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# Catalogue record: 987 — Quaesivi an liceret transformatione efficere si non tractabilem, saltem divisibilem per tractabilem. sed video si illud non licet nec hoc licere
# Incipit: Sit aequatio differentialis data, transformatur faciendo
# Date: 1691-10
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=987&katnr=58569
# Work: https://leibnizlegible.com/work/00068051
# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 6
# Folios: Bl. 25 (from the shelfmark LH 35, 6, 6 Bl. 25): 2 pages, canvases 49–50, page ids 00068051:0049 to 00068051:0050
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068051&page=49
# Lines: 109 recognised on 2 of 2 pages, mean confidence 0.82
# 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 25r — 00068051:0049
# Source image: https://digitale-sammlungen.gwlb.de/content/00068051/jpgs/default/00000049.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 75 lines, status machine
quaesi vi an liceret transformade nfficeres: non trachabilem
q
saltem divisibilem per tractabilem. Sed s video si ilad non licet
mec hoc licere
sit, aequalio differentralis vata, trans formetur faciendo
X = 1Z+C 100 + 110X+ 101 V + 111ZV + 1207+ 102V et c
e
ve y=
200 + 210 Z + 201 V + 211 Zy +2207+202 V etc
200 + 310 Z + 30+ Y + 311 Z v + 32032+ 302v etc
200 + 410 Z + 401 v F211 ZV +220227402 V etc
habe
Hinc invesigetur X, X, etc item y y, etc. item Xy Xy etc.
item dx,  ly, et Xdx e ydx, x ax etc. ut substitui
possint in aequatione proposito, neque enim dubito quia regulae generates
sint proditurae pro calculi comnpendio
Explicetur intur proposita aequatio differentialis in qua
diferntialis est scd ex nullius potestatis simplicis
per hos valores. Et habebitur trans formatio max una
possibilis
L
er deamus hunc jam an haec aequalio sit divisibilis, seu
reddenda divisibilis per aliam in qua absit una exciteris
verbi gratia V. manentibus dZ, de, et Z quo facto habobetur
solutio.
oportebit terminos omnes in quibus est
v et Y, et V, e et V, etc. habere divisorem communem
qui erit quaesita aequatio
e
possumus etiam ponere aequationem datam esse
V et q inter se servant leges homogeneorum, quo facto oportebit
tam . v +. q, quam V +. 7 et . v +.. vZ+. Z, et
et . x +. vZ +. vZ +. Z3, et ita porro habere divisorem communem
prior modus est simplicior; et video si oste duarum
aequationum tollatur do: de nempe ope moducto
(vec alteris pareutis ubi v et Z Servas homogen
et ope assuntae carentis litera vm debere omnia destrui
seu fieri aequationem identicam, destruendi autem erunt
termini formati ex V et Z
Et hac methodus etiam
fuccedet si in assumta sit non tantum simplex do:dz, sed
et utcunque ejus quadratum, cubus etc. simulus viqu, ejusque quadrito
cubus, et C.
Atque hoc videtur esse valde ginerate
sestimemus ciier sufficientiam arbitrariarum.
v.g. aequatio
X
dy = axd-aydus Explicaa siat
Namer aldremer
per 6g. 1 et 2. In (59. 1 assurgat valo v et Z ad gradum b, in
 iiumeratore aequ. 2. ad gr. E in nominatore ad grad. e.
et
Xtus ussumet ad gr. 2b et dy ad gr. c-1 et  Xdque ad gr b + b - 1
Ergl i dy ad 2b+C - 1 et aXdx ad gr b + b - 1 + e et aydy ud e+C1t
aquod si b non sit minor quam e aut C erit altissimus gradus 2b + c -1
si jam sit aequ in qua sit et do: dx= b:Z
e+ v77. p: y et
+ . v. Z F.. v: Z etc
 V :7
habeu et assurgat nunerator ad f, nominator ad q
assurgatur
= ad 2b +C - 1 + f, item ad 2b+ c- 1 + g abrum horum sit altissimum
Eporro abrtrariae sunt ex aquent numeratore, sunt
b + C + C + f + 9 quo a quo numero si auferas 26 + c - 1 + f restabil
ut q +C + 9 - b cobsit) seranm santest, sed si loco X ay fuisset x dy
tn mb + c-1 + f anterenda a b+ et e+f +g, et restabit quantitas
etg eec ∞- m 1b.
Est auteur bmajor quam e
semper res et
reducta intellum
ppotest uldua
tiones
ui
natoem coem
2

## Folio 25v — 00068051:0050
# Source image: https://digitale-sammlungen.gwlb.de/content/00068051/jpgs/default/00000050.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 34 lines, status machine
utZue + 12dv
sit X dy  axdx - aydy. sit
C
X- B 100 + 110E+ 61+
200 + 2p0Z + c01 P
sit x = 10 + 11X + 12v, : 20 + 21ZF22V et y = 30 + 313+ 32V,: 20 + 21 Z+ 2lV
siet dX = 20 H 1Z + gurdZ    2. 1 q + 21. 1( 2  d Z t
ren
-adq - 2 dv (iY dx - + 20. nd Z + 2t ( 21. 11 Z dque + 22. u1 vd2
+20. 12dv + 22. 12v dv + 21. 12 Z do
= 10. 21 d - 11. 21 7dZ = 12. 21 vdZ
=10. 22 db = 12. 22 vdv - 11. 22248,
(2) 20+ 212+ 22Y
eto
su queit est
nommoxur elggoas ahamen ggchtob vonz achron
Ewahn nogor orede lunzg nevet dh. et rungussuno
rsgnons elos nogar slib euvioif nede nm ponot 97
X dys axdx- aydy.  dyldx seu dy: ax = aX: xx+02 faciamus
pas
si effet fiat ria V e+ aZ m
factamus Data sit aequ. differentialis intractabilis, ut xdus = axax - aydy.
(4) L
sit M = do - v: Z et a n = do: dz  et fiat x = m. n et y emon
quibus substitatis in equ. posito valores aegug. 4 et 5 esse rationales. exsolicar:
Hmo jam explicaione fccrta aequationis 1 per 4, et 5 et hinc deductas, interet
Ponendo d dazue =0, tantum opus erit destruere terminos, in quibus est t ddv
sit X = v: 
et y = v. Z +1
va
pur
n
vvul)
asibet horves ysinsso.