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# Piece: https://leibnizlegible.com/api/records/543/text
# Catalogue record: 543 — Lex Homogeneorum in Transcendentibus
# Incipit: ∫x | x1 | dx | ddx | dddx | ddddx |
# Date: 1691
# Akademie-Ausgabe: AA VII (assigned, not yet published)
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=543&katnr=58120
# Work: https://leibnizlegible.com/work/00068104
# Title: Leibniz-Handschriften zur Mathematik LH 35, 8, 9
# Folios: Bl. 3 (from the shelfmark LH 35, 8, 9 Bl. 3): 2 pages, canvases 5–6, page ids 00068104:0005 to 00068104:0006
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068104&page=5
# Lines: 180 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.
# 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 3r — 00068104:0005
# Source image: https://digitale-sammlungen.gwlb.de/content/00068104/jpgs/default/00000005.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 113 lines, status machine
Hic nota ada pro diversis
summovibus ejusdem
Lex. Homogencorum
replicatioprima ida
3tia
lx) ddx dddx ) ddddX)
2
dx d(i)x d0)X
= 2
((3) X
XddX et dx X dddx et dx Xddddx et x et. charrcter est o.
Haec igtur  ducta in formulam angent gradum non characterem
idem est si in ex, de etc substituas pro dx, du, et. eoum atorem
X ddddx et X dddx et ddx Xddddx et dddx esc homoy. charautex est ∞)
sed quia non est idem numerus literarum, adque = xay +y dx
ideo tantum semi hogegenea sunt, seu non accurrunt adaque- Xadus + Edxdy Fydax
in calculo Idem est in caeters hujusmodi
X dd ddX et dd
adttZ. Xalbys Ayexddyst Sddx lus Liulx .).
Non est opus examinare
feupas cub uti abest x nam eadem prodit
foruntat in quibus plane abest X nam ut in
iis appareat rex homogenectum sufficil maximam
scim Nam x sequentem pro albissimam sumi pro X
proxinam pro ax etc
 X 
ix et dx Xdex et dx
LH et d: Ad  et doX
X desx et doX
Ka up (XdB) X et dlB)X
char.0 char. 2
X d (IX et dC)X
X d3)X et d CZ
X AOX et dcos
X d 5) X et diE) X
X ABIX et d6
X d5)X et
X d5) X et 2)X X d(8) X et d
X
char. 4
X d(gX et d()X
4. 1.
X d(1) X
coi danscendentu
Nobis in calculo nostro non est opus
ire ultra d(2) X seu dddx. Nam
_ tollendo y imus ad ddx, tollendo du ad dddx,
ubi ponendo ddy =0 sublatae sunt affectiones ipsius y.
opus est calcula summationum
dx = 16)X dX = 2 X d (o)X dX = 3X d (o)X eu
dsex = d (uX daoX = 2 d oX d()X d d( )X = 3 dbX dl. IX etc
et C
ArgaZ =. Xy
=C
etL
di dex = X ds)X + d C)X dX d CoX = X d ()X + 2 x doX c(X d (esx = X db) X + 3 X LC)X et
dX duX = X dE)X + dlgX d(X dx d6)X = X d(3x + 2 X d(oXdeX d X deyx = X diX + 3 illox dyx et
l
ec
d X dxx dG)X = X e X d(2) X + X dC)X + d(X d C)X
L
Generaliter analysi effici poterit, oblato Cummando, ut
reperiantur 908 ipsi adjungenda, ut bimul omnia summari
possint d de (s, =  d ()(5) d d(f +1)
 d dex d(SXD = h a (esX  d (etIX, d (f) x D + 9 a(eX, d(f+ 1) X, d (3 
Et eodem modo facti ex tribus gentraliter; quaeri pt differentia. Hinc colligitur
otiam feri posse ut idem summetur pluribus modiso.q. d(1) X  explic per ale
fer (VX doX explicestur per d(es
()Xd a (+ CX. st sit dxh L 5) Xd = d (1) X3. fiet h Ia et D = 3 et f = 1
posteniore casa fit e- it sit d ()k d sosx =  A()XD. Ergo h = 2 et 421 1 922 (verte)
X d4)X et a b)X
X il7)X et d (3)X
char. C
a X ddx = X dddx + 2X dx ddx
et 2Xdk ddx homog cum ex
Ergo X addx homog. cum du
den q =X dddus +3 dxdaytgaes
Et ita pooro
Tabula Homogene.
tatis Tanscende–
fium
X dBiX et d (O)X etc Homognitur o.
etc
etc
ubi varius possunt
permini homogenet
pro quaris potentia
alicuius differentialis
nostiinendo valorem
explicando eam potent.
X toto vel parte
ale sX = X al X dlX = d  y
d
= X d(1)X = dX
=X dosty E ale)x d CeX
Item Eadem fabula linasmutam pt
omn
reuttates ingredientes literas
commuu o charactare multando
t videntur etiam nunquam
nemyle
binari termini per altum
et
imnes multes
amtate, seu deprimas tunc par
NCcpir X
d(o)x d)x eci
stabunt d(o) d()X, a(2)X et
id ea est in lolo hoc argumento
X = d(- 1) X
a(t X a(tIIX
 debet esfe d 1et 1)X vel ox vel duyX. Priore casu sit e = -1
ica

## Folio 3v — 00068104:0006
# Source image: https://digitale-sammlungen.gwlb.de/content/00068104/jpgs/default/00000006.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 67 lines, status machine
Consideratio haec summandi pe plureseque modis utilis esse
potest, nam ita possumus  aq aequirere arbitrarias
nam f t+m et na p. et fer este  pro
Ph porerit poni  +1L + Am _ K A- K p, atque
Ita obtinuimus arbitrariam K. Et hac arte possum,
etiam excitare terminos qui absunt verb. g
fh non adsit scribemus KL + K m = Kn = K p
Sic 4) doix d(IX = X a (1) X = 3 (X d Ex dCyX
=  dCox deX  (doX dux de)X
2
similiter possumus datam quemvis formulam conferrie cum
Himinum conferre cum termino aliquo ex differentiatione
terionis sut proxime binionis ve ita et ternionis, ael
qualeminis orto, sed tunc, duae tres etc summae assunua
Hac methodo etiam
sut pro finione una
possumus excitare et impune ascribere terminos
qui non adsunt. ut si non ad sit (dee) X di seu sax dax
poterimus tamen scribere MX dax - 3n X de a x
m X ddx - 3m sadx adax - im dx sdx + M sdx dax dadx = 0
et ita plurimos poterimus obtinere terminos cum arbitrariis
ut tandem ex omnibus formula tractabilis componatur; qualicet
quoeda extalibus sunt sua natura seu per. se tractabiles
sed quid jam d
ut (dax dadx. = d aX
l
his, quae byposilis quadraturis linctabilia,S.
Ax ddx 3  + 3X adx dddx addx
x
X ddx
eo log. X ddx = d alius a adx. tx
Xy dx = d X y = P. X
2 e =3. h = 1
v’x
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