1c. 12( 4 3. 2. 10. 13 1 Cx
seu 410)  =a, 2. 10 aX + L. 2.10 ydx.
residuu inaeq. n 4 non differet ab aeq. resullante
ex explicatione aequationis 5
sed saccurit alium scilicel 5g. 1. et 2 habent eandem diferentiu differentialem; per
quam invenietur, a quatio communis utrique et valor etiam ipsius X Mer y ubri
fiet
 quatio ota
et X   20 +21y 4 22  + 23 y et.
 primariarm albesariar
, 2 21 4, 22y + 3. 233 +9. 24y  tC ixdx data Ny dyt agdx axdx tyx  -aquo
 104 11y. + 12 y etc
dp cndy + 2L vd 5 13y a dy estc
+ Xyay = ( X 10y + 11y + 12 y + 13y714y+15 y etc
etc
fdx
L.HI. 2.12. 3.13 4. 14, 5.1 6.16
= aXdx = -a, 1. 21 +2.1 43. 23 1914. 45.25 6. 26 7. 27
+ aydx =a,
etc
dlax = 1. 2 diy + 2.3ydy 13. 4ydis et
quatuor
+ aa = taa
+ yixdu-
121 222 323 4. 24 5.25 6. 26 ex
+ y dxx 0
a dx dya.
Fagdax =a,
=ar  1
Unde talis oritur aequatio
10 tuy tres r3y ay +15y 116Y etc
3.13. 4.13 3.18 si 6.19
121 23.13 3.4.14 4515 5.6.16 6.7. 17 etc
1n hE
Hinc jam per aequationes compa
+22 2323, 3.414 4525 5(. 2b 6.7. 27 7. 828 eH.
1.21, 22 3.23 414 5.25 6. 26 7. 27 eu
ratitias O destructitias
sunt tolli
O. D, 3 et G invemeri possu
tres literae 22, 13, 24 bm
et habetur relatio inter
Notui autem
11, 12, 13
10 et 21 ingredi calculum,
ut scilicet servatur generalitos
12 2 2315 3414 4515 5616 67 27 ete
Txdxdy = h
ut vero hic calculus sit communis omnib  ribus literis impediatis (demta
prima 10) sit 2= n siet 3- n+1 4 = n + 2 1 = n - 1, 1, 2 = nn -n 23 = n + R 3.4. n13, +2
cmttishi term
+ylx = 2
Et ita putcherrime et brevissime habetur regula generalis fimita pro terminoru e
potuissernus etiam aequationem multialicate
per vel per ay vel per simile aliquid, pro arbitur, unde rursus novae aequationes. Et videndu
an sic  obtneri possitet tales aequationes ut composititae quasi ntio aequatio esset ordinarium
Quin et certo casu poterunt
aet
progressione, non alioqui facile epuenda
Javenori quadraburae speciates, ~ o. Esto aequatio axdus + aa axdus + baxdy
explicata, ut si semper simul exearent a1 + b21
ax dy + baxdus + caedus + Cy dx B.5 a dx =  ft Ergo in casu quo es -5 = 0. sit
iterum Ax + Iax + ca = 0 quod quidem ex ipsa corva patet eu differentiali: sed securius habet.
ex seriebus. Et ita invenitur tam y quam X. sit a Saa + XX a et + est dx: fdX
E 2Xax = Cy Ax - fdx + Saa + xx dy = 0 Ergo quo casu ey - f -
tiet y Fa 2Xdx: taa + XX = dus sed an licet hinc inferre? esse aatX = y + b
eo iesa quo ey - f 20
videndum quo modo ex seriebus talia jastificentur, a quatenus
x = 1. 2. atus r2. 31. 19 iy 13 424y dy = (p 6) 2. (X ax
Fairdys
quae ratur differentio differentialis aequationis B1 fiet: xdus Fi dedtus + a gadx = a d pdx A Xte ay
+ yxdx = 2)
+ Xdu =
et

## Folio 372r — 00068196:0739
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 371v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000739.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 99 lines, status machine
Nunc reaps tractunus aegy. 0 6, G.
2cC nc
sed sufficit pro 40 scribi 4(7)
 1 + nFC13+ nf2. 24 + nrI. 23 = 18,
3(1)  3 (6) 6(8) 6(3)  0(7)
nec opus est numeros
1+ n-1 seu n 11 + nn -n 12 + unn 23 + n 22
Ino sic
quaer ut 0 et ag.
(3, 21 + Ena EnaN - n + a 23 + n22 = BC3)
4(0)
3(4) 3(8) 8(2)
M, +29h12)
8, 2.I1 (+ 2. A- B11 (+ 2. nn- A1 - nn + gus + 2M 22 + _ n  - n22
seu 2na + 2 nn - n12 - nn + x 23 + nn +n 22
mo jam video C et s 8 et  uut prodesse, cum uit aliud sint, quam
sipse Y 0 cD duplicatae
1X+ =9y
1. 73
uonjam ex quation ex solis ( 10, 1, c2 etC et 20, 21, 22, etc. sine exali
catione horum ductum ex se invire video non posse rolationem inter ipsas 10,1,12, et
etiamsi ad diffenentoas differentia s idas descentas vel aliter, ideo it aliud superest, quam
merale constitulum pro evolvendis latibus
vut ex 2g. 4 in vestigemus 1 vE, 13, cu.
aequationibus ubi sit a, b, C; d, e, etc. He quod tget 18,13, 19, 15, et C.
2a ton E te H
20nt2 ue 20+21. 12 m = 0. maneat
30 = 114 2. 11. 1 31 = 2a - 3a 11 - 210 32 = - 3a 10
ist videa non sire valere nostram regulam
quia 12. pssurgad quadratum Nostra regula antem est pro simplicibus
X_ 10Y a  a2.) ( H 110N 2 = 2 10. 12
ga 2o 39 = 11 + L. 2. 11.11 31 = Ea -3a)1+ 2.10, 32= -5 a 10
ovu ergo hcorema fabricandum
quoratur cursta avax + XVdv + ex dx XV dx = a dv = 0
40 0 41-7
du: dx = x av +XV ,:, aa - x v = E B 1+ X. d,, av = x. a
Aequatio
Ostendiuus prag. 1. hanc curvam esse unicam specie sit  sit do: dx = b: a siet
in eo loco, ubi hanc habet tangentium proprietale,
et tales curvae sunt infinilae, aliae scilicet atque aliae, quam aequatio communis
curva quae curvae nostrae occurrit
aab = bxY = aav + axV ubi b est parameter variabilis
itaque data b, curva est positione data, sed non locus ubi occurrit, quaesitae
patium cumer quaeramus tarum curoapum ocursus videamus au haec
curva nostram tangat, et fiat lineam concursuum hujus curvae, quaeet
differentd fiet patet geu sed video non posse eam invenini
quid si (crbaious av dx + xvdo + xvax Ck + v do = 0 undm
quid si 2 ex aqu. nadus + xydy 4xglx taydx = axax faciamus
prod
ponanus addus 4 Z ydy + Zy d + aydZ = a Z d.
 G = a = rertus + g d gg + d Zyt = 22 dy + Z ydz + y dZ + 2Zy dy
d X = 30v + 31 Vte + 32 vndre + 30 v n+3e etc.
dx =n. 30 V= + n +e. 3) Vnte + nare 32 vntie - eic
+ 2. 30. 32.
X X = 30.30. v + 2. 30. 31 v
n. 30. v + n +C. 31 v + n + 2C 32 Vntre
fiet
2n +2e
avdx -.
etc.
intre etc
XVdv = 30v + 3, v L 3. vmea eu
XV dx = L 2n. 30.30vm  + 2. 2n +C 32. V  + 2. Ent2C. 30. 3E Y
+
explicandae n et e, ut hoc in seriem contemperentur, sive adsit, sive absit a
a do =  sa v
Et aute omnia distinquendum sit ne ne untus, an vero major an minor concute quam o.
Efficiecta a est primi termini cujuszue seriei particularis sunt
ponauus esse
Quaeritur quomodo
 exponentium R, n+1 2n; 0. Ex his duo minimum sibi aequandi
Restat nat seu t. et qui terminus ne evanescat, aequandus est cum
ita res
si n ~ 0 tune res omnis redit ad 
n = 0 etiam v., et  hic evarescunt. sin n + 1= 0 sit n L -
quocedit, ut paulo ante, sed in casu quo adest a D, si absit non procedit, ut jam experti sumus, nobis autem
 n = 0, fiet et n = 2 N
quaerebatur aliquid generale. Si n+ 1 = 2a = fit n = 1 ponaus C sed cui aequabitur n, Cin 0
sed video nem non procedere quia
ebet n ooar seu cui 1 et 0. nempe V est Y, ut et tost ete Nemps habemus consiveremus cud aequemus n seu
equabuune un nt et seu 2tC. Et siet e= -1 ita simul cm 1 aequabitur 21 +C. sed 0 dabunt n +C, et 2n+2
ex ita porro. Res ergo sic adetur sucredere ergso reipsa tentemus pagira versa
et pro 18,6. et ila pcorro
ad hibeacuu autem abjectiones nogenaria, et
fundenaria, et ut succedant; ad habemus adii
ciaaqua
sit n aspt pro
1 + n1E + n +123 + n. 22 =
(0) 6( 6(3)
8(2) 4 C7
22+ 2ntr a. 13 - nti 624 + nF1 15 = 36
6(2) 6(1)
1x+ Z= gyta
11X13= 99
Ergo quaetali
algusid
alsit
et sat
11. 317 =49

## Folio 372v — 00068196:0740
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 371r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000740.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 84 lines, status machine
X = 30 V + 31 + 32 + 33 etc.
ex = 1, 30 -172 - 2. 30 - 3. 24 e2t
X = 30 v + 2. 30 3V + 2. 30. 32 + 2. 30. 33 + 2. 3034
+ 31. 31 +2. 31. 32 + 2. 31. 33
+ 32.56
-. 2 38. 34
=2 1. 31 
E P. E2
Xdx = 1
Xdx = 1 2. 30. v 4 1. 2. 30.31 F X 9 2 30. 2
1pr. 3.3
avdx = a
XVAV = 300
130V
3IP
XVAx = 1 230 4. 2. 1031
32
br
-1. 2.30. 31 -2. 23034 - 3. 2
2.31. 32
3
32.32 2. 32. 33
ubi video 30 = 1 et seu 30 + 30 = 0, ubi 30 vel est1 vel est 0. Et quidem pt esse o, si adsit
a
ado
et etiam 31 fiet 0 habeliturque 32, et 33 etc. sed si sit 30 = -1 fi quod necesse.
foret, si abesset t-a, fint -a + 31 -30 quod absurdum, nam fieret a = 0 Ergo t
si in aeq. aadu +Xy du + Xyax + aydx = a X dX  sit X = b Z C V et y- m Zu tiB
nihil to frustra haec agimus.
fint dx 3 b dZ + edo et du - maz + ndv, et fet Xya b bn Zb
i
as maq cxy
nav
aa A3
 xde
  = 4 du Xy Adydx Xy = 
ad sit v - t  b bet  c0v = 2 ay ci, atb = C - cb + Cb sit ergo v = 30 + 31X + 31 X etc
a
  fibt 1. v = 30 =
go 30303 30
quod si jor. pro v sumamus co
+ 3,2 X + 2.31 32.
303
303
+ 3pX
3
et sibaequalio - a du + adx + xdeo =0
 patet co- 40= 41X-42X, eodem
modo haberi ex Y, 40, 41 42, X, a
ut N = 30 =3(X = 32X habetur
ex 30, 31, 32, X, a
et quide 31 et 41 =0 erqu
0
est solum in 40 etc2 in 32. unde
si lollas a per 42, et  inquiras
ex X et gr (uti haberi pb ex X et a)
titues in valore ipsius co per X et q
30
cujus itide datur vator Ecyb suberesset ut inveniretur haec
Pro 32si u 42 substitues 32
curva: adco: coa na + adu 4Xdio =0 ubi jes licebit
explicare coper X, et habetur solutio facilis per seriem infinitar
ana
a ded=
sit lenim X = 50 +G eo + 52 Co +53Cd etc et tt ao x a,
Hik aee
31. 33
en Do
as ryta
X
= 317 X = 31. 32 X - 31, 33 X
12
+ 3D X +3141 31 X
2923
303
+ 313 7
.31. 32
L2. 31132
+4. 34 ero
ara, 
xAdo=
ressix-
M. 51Cd + _ 32Co + L. 33coe
+50 S 52