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# Piece: https://leibnizlegible.com/api/records/25449/text
# Catalogue record: 25449 — [Tabulae divisionum]
# Incipit: 20x_3 + 21x_2 + 22x + 23
# Date: 1694-01-15
# Akademie-Ausgabe: AA VII (assigned, not yet published)
# Other printings: Knobloch: Beginn der Determinantentheorie, 1980, S. 316-323
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=25449&katnr=56699
# Work: https://leibnizlegible.com/work/00068013
# Title: Leibniz-Handschriften zur Mathematik LH 35, 4, 16
# Folios: Bl. 6–7 (from the shelfmark LH 35, 4, 16 Bl. 6-7): 4 pages, canvases 11–14, page ids 00068013:0011 to 00068013:0014
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068013&page=11
# Lines: 321 recognised on 4 of 4 pages, mean confidence 0.79
# 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.
# Licence: CC BY 4.0, Leibniz Legible. Machine output: do not ingest it as verified text; keep this provenance with the text.
# Wording rule: say "the machine reads it as …", never "Leibniz wrote".
# 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 6r — 00068013:0011
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 7v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068013/jpgs/default/00000011.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 57 lines, status machine
tsi des
 per _10X3+ nX +1IX + 13 dividi debet AX + BP
p-2
post primam divisionem feri debet A X  = 1B XD. 2 C  3 +I DX A et post secundam divi-
n-2
sionem 2A.
+3C x , poste tertiam 3A 3
B. - 4
fit . H:_ 1A -  B- AV: 1 = 1DB AN,: 10 operatio stahit: (4) X +(G X + (C) X
+(6)n 3 ete
sit 1A- B. A. 10  B1O - AII, . 1  CB - C Rie-(AA131D - C- A1. 10-BM. 1O +A11. 10 seu Ar. 1OA - 12. 10B
.IL. 105
(D RSLIAO- (ALA (B) = AB = A11. 10+ B (9 = LB) 11.10 - A12. 10 + C = A11. 10 - A12.10- 811. 10 +C
)= D_ -(O)11. 10 -(B)12: 10 - (A) 13:10. Et ita regula gereralis ( (A) = A 1(B) +() n  B10 Nempe
Et ita porro
t
10 (A) = Let 10(B) + nA) = 10B 10(C) + 1(B) + 12(2) = 10C 10(D) + 1(C) + 12 (B) + 13(A) = 10 D. Hoc est principium egalaritatis
10A
et hinc jam uniuscujusque literae valor est investigandus
OPENRHOCONTRACA, 30 + 3
et
devlockdi
+I1X
122 1
X
X
32
X
20+ 210 +27
+33... 43400
a
X B
e
424.0240 0 10 +20. X
aesse
= 10. 12,710 13+ 10. 21 1X
mume
e
 10. 10. 11.22 = 10, 10. 12. 220)t 10. 10. 22)0
4 10. 11. 12. 20 + 10. 12. 12.20 0 - 10. 12. 201
+ 10. HI. 11. 21 + 10. 11. 12.21  = 10. 11. 21 (10
=Il. 11. I. 20) = 11. 11. 12. 201 0+ 11. 11. 20
10
D
Sit Divisor rtI IOX FUX + M2. Dividendus 20X + 21X + 22X + 23X + 24. Operatio con-
tracta 30X +31X + 32 X + 33 X + 34.
(4 11. 11. 29 + 11. 12. 20, _.- 11. 205107
I0 ( 10. 11. 21
a
E
acmerius
c
C C
eda
a
= +1OX _ (- 11. 29 -12. 20
es

## Folio 6v — 00068013:0012
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 7r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068013/jpgs/default/00000012.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 61 lines, status machine
Dividivisor W0X+
Residuris ar, n
Quotiens, X, 10
Residuus 
9uOL(X, 1020
1
X, 10. 21 - 10. 11,203
Resid. X. 10.22 - 10. 11. 21 + 112207:
(X, 10320
X, 103,21 - 10, 11. 20
Cx, 10. 22 - 10, 11. 21 + 10. 11. 20
Resid. X, 18. 23 - 10. 11. 22 +10. 15.21 - 11. 10,
(X 16. 20
X, 10,21 - 10. 11. 20
X, 101. 22 -10. 11. 21 +1011.20
N, 10. 23 - 10, 11. 22+10, 11. 21-
=12 2, 11OK
24 Resid. X, 10. 24 - 10. 11. 29+ 18. 11. 22-10. 11.21: , 110.11
=X, 10,20l
X, 105, 21- 10. 11. 20
22X 4310F) X, 10. 22 = 10. 11. 21 +18. 1120
X, 10. 23 -10. 11. 22 +10., 1121 - 1. 20
Resid
Xo 10. 24 - 101. 11. 23+10. 11. 2L 161. 211
10. 25 - 10. 11. 24 + 10. 11. 27 10 1 VoIL. 25 2
dividendi
LIX
22
2
9201
k, 2I)
LUOF. X, 10. 20,: 10
(X, 10. 21 - 11. 20
Resid
Quot
0,10. 22-12. 2032710.12
IX, 10.20
X, 18. 21 -10. 11. 203
Resid (X, 10. 22,- 10. 11. 21 + 112 207 118x
1+10. 11X
=10, 12. 20
1X, 16. 23 -10. 12. 21 711. 12.20) l+18. 12
 1020
uot X, 10. 21 - 18. 11. 20
lX, 10. 22 - 10. 11. 21 +1011.20
= 10, 12. 20
esid Sx, 10. 23 - 10, 11. 22 410. 1- 21 - 1. 20
= 10. 12. 214)101 1710
LX31024- 106. 12. 12 L1∞. 11 1L2I -1. 12. 25+16. 
(X31020
+142
XE, 10721 - 10311.20
X, 10t22 = 103, 11. 21 + 10,15, 20
- 102. 12.20
X, 10. 23 - 16211. 22 + 10, 112 21 - 10. 11
=103. 12. 2 X ) 10. 11. 12.20
Resid )X 116. 24 - 15, 1123 + 18, 15. 22 - 1018.28 +11 20
-103. 12. 22 +) 10. 111221 (3)1015.12.20
+idt. IL. 20
1X, 1835 - 10.1223 +15.11.12. 2 - 1 0. 1F. 12. 2111132C
1152 122. 2. =(2)10. 11. 12. 20

## Folio 7r — 00068013:0013
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 6v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068013/jpgs/default/00000013.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 132 lines, status machine
Diviser IOX FIX +I2X +B
ioxuxuXt
X, C0 sidx
X, 20) (10X
Resid. X, 21 c: +I2X
X: 22 (+ 13
opotiens X, 20: 10
(X3 10, 21 - 11. 202 (10X3
410. I1X
Resid. X, 10. 22-12. 207:711012X
X, 1023 13.20) l+ 10. 13
X, 10.2C
(03 10. 21 -11. 223
12 - 10. 11.21 + 11,204
-10.12 20
(10)X
X, 10223 - 10. 1X2] + 11. 12. 20L. HHFUX
-10. 13.2
45. 1IX
X, 1024 - 10. 13. 21+ 11. 13.20) (+10.13
X, 1020
X, 10321 - 10. 11.20, 10.
X, 10222 = 16. 12. L0
X2 10.3
l: 10
10. 11. 22 + 10. 11. 2 =11320
10. 12. 21 H2 )(C Il. 12. 20
=12 13. 20
10.24 - 10. 12. 22 +1 011. 12 2 - 1F. 12. 
-18. 13. 2]+10. 1220
+V. I1. 132
X 3 10.25 4 1.13. 2 + 10. 1I. 13, 21 -13
X, 101.20
X 10. 21 - 10 11. 20
+10. 12. 13. 20
ud. X, 10. 21- 18. 11. 21 + 11. 20
1612.20
X, 10.23 - 10, 11. 2 + 10.15.21 +11. 20
10=1X2 Y2) 10. 11.1220
101320
= 10. (.Et
a1420
X, 10.2 - 103. 11. 23 + 16.1. IL ( po. 1I. 12. 20
10312. 2E +6) 10. H1. 124
=103. 13. 21 4(1),0. 111320 4
+102. 12 20
Quotiantes semper deprimi possunt per 10. sed
Exaltavimus et progressionem cum residuis
servarent ubi divisio depressionem non patitur
multitudo terminorum divisoris
Si dividerau et divisor nihilo aequales etiam residuns
est niluo aequalis. itaque dum s aqruationes
esusdeia leterae semper habent commanes dvisores gradus
cujuscunque ambarum grade inferioris
idem est in omnibus quotientibus Eusddiviloris
coefficieus termini primi idem secundi et Nam divisio usque
ca resionos exclasire eadem est, quantae cunque altitudinis sit divident
usi qoo potentiae literae capitatis sunt ultiores.
similis est nexus inter resiuos auteriores et posteriores, non
Tamen ejsdem divisoris, nec ejusdem daidendi, sed earum
divisionum, qualum residua sunt similia, seu in quibus eadem
est differertia inter expinentes potentiae psiux a termino
primo divisoris et dividendi ita coincidunt termini in
residuis primi coincident ixterse; item 2di inter se et
Et geatralite, praevideri poterat. Cum eadem est differentia
exponentiam terminorum summorum litera Capitalis,
divisiones inferiores continentur in soperioribus ut casus
sperialis. Itaque priod coincidentia den potius ide aticitas
quotientibus. jusdem ordinis a sumno terminorum est
vanescentibus
resi ducron ion
speciei et generis
In omni divisione Residus est quoedam continnatio quo
perfecte
atioinferiore e
quibus dreti ergoisat
tientis siae divisibnis Nam si continuari posset divisio, er-
minus primus recidui daret, novum quotientem, quod
in dividendo altiore. Hinc eodem exstente
ivisore inter se consentiuntin Numeratoribus sciliset
et coefficientibus) erminus primus residui in divisione
proinde fit
et erminus altimus quotientis in divisione proxime altiore
isque consansus est omnimodus, non ut peciei et generis
 Residuus nascitur ex quotiente divisionis (ut ipsa
docet operatis) Ultimius terminus quotientis maltipli
cetur per totum divisorem, perultimas seu secuntus a
fine per per divisorem semdo fmo en primo ter
mrino minutum, antepenultimms eu itius a cne
divisorem duobus summis terminiseu primo et secundo
minutum; et ita porro. Agglegatum horum pro-
ductorum subtracum a fotidem novissimis terminis
Quotiens divisionis, intteligo numeratorem, compo
divitendi quot sunt ipse lapo
nitur ex Asiamis autercoribas ejusdem digisoris et ita
quidem  coefficiers primi termini quolientis tot ex inefficiert
primi termini prim. residui; coeff. secunti exioefficiente
secundi et ita porro. Tantum praeterea multiplicanda
eoeficientem per potentiam ipsius 10so blatum ad eum gradu
qui aegoetur potentiae cujus in quotierte dabet esse edesti
 eot Relu terinini e sumi possut ex
ab Termini quorientea et Relicui sumi possunt el
Tabula explicationum ipsius 20,21, 22, etc N spectives
per 18. 2 - 1. 25; 1021 - 12. 20; 10. 23 - 13. 20 et
contiovalis
semper explica tionibus peculiaris cheda exhibiii. Quotiente
sunt linea interrae corizontales ex tabulus qontae
Residui sunt tussdum primus terminus quoriena srima linea
Zoq Eda, etc. Resiavi primus tominus est proxima seipea intema
20. 112712. 2 F 1P.1220 5.
10. 26 - 10. 13, 23+10:11. 13.22 -10. 11. 132 H. 1320 puxina post lineam constimr entem inamum tergunum quotentis. Edue st dimidia
sequentis; tertius, nadax pximus ineae adm sequn
fis; et ita peare suntque olidem aleo colidem termini in quoves Ter
niman
3 1025 -10. 12. 23 +10311.12. 2 - 10 11313,30
103. 13. 22710212.1
1)10. 11.11. 20
+10212. 13. 21 (2)10. 11. 1. 13 20
I 1o. 18. 20
C calculo et Tabula quemlibet terminum dictae
mino residui.
ine calculo, etremporanea
scribi possent, quod erat
resideratum
pos
tbole 
tom priot mostantem iende
vidento
 et residvus
Resie
Resta
s

## Folio 7v — 00068013:0014
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 6r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068013/jpgs/default/00000014.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 71 lines, status machine
20X +21X + 22X +23(26
(10X3 + 11X + 12X +
20X + 2IX +22X+23X +
10X
+IIX
10
(107
12o 
+ 12X + 13
et
ut settre
H.20 12 20 M
10 10 h+1
3 +UX
+
2(102 C
10H.  f ∞ +2. 2
Tunro tuisere
a2OXF 2IX + =X +23X+24X+
0
+1. N122O
tou tnn
Eu
9. R. 20
10. 11. 21
120 ut (1 120)
110 1320
(=1019H 10. 12. 21 201. 21
+II. 1I.20 +H12. 20 FuI320
20X + 21X5 +22X,, +23X000 +24X,000 +25X10000 426001000720Y
1.20 =12. 20
40. H 21
= 13. 20
10. 12.21
I1. 1820 F 11. 12.20
0
= 10.13,21
+ 1. 12.20
1010.11. 22 = 10. 10. 12. 22
+ 10. 11. 12. 20 4 10, 12, 12,20
- 10. 10. 13. 22
+ 10. 12. 13, 20
+10. I1. 13. 21
+ 10. H1. 11. 21 + 10. 11. 12.2)
= 11. 11. 1I. 20 = 11. 11. 12.23 = 11. 11. 13,20
=10.10.10. 13.23
= 10. 10. 10. II. 23 =10. 10. 10. 12. 23
+ 10. 10. 11. 13. 20 410. 10. 12. 1320 + 10, 10. 13. 13,20
0. 12. 1
vc 10
+ 10. 10. 11. 12.2 + 10. 10, 12. 12. 27 4 10.
10. 11. 1I. 12.20 =10. 11. 12. 92.20 - 10. 11.1
+10. 10, 11. 11. 22 +10. 10. 11. 12.22 4 10. 10.
110. 212X
11. 20FH2
le
s
+10.10,22)
10. 12. 201
- 10. 11. 21 10.
+ I1. 1120)
+1010. 10.23
= 10. 10. 13.20
- 10. 10, 12. 2)
1
= 10. 11. 11. 12,20 = 10. 11. 12,12. 20 = 10. 11. 12. 13,20
10. 11. 11. 11. 2) 10. 11. 18. 12.21 = 10. 11. 11. 13.21
+ 11. 11. II. H. 20 +11. 11. 11. 12.20 4 11. 11. 11. 13. 20) = 10. 10. H. 22 (10
+ 10. II. 12. 20
+ 10. II. II.2)
= 11. 11. II.20)