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# Catalogue record: 1126 — [Aufz. zu Differentialgleichungen]
# Incipit: Sunt quaedam ita inquirenda, ut multae tentari debeant viae. Et talia
# Date: 1703-11-29 (03-11-19)
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=1126&katnr=58677
# Work: https://leibnizlegible.com/work/00068072
# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 27
# Folios: Bl. 12–13 (from the shelfmark LH 35, 6, 27 Bl. 12-13): 4 pages, canvases 23–26, page ids 00068072:0023 to 00068072:0026
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068072&page=23
# Lines: 231 recognised on 3 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.
# 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 12r — 00068072:0023
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 13v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068072/jpgs/default/00000023.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 81 lines, status machine
19 N Nemtr. 1703
sunt quaedam ita inquirenda, ut multae tentari
debeant viae, Et talia sunt inprimis ea, in quibus compendia quaeruntus
quae an omnino dentur dubitam ante eventum potest. Ut quarentu
numeri divisores Talis videtur inquisitio qreducendamm  infinite-
simalium affectarum ad puras; cum saperranda sunt in determinata
modo tractabili. Sane varia frustra tentavi dudum, sed videmm, nunc
tandem aditus se apernisse Ane ex eo quod
deo constans deo assumi potest per dc ddv: daz Unde in his evitatur
pvarbitrio
ninia depressio; ettell potest res reduci potest vel ad daxidat
per du: dZ vel ad dav: ddz per d dZ: duo.
ut vaglorem ipsius
primum autem hac usus tum via, utl dy: dx obtinerem sine ad,
itoque pro datis y et X, assumsi y: a = sadz: dZ et X: a = sadz. dx
unde fit dy 3dx = dv: d2. sed quia ita bis differen
Quid ergo si sumamus
bis differentiaadum est ante quam perveniatur ad solas dv, dz, ddr
tiandum est calculus fit paulo prolixior.
daz; ideo fit calculus paulo prolixior. Nunc ergo videor
mihi  posse contentus esse
quia in nostra potestate est assumere constantem per
sit ergo X: a ( sddz: aZ et y: a = dv: d2
ddv: _t ddz.
dz ddv - dvddz,: dZ d2
sit jam n
aequatio Taugertialis pata resolvenda dy: dx = XX  + yy, : aa,
s
et dy: dx (5)
erit 1Z tdr = dvddZo: d (= (ad Z qu saaz: dZ + dpdv: d7 d2 = (8,
quam aequod
da
de
d ddv: dd
daz x et fiet de ddv: sd
et lZ) dd v: adZ =
ded  =) dZ qu Sadzue: dz +de + dvdv de.
de
fietque aequatio
quam differentiato
erit dam= (Sad Z int Z) add : dE + d Z qu fdd Z. dz seu da sad. t + (. 4u ) i
de
+ dZdv
et faciamus
e
et fiet ex 10 fiet deo
+(Z. Lato)  dtZ du e seu, qus ddqe d + 2(ddZ. de)
+qudZ Au sddz: de + dd seu
unde extraheado prodibil
compendio du  dudo.due to vocemus di
et diferentiando fit o (13) (dZe 2(5daZ: AZ) dd 3 l: d3
ddz + L, seu qu sddz: dZ +2 sad Z: dz  + dd ip: de z = 0
1
2t
pe
qu sddq d3e + 2 (d ZdZ) fiaz. de + ad 1. 4  0
- 2 Y ( - ddip. dd2) = (17) (per Y7) tedqe di = dvav. da
(ddze. dz = X e dde t et d H
 est autam dd dd (per 1)   d, du (1+ (dv: di). Ergo ad e = de Ad vede) Atet dv. B siu
sddz: d
unde extrahendo
A do: de = d7 ddo - dvadz: dbdo seu = dao +.dZ dv ddx - ddd dd Z : dzdz  et  ddV
2-  dd ip. dezue
et des d(dvidz) = (13) d3 ddv - dodaz ide dv, adeaque per 12 et B ex 59. (7 fiet
qu Sdezidz L
2 - ddv, F 2 dvdav. duebus (- dvdp . d3 d  ddv: ddzu - dvi du - (dv dv: dz Z
fiet 2q (daz. d2) dl f. d2 + da qze qus dav. d + cax 0. unde
-, + ((, - ddv: ddz)
du(per (10) 10) - (3) des - do: d3
seu Adv: ddque = 2 + dvidque 51, 2, 1- dvid2
, -ddvidaz seu (ddz: dx
et differentiando aequationem i
et quSddz: d2 = 2, C. 4 d Lv: ddz - dv.: de - dua v: qed 3 le sit   lv: de + lodv: dide
qusddz: d, +2 sudz: 47 +
2 = ddv: ddque + 2 H1, - dlv: ddZ) C
pro compendio, idve du f dudr: dd vecetur 2d p: deo et fiet X H(1 - dax: daz) Z ddv: ddque - 2 dip: deo, adeoque
1- ddv: d dZ = dd vddo; dazddz - 2 dis ddd: deodu + dv d V: deo dld. et ddv: ddz =
ita res
seu  = ddv: da Z + 2 f (1 - ddv: dd
1+ dbideo 4 + + a   devat B) +5-4 dp. du ( B. 4 dy a p. deoduo)
vdttur reusi ad quadutures rationales

## Folio 12v — 00068072:0024
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 13r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068072/jpgs/default/00000024.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 71 lines, status machine
Resumamus calculum
X: a = (1) SddZ: dZu et y: a (1) dv: dZ et dx: a = (3) d d Z: d7
et dy: a = (4). dq ddv - dvddz,: dz dZ. sit jam aequatio tangentialis
ad resolvendum data dy. Unde dy: dx = d Zddv - d vddz,: (dZ d2:
d ddz (: az =  ddv: ddz - dvZ dv: dZ = (5) dy: ax.
sit jam aequatio tangentialis ad resolvendum data dy: dx = (5) exryy: aa
fet Unde per 1,2, 5. fiet adv: dedzue - du: dz = (7) qu sal Z: dz + dudo: dzdZ, quam multiplicand
pe C per dZue fiet dZddv: ddz -dv = dz qu (dd Z: d + dudo: d2. sit jam
dZ ddv: ddz - dv dv:dz = (8) deo constansi, et ex praecedenti aquatione fiet
et ex proxima aequatione, fiet deo =  (8) dque qu sddz:dz + du, quam
differentiando, prodibit erit 0 = 2 dZ (sdaz. d2) aa (idZ + ddzue qu saaz: a +adv seu
Ju Sddzidz + 2 Sddz. dz + dav: daz = 0. Et extrahendo radicem ex hac aequa–
tione, fiet fddz: dz L = 1 + Y(1 = ddv: daZ)  C unde
qu. Sdaz: dz = (11) 2 - ddv: ddz + 2 + (. - ddv: dd Z) per 2 = (12) ddv: ddz - dv. 4Z = dvdv: did
 seu - 2 Ysi - dav: daz) = Y dv: dZ + dpav: di dZ + 2 - 2 ddv: ddZ, et
compendio fiat  dv: dz tdvdv: dzdz +2 = (13) 2d: de et ex proxima
aequatione fiet - L + - ddv: ddZue) =4 fd: du = Zddv: daz, et quadrando
1 =- ddv: ddque  dip d: deodeo - x dr 3dip ddx: dZ deo dd Z + ddvddv: de Z dd Z
itaque ut extrahamus sl deda: deda Lc
iel
davat A- 3g) ddv + a () (1 - 2 de) = 1 - 4. lex + 4 d d.s) + 4 - 4 dis d) : 4
da t ad Z
o daz
deo
deo duodeo)
dodco
unde cum  dip: deo detur
et prodibit ddv = d - = + L ((5 - 448)
per do: dz ex aeg. 13, hinc ettam t hinc res redacta est ad qadraturas;, sed
des
hanc tentemus reddere rationalem, et 5 4dpido oetur e seu (per 13
144)  - 2dv: dz - dedv. d7dZ (, aequetur quadrateo (17) 1+. dvdg. d  t dode: dZ
ab 1 = dvdg: dzdZ, it fiet (d) - 2du idque - Av dv a2dZ = (1 - 2 dvdes: dy dz + dvardisle: dz
et divisis omnibas per dv: dz fiet + 2 v: due L  d: de - dv dedg: d ZdZdZ, atque adeo
(v: dque (17) 2 d8 2Z,  dz dZ des d et quia t5- 4+4. duo) - 1 dvds : did
ibi substituto varore ipsius deidq modibit B 4duido- didz des (Ed9+ 2de dZ:de
27 (def. dZ, 1): (de df: dadZ) t1) 27 d8: dz =1, id de: didz +
u du 2 dessie E A d2 : ds19
seu du: dz d   9aZ- dzdZ,: d d9 + did, quo valore
substituto in (5- 4dib), seu in 5 _+ seu Cer) in + 2dvidque d
a2 42
deo
1- dv des: dzdZ, fiet di d dq 2td ed edZ 1 d2 - = d9  d3 + 2d didz
sive dZ aZ - dv des, : d7dZ = (dZ d Z de d9 e d7 d Z + Ed( d73;d2 + dz u des l
yl17)
dyd - dvda ee ed 1 =2do: d3 = dodv: 47dz aequant debet quadrato ab
I= avda: dZdz, seu ad dZ dz = 2d0 d2- dudo aequari debet quadrato  a
ipsi de et dedz- d 2dvdg  + evdr d8s: dydzue, et
fiat - 2 dZ = dv = 2d9 + dv
1 5 ++, , 4 daq  a d 1Z - 4 das dg d 7. 7 - 3dg a 3 + 4 dZ
disdes + dea
4s + dq d3 (2 ( d5 + d3 dZ)
M
seu dest d tdatus ereque +ded ( a di d d9des adid + 4 d  adodestdsses
 itaque ((5 - 4 d p. 10) debet aequari quadrato seu per aquis a debet
ac aequari quodrato AtA) - 2dv: dZ - do dv: a2 dz (4), aequari quadrato
itaque 1 = 2dv: dz dvdr: dzdz aequetur quadrato ab 1- odG: dydzue  tvardfdes. d
erit S5 = + d 1: aco) = 1 - dxdg: dzdz. et erit (1) 2dv: dq Edvdv: d 7dZ = (1) - 2 d vdg. d7 d2
+ duav dedg: d3 et divisis omnibus per du fiat -2 -, dv: dZ = - 2 de: d3e + dxdis de: d3
seu du: dz = (19) (dg. d _  2, de: dz - 1,: d8 d8: d2d7 +2 His rntor substituatu
i 2d23,; d9 des +az d7
 seu du 
seu do = (20) dy dZ de - d33: ddgdyd Hic vator substituatur in 1- 2dr: d2
ia aqu. 17. fint
-ErdridZdZ (it Ac 4d8dBL  Zdc. = l3d82+ dZ + d3dZ d ed5 dyaz de des
s () dz  4 did de de + la) dides Edzat ttidzi didz rajar ds des + Mxt d des djurte.
ndrat dic +4ax dg  142):,  (lesdes + 2dide)
At seu
avid Z2
e 1zde d7d
ds +d2dZ

## Folio 13r — 00068072:0025
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 12v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068072/jpgs/default/00000025.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 79 lines, status machine
seu dv: dZ  Ed Z d - 2 d idq, :,  de d + 24 Z dZ.. quo valore sobistituto in 1 - rdr. a2
19
- 2 dvav: dedz, fiet dG + 4. l des dz d z + 4d3 C542- 4 (d7 d9 - dz 2) (d  de + 2d7 d 2)
- 8 qu (dZ de - dzdZ
 qu (d9dg +27d5)
seu ex exponendo,
e
d92 + 4 de dis did al ag3=g2 dgo az a deaus + 4t 4 d   + 8 d2
sad
(=8 dZ dz d9 de) +6 dZ3dG= 8 dz
s
1 - 2dv idq -2
3: ququ dde + 2dZd2
dlae 247Z seu do
44ue d03+ 8 d3 de +4 d732
 eiAEiqu.
itaque 5.= 4 dp: deo debet aiqual quodrato est aut Ergo per 13, debet quadvato aequam
divitione facta p
(A (+ . dvidque  2 duav: +8. 1( 4, quadrato aequari (17) quadrato ab 1-dvd8: dzaz
-2 = 2 dv: dZ,  2 d : dZ + do dedg. d3 seu dv: d 2 deg: dez - 2, :, d Gde: djdz + 2
seu dv: dz (7)  231 d21:, da des + 2d7 dZ. quo valore substituto in
seu qulcato
1 - 2 dv: dz - 2 ardv: d3 dz fint dc) A ade d c dz d2) + 4 dy 7 - 4 d Z de3  8 d 23 dg  4d2 du desdes
(+ 8 d. 5) = 8dZ d7 d d + (16) 47 d9= 8. l8
:, () (ade + 242 d2) adeoque
erit 1 - 2 dudzue - 2drdv: d7 dZ= (2 X (d ep des - 2d7d2 3:, des de + 2 dZ d Z
A
seu
quod et patet substituendo vatorem ipsius dvidque ex aequ. c9 substituendo in 1- dvage aza)
nam fit 1Zd + 2d7dZ C 2dZ454 + 2des d3 d7 9: R. de d8 12 d7d3 (2d2a8
8 -d dc + 2d7a3 (deqas  d
a
2  21)49-2d3=
q+ 2d7aZ
dcdes +2d2d2
2dZdg dZ4Z ag -48d9 + 2d7 aZ - (3) d8 de + 2 d Z des
se A Z dd) F 2d3 (2 d dgds + 2d
qu Hoc et comprobatur si in insin  -d i: deo) valore invento 1 - dvde  ddque por av: d ponamus
dade +2d7 dZ Z
valorem, nam prodit 182d 2d2 ax de = dZd7 (+ d4 dD- (2d4 d8 + 2 dZd9 ut ante
1C 222
dz dzu +des des
quia ergo de: dZ = 2d+ dd et ita possumus obtinere quadraturam rationalem.
d2a3+ d5de d
 sed prius absolvamus  jam obpiam ex aequ. 16.
Iddo = ( dx +  axay + 1 +  (1 - 2dy - 2 aodo)
d3d2
fice eta  Ex Hoeri est
duar (+do +1 Ao =  - 2dx -2 dudo
d2 t2
22
Unde fil ddZ (22) =d (dv: d3
sed hinc jam video methodum istam procevere non posse, ita enim Cum enim
de
dZ
dzdz, d2
per initio posita vit ddz: d7 = dx: a, et d(dv:d Z) = dxta dy: a. Ergo
ex aequatione 22 haberetur du (ddz: dZ): d (dv:dz) seu du: dx. sed
jam ante haberetur dy: dx ob aequationem tangentialem datam. Ergo si haec methodis
procederet, nimium praestaretur ham duidx habertur bis, atque adeo relata haberetur
inter X: a et y: a sine quadraturis quod non licet.
An ergo licet facere X: a = (1) (ddz: do et y: a ( )4Z: dv et dyr a = ddz: dx
et dy: _ = (4) dvddzue - dz ddv,:, dudo, et dy: dx = d vddz - d Z ddv, : do dv) : ddz (: d v et
= d B daqu - dZ ddv,: d ddzue  du: dx (1 - dz ddv: du ddzue. sit jam aequatio
data dy: dx = XX + yygiaa seu dy: dx =D qu 5ddz: dv + d2dz: dudo et multiplicata, aequa
tione per de, fiat du dy: dx seu dv = a Z ddp: daz = dz dz: do = (9) quav qu sdaz: d0.= deo
d (dv) 2(5cdZ: d0) ddz (: d v) + ddvies aaz: dv =0 et ita forel
2 ddze + ddo (ddz: dv 20 sed vereor ut hoc liceat, nam ita fieret et X: a = - 2 dZ: dV sed ita
data aequatione unde
dco, sd  Et erit
per compendium; et fiet
per 6, daretur relutao dy: dx ex y: a et x: a ante data, et ruit
nimium haberetur. superest Ergo potius facipium +dZdav: daque + azdz: du
dp - deo ~) dv qu sadz: do et dferentiando posita, et differentiando ddv =(12) A dd: dx) age
4 dav qu sdaz. do sed ita video similiter  saz: ax hat seua seu xia haberi dav: daz vel
viceversa. A adeoque haberi quaesitum ante quadraturas, Ae proinde redire oportet
ad methodum ubi X: a  Sdaz:  et H: a (2) Sddz: dZ. unde dy: dx = B) dv: d2
cui melhodo nondum habeo quod objicere possum. Etsi suspectum sit, quod non
possum fabricare exempla apta

## Folio 13v — 00068072:0026
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 12r); this page is its left half.
# No recognised text on this page (recognized).