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# Piece: https://leibnizlegible.com/api/records/9143/text
# Catalogue record: 9143 — Problema mihi ab Hugenio propositum in literis 7/17 Septemb. 1693
# Incipit: 2aadx=2aydy+xxdx+yydx
# Date: 1693-09-30 (93-09-20)
# Akademie-Ausgabe: AA VII (assigned, not yet published)
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=9143&katnr=42054
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 197 (from the shelfmark LH 35, 13, 1 Bl. 197): 2 pages, canvases 389–390, page ids 00068196:0389 to 00068196:0390
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=389
# Lines: 256 recognised on 2 of 2 pages, mean confidence 0.80
# 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 197r — 00068196:0389
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000389.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 138 lines, status machine
in 205 f 1643
24 2da= raydy + 2 xx dx + yyd
problem
in literis
Sit y = ( x 10 + uX + 12XX + 1X + 14 X + 15X + 16 X 117X ri9 X cu
2 γy =
Efytus
Kyduy -
+
XX =
- aa
seu 1.
11010 +ll10 X +, 10 M2 X , H 1on X + 10IE A. 1015 X _ o. 1BX  10. 17 2 L. 0, 8X et
7, 71. 11 +  11a12 R. V1. 13 2R. 11.14 X. 11.15 + X. 11 1b R. 11 17
.12. 2 B.12. 13 9. 12. 14 f. 12.15 A.tE 1b
13.B G1514 4 13.15
)4. 14
+2 11. II 3. 11. 12. 4. II. 13. 5. 11. 14. 6. 1. 15 2. 11. 16 B. 11. 17. 9 11.18
FLNX
4,12. 2, 5. 12. 13. G. 12. 14 3, 12. 15 812. 16. 9. 12. 17
R13. 1 13.14 8.1315. 9. 13.16
 assumamus 10 ut datam, quamtaalibe quale in libet
si sit 14 = 1 ecit H= L
Melius sic
Gsi, gy = 20 + 21X +22X + 23 X + 24 X 125 X + 26X + 27X+ 26y
et
+ ydy = 1 21 +2. 22X + 3. 23X + 4. 24X + 5 25 X + B. 2bX +7 2y X 1828X + 9. 29X ex
4 ZXX
= X
sit 2 = - 20 - 1 . F n=
Ergo 2 a 10 B, 1. et 10 24 = -23: 4 25 = + 23:A5
et 26 = 23:94, 5, b 27 = + 23:, 4. 5.6.7, etc. 
Ergo fint Lgy = 20 + 21X +22X + 23X - 2 X + (23X- 24
4
sit 0 = (D ( = X. L X + L X - L X + 1- X - .
Ergo
1.2
23 23, 23
superest ut iaveptigemus 20, 21, 22, 23
possumus antem jactie 20∞0
fet: 21 = -20+ 1,: 1
22 - +201, :2
23 =13) = 20 (+ L. Ergo si 20 -0 fit
2.3
23s 
1.2.3 1.23.4 18 3.815 1.213. 4.5.6, 1.23.45.b.7
Lyy - 20 - 21X - 22X ( X) = 12. 3V - 23X + 3X (=X3)
2.13
er
G 10. 11 42. 10. 12F +3. 10. 13Y, A1C.14 PY. 1D. 15+ + B. 10.6 +y, 101, X + 818. 5X 9101g et
 1.
8. 19.14: 9,14. 15
4.5 4.5.b
5
ui etiam 23= 0 (do- 1X+  X etc. Ergo do= 1n B. dp 72 1. 23
et si faciamus 23 =0 res redit ad ordinarias
et fit 2gy = X - = X seu yy = 2ax - xx quae est aequatio ad Circulum
Nam ut expediamur: G ydy = adx - x dx visen dy: dx = a -X,: y
ag =  - 9y = (E aa) - 2ax
seu fit 2ax = XX = gy, ut ante ergo haec aequatio resolvit, problema
Hugenianum nempe  dy: dx = 2aa -XX = yu Idem tamen
i
pote  2 g 30 Cyy 4 +2- 1X - 20-1 X = 20, v. posito 12V ess num.
qe 
egus =ogux Aux(x 21. 6V - 23(6X + 13. 3XF BX seu Lry - 10 LL 1X C. ur 2X -20B trox. 2X
seuseus q sit
dy 2.20+2X. X 220
tr umme
+67
ecipitire
Htinc si saciamus y 20 = L firt y9 = 1, + 2X. XX-  Eenmlit sd B, 2. 20a = n fiet gy-
220
a+
et - x esse loga.
mum
Senpotibr  sorid 10+ 
Sin fatiamus 202 1 46 = 0 seu 20 = -5. fiet 2yy + 5 = 5
Hliud problema ejusdem:
e Gu
posite Lex esse Nonmasius loqe est x
uia s
na dy  dy: dx = ax - Xy -ay,, aa + Xy
l, aa dy + xydy =_ xy dx + ay dx = ax dx aa dq + du + dx + dx = a dx
sita y a50 + nX + C X + 13X+ 14 X + 15X + 16 X  + 17 X + 18 X
E xHds Am +21 1323 1424 +5e 16. 2 1x 17 12 1919.
xydys = , 1. 21 +2. 2 1B. 23 + 434 t5. 25 tb. 26 +7 27. +8.28.
y
sit yy = 20 +21 X + 2L X + 23 X 424X es
2ydy = 121 + 21X +3 23X +42 4X 152sX e
222 = f.11 +al2 13f. 13 = -7.1. 12- 2. 19. 1
-13. 23 = 12+ 13 +4. 19 = 8. 10. 13-11. 12
10 +1 +12 + 13 +14 + _ + 1b +17 -:4. 24 = 13+14 +5. 15 = -1014. 3 11. 13- +12. 1.
aadue = 14 5. 11 2. 12 313 4.14 5.15 646 717 8. 18. 9.19
Xydx =
= axdx
a
6
5, 25= 14+15 +6. 16 = -15) 18. 15 - (5) uil 47 1)1219
s et puo  ponendr rutorem fiet dea rursus ex aeq est rydus
seu ydy = 5dx 1aD  sydy = sax - s v dx, seu
Cr 2ytoa na ampex.  ds = sdx = hyyen  Gd
idx -Exdx de vii hud e rytus. id rxdi xdx
5u = 2yy-
d
yy = -10 + 10V. et x = (dv: 1 -  Eydy = 10dv. seu ydy = 5dV
yy21.+ 2x - Xx - v iam dxX = 5dB: 1. B. Ergo als  dv= dx- vd 
2
17 Seplemti. 169
1d 20geF H = H+ 1
Fia   = 11
 1 = 2 = 103: 210
48
2
si I1 = 0 fit 10 = 1
Ergo peuinde est ac
si fecissemus x+ dZ
 Zax + aZ + Lxraf aadx
Z+ x
si 1tn num,
et l log. fil.
fieret Z=yy
= posito 1-H
esse num. et
X esse logs
 log. T+X
num. et
1X LX +L X etc
=-N. posito
l+ P + (3 +
Ergo si con numfils 12 123 723.) eCC
4. 5. 6.7
6+ 1 -X
si 
CA=
BM=-- (
1. 2. 7. 23
Ergo 1-n num
et al log. fiet H= l- 12

## Folio 197v — 00068196:0390
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000390.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 118 lines, status machine
a10. =
aCa)
HR 1I +412 + 3, 13 = 4. 1B 12 = ~. L, IL. 1 .. . . ..
12+ 43 +4. 14  - 10 t3 - 8, I1. 12.. . ..
13 114 tas. 15 = A 101 +-8 H 13 4 L1212
= aa 11
= 1. 10. 11
= - 2. 19. 12 = 2 11. 11
= = 3. 19. 13 = 3. 11.12
= - 4. 10. 14 - 4 11. 13 = 4 12. 12
= -5. 10. 15 = 5. 11. 14 - 5. 12. 13
14 +15 +b. 16 = 0. 15 (() 11. 14 -(F 12. 13 
15 +46 +7. 18 = -(1c 16 - (12)1. 15 (12) 12. 14 - (12) 5. B3. 13 = - 6. 1016 - 6. 1. 15 - 6 12. 14 - 6. 13 13
16 17 +18. 18 = - (14) 1017- 14. H. 16 = 14. 12. 15 fr4) 13, 19 = - 7. 10.17 - 7. 11. 16 - 7. 12. 15 - 7. 13. 14
ed AF p 13 = H. 3 =12.3. Fortasse tamen praeptiterit calculum eadem opera instituere generaliorem, ut
+
X = 10 + 1Y +n y +By, dx = 1. 1 + 2. 7 + 3y et c. X = 10. 10 +2.10. 11y + 2.10. 12y + 2.12 11 + 2. 10. 14y
non considerando, adernt dx, X, dx, X, XdX, id est
1
 11. 1 +2. 11. 12 43. 11. 13
et 2Xdx = 1. 2. 10. 11 +3. 2. 10. 122+ 3. 210. 13y + 4. 2. 10.14 y et. hoer, mullipli.. F12. 12
2. 11. 13
1. 11. 11 S
2. 11. 12
Hoc per se vel mulliplicata per a, vel y, vel eorum potentias, Certosue numeros datos
catu
2. 11. 12
Nunc illud considerare placet, videri i maximi momenti, quod et calculum mire contrahii. Nempe
inter se cdduntur, de quo in scheda seculiari.
pulixum serierum infinitarum abscindit; ac praesenti exemplo inelligi potest eempe estosuimus
13 _ explicata deide data aequatione differentiali aadus +xydus ete
pi et pontudi coefficientes terminorum nihilo aequales. incipiemus a ciefficiente lipsius t
y( igL yY +92Y
ubi aequabionem destrnctitiam inde natam ingreditur 13 et quidem 13 determinaum est 13 deter-
bi
aeq
icata s
minatur per 10 et 10, 11, 12. Nempe hinc incipimus quia ab eo progreditur series lege umforum
ex aeq. 
cum in termino p praecedente sic scil. 2dodjiciatur ubi X, adjiccatur-a. onp
 p Fingamus ja 13, 14, esc hoc modo dari, seu seriem esse inventam exit gu
et 13X + 14X + 15X et sit =  fiet _ y = 10 - 1X - 12 X = V ex 4et 3. assumatur jam
alia linea, ubi prro y sumendo co, fiet co = 20+ 2X + 22X +33X + 24X et. orr Et aequatio
sit dissecnlialis sit a +Xd+ xddx + aa ded = 0 (omisso -axdx) Ergo. Itaque hanc
aequationem explicando sere aquatu inde producta inde ab xt inclusive per omna similis erit
aequationi nostrae, et eodem modo plane lifere B, 14 23, 24, 25, 26, etc invenientur ex 20, 21, 22,
quam  13, 14, 15, 1b, esc. paulo ante repertae sunt, ex 10, 11, 12. Sit ergo 
Ergu. 23 X + 24X +25 X etc sit = (v) et fiet Cd = 20 +21X + 2  pvad
(V) = C0 = 20 - 21 22X porro 20, 2t 27 11 et 12 possunt invenir,
seu feit
nam
49 s
112 10:a, et 1E∞ hinc (10 = 10) + a. 212(= a = + 10.10: a. Ergo 12 = 16+ (: 2aa
Cu
ad impleniam legem homogeneorum debet 11 concipi ut litera divisa per a, et 12, ullitera divisa per a
sed 10 est laterne absoluta. porgo. sed pro 21 et 22 omittenda a, vel aa, ubi lineolts sub-
sex
22
 20: 2aa
.Ergo sequitur
() haberi Cx X, a, et 20; uti v ex X, a, et 210 + a ponamus
ducta nolvimus et fit 21= -20: a
16
artitraria ostantet  tunc fieri potest
ut
uod
Jam aeq. 7. esse resolutam seu haberi C ex X et eta et  quoq ipsa 20
ad habendam . generalem resolutionem. Ita enim apparet quibus lam a
se
q quantilas illa
quae arbitraria quae angere pt. Quae ut eadem se quam exhlbetau. e contelligi sit
ex aequ. 77. ducendo unubi quaedam arbitraria intercurret, qu ad habendam resolutionem
gentralem, quae at comparetur cum ipsa 20, effici  si resolutionem illam
in eu reducen itide por sercu infiutam 6 explicemus. Sin si adhuc et
ut relatio habeatur inter h si et 20 per a, oportet solum aeg. 17 explicare, pro c substi
mendo valorem ex ag. 6. et ita ut g destrnctitias instituendo habebitur 20 per h vel
otra. Et fit  aeq. (1) ubi sit qua ergo cd jam habetur per X, 20, a,  quaesit aeqs
sequitur inde ope aeg. 10. etam (V) haberi per (explicata per 14 15) etiam ) haberi pe
1b
aC
Et hus ope
 etiam (V) haberi (19) per X, a et 20. seu (3, 2b. 2 = ati per X, a, nt
ex aam et in jam in f. Et _ per 16 similiter habetibur V ex X, a, 10 +a
Ergo pro (V) substituendo v, et pro  n substituendo 10 + a: a, habebitur
ex 10 per 1 propter anclogiam relation anaphoriam homanaphoriam seu analogn
hyper analogiam habebitur aequatio, (20) pier quam daturexa v ex  X, a, 10.
quem valorem substituendo in aeg.5. habetur relatibanter y et X, quaesitqe
 jam curva aequativns Z, novam reductionem ita pati videtur: faciamus X= 3e+ 31CO  32 C23
et dx =1g + 132C ++k. ABXdX= L 2. 42W 4 3. 43dC fiet
aut =q . c = a, A30C + nI. Co +nF COntL + n+3 Co
31
30
in
2n+
32
in schedia sequeatis die pro  posii V, et eadi
= L)
e cer 243
Xdu = L, 2n. 40c3= AH+ 1. AI C 
deprehenti si faciamus x = 30 +31 + 32 etc. C quadem
et adio a pon
sublon sed pb aponi sub cn alioques
et soret A 112 sed une sentur n seu
sequitur  co conete s
 duetu
non hibem axitus res exactu in spicienia
ut jam nalio schediasmale
posentem avda + X dv + X xdx Baa. =c posse quidem
exalicari sed hac admita ipsius X explicatione, non posse
eandem retineri explicatione ipsius Z. si resecemus aadv, seu si faciarum avdus+ v2d B, vo -0
seu aZ c dZ + Z d X+ Zc Y = C. pam non licet fatere, Z sed viatur la qu +4 + t ec. video
ia
plerydi
lanem rem seneraliter sucaneie si saciamus X = 29V 4 30 + 31 43
et 2- 3e + 41+ a etc Hoc
coxduo-
RWodx =Z.
aadeo =