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# Piece: https://leibnizlegible.com/api/records/60532/text
# Catalogue record: 60532 — [Aufz. zu Differentialgleichungen]
# Incipit: Putabam tali via differentiales primi gradus transformari in alias eju
# Date: 1693-03-24
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=60532&katnr=57128
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 367–368 (from the shelfmark LH 35, 13, 1 Bl. 367-368): 4 pages, canvases 729–732, page ids 00068196:0729 to 00068196:0732
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=729
# Lines: 304 recognised on 4 of 4 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.
# 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 367r — 00068196:0729
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 368v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000729.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 74 lines, status machine
4 Martii. 68 1093
putabam tali via differentiates primi gradus drnus formari in alpas
EX:a; y:a sit X: a = v. Z
ejusdem gradus constante carentes dy: ex
fiet xia 3 Zdo - Vdz, : ZZ. fiat ergo g: a = (2do -v + 3: coco
(5
us
tutcunque. itaque dy: dx eut  (per 3 et 4) erit  10c0: 22 Itaque tribus
vuloribus ex 2, 45 substitulis in aq. 1. habebitur aequatio cujus  cujus
ope quaerendo valorem ipsius (Zd0- vdz: coco, et eum utrinque aequationem
valais differentiando, habebitur aeqe (B) in qua tantum dv: du per v: Z
)
habebitur. Videbantur haec pulchre procedere sed ecce difficultatem
aut potius obstaculum, ex sola per aequationes2,4, 5, tollantur v:Z 2b
s
Zdv - vaq: coco, habebitur aequatio nova (9) ut qua dabitur dus: dx
per _ x: a et g: adifferens a data quod est contra hypothesin.
ut scilicet aequalio determinativa habeatur, aute adhibitionem
datae unde patet aequationes 2 et 4 non esse pare arbitrarias.
dem est si generalius sic procedamus: dy: x= X. a: y. a
l
H
(3)C
et X. a = D: Z, S2do - vdzue : 10no et _ y: a = v: Z5 (2do- vd5,: c0c0
H
Ergo per
v: ZV
a
ex
Ergo si co = vs2 fiet dy: dx ( ex 2 et 3)= u
(4)
oo
23, 5 tallendo v.2 et (... habetur dy: dx X: a ; y:a differens
non dum licet assumta aequatione data.Itaque si liceat datas
fabricare pro arbitrio poterimus Ron poretongare i finitas, it
at ita utaC sit ipsa 1. poterimus ag. 6 vel 1. redacere ad qua
draturas Nam pu aeq in aeq6, vel 1, Y pro duidx, x. a, y: a
Su
substorendo valores ex aeg. 2 3, 5 habebitur aeq. (7) quam ita
ferentiando praeformataque differentiando ut tollatur summtri
summatoria quantitas habetiur aeque.  (8) in qua solum datitur
Sed si aq. 1. sit ab alio datus difficile erit sic assumere
Valorem ipsius co  relationes in aq. 2 et 3 ut et coimcidat aeg. 6. cum
Cdo: dZ per V. Z.
n6
Redeamus ergo ad hocut ipsam  aequ. 1. nobis
ipsi detnus seu formemus ex D6. sit X. a = v: Z
ipsu aeque 1.
et y: a (ydv- vdz.: vZ ut adeo fitt coco vq
 et g. asv
 ideo ex 2 et 3
fiet dy: dx = Z. v (nam dx: a = Z dv vd Z3: 72 et dy: a = Zlv - raz: v2
s
Ergo dy : dx =  1. 8Z : l: 1Z = 42. vZ = Z. v) Hinc ex aeqe 2 et 5 fiet
c
ne
du: dx = a: X seu dy = dxa: X quae aliunde jam habetur, sed
S flud ergo sumamus exemplum: sit X: a = v: Z, + 10si d vdque : _ vZ + 11
yia  AES2dv - vdque: vE fit dx: da Zdv - vdzue, in, 1: 2Z+. l: vZ
et dy: a = Zdv - vdz in J: vZ. Ergo fit dy, dx = 1: DZ: 1. 32 + 1: vZ = 1: v. Z +1
viZ +1 a jam ex g Ergo ex aeg. 2,3,0
(3)V
fiet X: a = dx: dy -1+ 10y: a + 11 seu dx: du  xa sit 11 - 1 = 12
9
(6
fiet dx: du = X: a - 10 y: a + 12 seu adx + mlus 10 ydy = X dy + nady
Hiae aequatio igitur est resolubilis per quadraturas, seu cont parameter tolli potest.
(0
Nam in aeq. Y. tollantur X: a, y: a, ay.dx per 2, 3, 8, fiet
sed haec aequatio est
cum frnctus sec proinde
V. Z + 10fZetc +11 = v:Z + 10)Zek + 11 
identica, itaque non possiunus tollere n. a, qua, dy. dr C
hac methodo quicquam assequilicet

## Folio 367v — 00068196:0730
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 368r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000730.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 83 lines, status machine
ili sit dy: Ax = h, et X: a = m, et y: a = n
seu hamin
habotur aequatio ex h, m, n. Et simnl datur farcenta
rationatis Quod si _ habeat valorem rationalem, datur
l
v formula ex m et ni
sit m Pq et n = p 9
et subsituendo et tollendo m et n exaeq. 1 per aegg. 2 et 3.
AH
ia i= piql  posito Redari ex p; 4  p; 4
omnes evanescente q Hoc si paeplani posset haberemus
quaesitum per praecedentem staginam, ponamus
et rursus alium valorem ipsit 4 quaerendo ex2
 X
et 3. quia h amidu  seu = p p: 9 per
H C7) X
1iq= Pig
quae aquatio eo reducatur ut
sint tria Z. v p, SZdo- vazue : co0 24, et Zd0 - vaz:   Y
1) 
formmla generalis ex tribus M= p; qr rationaliter integre
n = pi qir ravaniter integre (id enm in Trubus semper licet salia
H
gneralitate  ut puto) qui valores slectutum s dm: dn
Ergo in an: Meldur evaccescet differentialitas restabunt
tamen duae summae seu tres qu numeri, p, q,r. et fiet
an: am = pi 4;r =h, nam hest duidur ut patet a
CA 4X
ex intio positis.
Purtus in aegu. 1 pro mn substituatur
I
valor ex aeq4. 5 et et 6. fiet h = p. 4
 (91 Au
quae duo
aequardo ex 7. et 8 fiet piq. r = piqir ubi evanescere
debel aliqua ex literis ut qet fiet p:1
Unde
Ex sequitur dari posse m asse (per 5 et 10) in exsol qir, et
n ex solis qir, adeoque et hexsolis qii; et quidem bis,
ex 7 et ex 8, sed ea coincident inter se
si expliantam
conferamus acont aeg. Cau ag. Z pir 72 23 ( a 462 90 n  61 75-
72. 73aa = 89 - 2. 6,4 2.73. Baa = 88 26,  et equando 2. 61 fit 88 - 2. 75. 13a- d9 32 3a
Confenm ob XyAX et Xdy patet fore 87 = 3. 89. rursus 561 = 86 - 72 - 2.  = 3. 99 - 3. 72. 13.
Ergo per aequ. 3 et 6 habentur 72,73, ideoque H 61. per Xaxet Xdy d habetur 6 et 71
comparando x du et XydX fit 83 = 2.85 = 3. 61 71 aa +4. 61. 73a et quia b1, 71, 73,am,
habentur, itaque hinc habet et 83 determinatur per 85 et alias superiores. 1
70 habetur per Xdx seu per 82. jam 81 = 82 - 2.70. 73.1 70. 75 + 2. 60. 15461. y p (C
2.61.734 61.72a
denique 80 = 70. 71 a + 268.61 + 60.71a - 2: 61. 70a
si hanc aequa tconem C comperrium cum aeq. n vet 3 iusleda disferentialir quabis xer ad BX Port e 
2o1 3
determinaves tantum quieam in a. C efficienda. Nam in calculo peculiari pro, hac aeq. D reperimus gb3. 89
ob priorem fier: 372. 73(4 364) 
(72+ 2.73(+3.64W
ob posteriorem fit: (2.72. 732.60+ 2.78643) n
2. 73 = 72. 73. seu 728273
4. 73+
 quo bulore substituto in aeq. 3 sit
quod succedit feliciter. Restat jam uniga aequatio
t raari - 2a415
+2.617 -26,4
ex explicatione. sed ita in
idem absurdum recidimus ut scilieri
tam co quam  dentur ex X+G, si sat
72=3. 93. itaque impegimus in scopula
quem vitabamus
13
87
omnes literae
praelen x, y, 
sunt rationes an
quanttates
2. 89 gC
est habetur ratio t72: 3
Ergo fit
B. 
34+85-83. 2.85-83
er1. B43. 71. 13a - 62. 71 24. 12a
+650.61 +606) ∞26D 6( l
(+2aa. 61. 13 +a. 61. 2.73 12. b1. 2. 13
=244. 61. 73 a1a. 6173) 4  2 651. 13
P. bie3

## Folio 368r — 00068196:0731
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 367v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000731.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 110 lines, status machine
sit
C
1Z. du =  10Z + u V,: 20Z + 21V
et v 40+41X + 42X) + 43Xy
50 +51 Xt s2 x +53Xy
5
fiat Z = 30F3IX + 32X +35+y 3)
1
 2. fuit du 848 0d
  et du =8d8- 9de
Ergo dz. d v 8d0- 0d8; , 9d3- D+8 et 10Z + 1V, : 20Z+ 21V   100 + 18, 3 2061
(10)
Ergo ex 1,8, 9 fit 8d0- Cd9:, 2dD- 2d8 = 100 + 18:, 200 + 2D
si
19: dD = 10O + 11D, :, 200 + 21D seu 200d0 +2D d0= 100dD + 1D d D.
fret 4 = 3 31de + 2. 92. X dx 4+ 33 Xdy
ydx
ydy
Odo 2 3031 dx + 2. 90. 32 Xat  30. 13 Xay
+ 31. 31 431.31 + 231 32X dx + 2. 31.3 E Xydx  3 R
42 35
X 31.32
+3133 +31.33 X dy
1
Y. 2. 32.32 XdXx + 2. 32. 32 Xydy +y Xdy
pr
x 33.33
+31. 32
+3233X dy +32.33X y dx
4. 2. 32.33
odem modo si  adjiciamus reliquia membra aequationis t explicata
 habebimus in summa terminos decem arbitrarias vero 30, 3, 32 33
4,41, 42, 43; et 10, 1,  20, 21. adeoque sufficientes, ut uilibet
ut productum curilibet datae simili, ejusdem gradus comparetur
Quod si voluissemus curare homoeoptosin haberemus in O arbitrarias
et 8 ut et in D, daret 16.  et cum 10, 11, 20, 21, fierent 20
Et ita eaequatio data
et fermini, fierent similiter 20
cum rit posset comparari.
videtu res procedere, et si sollum sit.
exferentralis
proposite est curva Cu aequatio a  coriaaco halis vaco
aals d+ re a + aacd d= raa de quaqe aegs
sochu
Si ego Y = 70 +71X + 72 Xy +7357
E1n
b belyik, 8
11 43.604
771. 72Xdy + 7C. 72Xydx 45 7 1.73  x r71 7 Xyayinnxi lus haptigu
71. 73
+71. 72
71. 73
1273. 7 111
gog L A 3X dy +2.p 21X
fiet 2 d  - 1 70. 71Ax F 30. 72 Xdy 12. 70. 73 Xdy 4 71. 7 vlx + 3. P1. 1. Xdx ry. qua tust tipdus A  ax + 2772 F1
71. 71
71. 71
20 = 23 60 s1. + 960 5 +3. 606
e
4
aacod =aa, 60. 71 460. 72
+61. 71
aa (dco =dua, 61.70 61. 71
n
n
+210. 73
+b1. 71
61. 71
podx +oixdus +82xd
13. B. 72
+660.61 +5. 60. 67 +3. 60. 6 + 3. 6s +36 +6i ubi
2. 71. 73
+61. 72
2. 61. 73
s1. 72
42. 61.73) +61. 72
∞61. 73
nepe  est mone
+83 xyax 184 X dy
61. 73
195Xay 186X pay . 9 Xy 1 B 2 B
C9)
Equalionis 90 + 91X + 92 Xy + 93X 9AXy +95X 196X9 +97X y + 982 =S
98- 89. 82 -31 quae proinde etiam succedit, ut patet (einus disralis est. 91q. + G2Xdy +293K 794X) L4gXx ib Xyly 13.91X X97Xdy + 4. 98X x = 0
du
sit c0 = 6C e + 60 + 61_ et 1= 70 + 7IX + 72Xy+ 73X +4 X
et WPdP = 70. 71dx +70. 72 Xdy f 7073 Xdx
+71. 71.
+71. 71
yox
dy
102
quim lle to ustiatius
ur
pay
 quaeratur conferantur aequ. 7 et gR ut quaerantur 90, 91, etc. fiet 9 = 80 92 = 81 93-82. ypara
94 = 83:2 = 85 95 = 84 96 = 86: 2 97 = 87: 3  = 89 98= 88: 4 3  lt pocet
tantum ergo in aeg. 7. opu est esse 83 = 2.85 et 97 = 2. 49. 9b8) = 3. 89 fil 3. )15 = 3. 12. 73a
 eti fu i ( 8d = 1.95 (t porere oportet 60 a osi pendicus snum socla est
aero
et ergo per se suac e  151.7 3.61 Y. 2
2. 72 seu 72= Z.
ad a M
y1t73 = 
bulla hobuctur ie cer ntus ab ioi lata tenncat 6oti
sed video sun nihilomun
prodire,
in propte
itu
ex hoc

## Folio 368v — 00068196:0732
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 367r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000732.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 37 lines, status machine
Ex aequatione aa H do + aaco di = 2a aldeo rundo a coc
Jedaz 2deb 7 2co ded raco L= a)deo et faciendo  = 2+ su V et  co = Z. F 1V colluntu
q Li et VPdv. porro faciendo  11xdx +1Lydy + 2D Xdy + 22y dx = 0 x = 6.6 92 g. 0
30atum
Lis 50 maetiritur +, et 81 v el 3 2mbino
3
 30 3 50 + 3 31Gd +
30 3 51 -
30 30 sL
 3 50 +4 32 n 50 +B 2 Y 5) + 13 35
30 31 51
L 40 4150 + 40 q so + 41 41 50 +
 4b 4 st = 40 c0 et
 30 4 5+
30 40 s
3 4 50 +
Gd  51 
31 41 5e 4
30 Ep st= 30 41 si
30 32 5b 2. 30 31 52 = 3
42 42 sd 4
4B HSL. -2
32 4250 +
3b 42 st
32 4d st
3d aB s 30 40 At  31 40 si
2 41 sit 3 a so+ 31 41 5o t 32 42 50 +
3 5
  Sib + 4 4 3
40 45 4 45
3 Qu Sb +
3 4 5 F 3
4. 5
4 5
G1 fe 51 = 3 4 5
A
5 + 34