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# Piece: https://leibnizlegible.com/api/records/28139/text
# Catalogue record: 28139 — [De sublatione literarum ex aequationibus]
# Incipit: 1mo 10a + 11x = 0
# Date: 1683-05
# Other printings: Knobloch: Beginn der Determinantentheorie, 1980, S. 84-91
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=28139&katnr=56655
# Work: https://leibnizlegible.com/work/00068201
# Title: Leibniz-Handschriften zur Mathematik LH 35, 14, 1
# Folios: Bl. 154–155 (from the shelfmark LH 35, 14, 1 Bl. 154-155): 4 pages, canvases 307–310, page ids 00068201:0307 to 00068201:0310
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068201&page=307
# Lines: 299 recognised on 3 of 4 pages, mean confidence 0.84
# 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 154r — 00068201:0307
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 155v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000307.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 100 lines, status machine
RN= 1+ + I 10D_ERyB e X a 4 21XC + 22 y = 0
(1mo)1a  10 a + 11 X = 0 Espd.) 20a +21 X = 0 fit 10. 2 aF 1. 2IX) = 11. 20a (411. 4)=0
seu (3tio) 10. 21 = 11. 20  seu 10. 21 - 1. 20
(1mo) 1oa + 1X + ny = 0 (2d0) 2a + X + ny =0 (tio) 30a + 31X + 32y =0
+10. 22 a + 1. 22X = 0
Undel   10. 2a + 1. 24X (F 12. 22 7) = 12. 20a + 12. 21 41. 22y) 6 seu (45) 2. 20 : - 12. 21.
ex 1. et 2 fil F
et 5t) ex 1 et 3 fil 10. n a + 11. 32X (4 12. 32y) = 12. 30 a + 12. 31X (F 12.329) senfto) 410.
12.30.. = 12. 31.
et ex 2 et 3 fit (4) 20. 32a + 21. 32 X (+ 22. 329 = 22. 30a + 22. 31X (+ 22. 329 se (6t ) +0. 31a + 231X = 0
- d. 203 ssn
n. 30. = 21.32.
ipso multiplicatione fiet (+10. 41. 22.132) ∞ (= 10. 11. 2232 =0 seu +10. 22. 3+ =11.20. 32 + 12. 21. 20,=
=10. 12. 22. 31 , +, 11. 13, 22, 30
= 11. 12. 20.32, + 10. 12. 2l. 32
- 10. 21. 32 = 11, 22. 32 - 12. 20. 31 - 0
prodeunt ergo omnes combinationes hau legibus sequentibus faciendo
+ 12. 12. 20. 31, 2 12. 12. 21. 30
Unum quodque membrum habet  factores diversos, tot quot sunt aequationes initio datae
2
scilicet
unum quodque membrum tres habet factores diversos tot quot sunt literae capitales ar, y
sca
seu t
. Hinc Notie Aequationum positis
eodem cordi
2 3 seu sinistris in omni
Variatio superest a sola possibili transpositione
62t, 0. 12 et 0. 2. 1 et 1. 0. 2 et 1. 2. 0 et 2. 0. 1 et 2. 1. 0
40. 21. 32 membio positis
1253,
-I1. 20. 3.. Vel quod idem redit, notis literarum capitalium (a, X, y) nempe dextris O, 1, 2
dextrarn
lo. 22. 31 sinsttarum
ilc. 22. 30
+I2. 20. 31
in omni membio eodem ordine positis, variatio superest in sola possibili Transpositione
deptraru sinistrarum 1. 2. 3, et 1. 3. 2 et 2. 1. 3 et 2. 3. 1(et 3. 1. 2 et 3. 2.1
Et cum tot sint aequationes quot  literae capitales, pat hinc idem
10. 21. Jesimem Hinc patet combinationes demum reduci hic ad transpositiones, quae habent
-k. 21.30
Get oppositam
tum
ubu tantum ipsi
utroque modo provenit.
aliquid simplicius, cum eadem retineant.
membra quae unum tantum factoren
Lex signorum haec est, ut quae unam tantum citeram communem habent
3. Lex signorum haec est
tatio etam in e differant signo quae duos cqueniant, quae tres rursus differant erc
10. 32. 31, sed e
num cart. h s placto autem calculum adhuc peragere in quatuor aequationibus
ganda curi lexsignaum
(Imo). 10 a + uX + 12 y + 137 =0 (2do). + 21X F er y4 233-0
Btio) 30a + 31X +32G+33 7 =0 (yto) quat aegue 4. s a
unde pes icumq ex 1 cum reliqui et per icum aequationibus et per combinata tit
sibi nunquam cot
o) 41043 a F 11. 43X + 12. 43Y =0 (6L )+10. 322 +1133X + 12. 3y =0 (SH0) H10. 23a + 11. 23X + 12. 237 = 0
per, = 13. 30.. - 13. 31.-13, 32
- 13. 20. . - 13, 210 - 1322 u
per Icum 5
per ten  13.40. - 13. 41. - 13. 42
uende ex ag. 5 cum reliquibus 6 et 7 conbinata fit
cum3
(800) per Scumbsit + (+ 10. 23G, 12. 33) a H + H. 23 )u 2 31 IX) 
2 - 17. 20 5. 13,329
7 - 13. 21 5. 13.32).G = 0
(+ 1d. 23 140 332a - (+ 12. 2,] L 1133 2 X
(= 13. 7 = 13305
l 13. 22)= 13,315
(Gno) per Scunit fit + sco. 2311. 437 a HfitiD3 2A 1AL X)-0
P. 13.203- 13. 42)
J. 13. 415-1342)
-(KI. B3 24 10.137- (41223 711. 43, Y
unde musus ope aequationum
8 et 9 toc demum
sed hoc calculo Xeqq. 8 et g non est opus, cum catculus praecedens jam, praebeat conpendium
l- 13. 225 - 13. 405)- 13. 225- 12.41)
ex ag. 5. 6.7
penge, et quia alie olim jam ress sucm essecutus, non
est ut repo repetam hoc loco
L2.
22.
0.
1
32
d
3
L1. 30
etiam sic potest
replosentari
pott
Ei duo ordines hoc loco
sibi in plerisque respon-
dent non tamen in
omnibus nam loco
et differunl in eo quod
penultinum et ante
penultimum inter
transferuntur.

## Folio 154v — 00068201:0308
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 155r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000308.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 72 lines, status machine
loX + AX + n =
10. 30 X + 1. 3oX + 12. 30 = Coinced. cam ( 20. 40 X + 20. 91X + 21. 41 -.
20  + 21 =0
40 X +41
20X +2IX +22 =0
+0X r41
10 X + uX +1 =0
30X 17+C
10. 30 X + 11. 30X4 12. 30X
4 1091. + 11. 31. u + 12.
1073 + uX +12X + 13
 30X +3
10.30x = 11. 30X + 1230X +13, 30X
+10. 31X F 11. 3IX +1231XF131
10X+ XFLX LBXL4
4 30X+31
1030X + 11. 30X4123OK + 13. 2X + 1t. 30X
+ 10. 31. 411. 31.. + p. 31. 4133, - +1.3)
28
Et itue forio
co. 40X + 20. 41X + 20. 42X
+ 21. 40X + 21. 41 X + 21. 92X
+ 12. 40 X + 27. 41 X +2
40X + 4IX +42X+43
∞X + 21X +22
20. 40X 120. 4( +20. Ae X + 10. 13X
+21. 40. 2.91.. 2. 42 - + 2. 43X
2.40- 22 4 = 2042. +22. 43S
Ad reddendum calculum uniformem possermus pro 10, 1, 12, 13, 14 ponere
11, 12, 13, 14, 13; et pro 40, 41, 42, 43, ponere 42, 43, 44,45, et ita pro
novissimis duab quasi fuisset 11 X+ + 12X + 13X + IX Y15, et 42 X + 43N + 44X + 45
Et ita pro duabus aequationibus novissimis scribemus:
1P. 20 X rr2. 30X + 13 30X p 1430( 4r5 30 X L 1 30
urgo X + 1230 Xti3. 30 X+14. 30 X +15. 30 X + 6. 30 X2
103] - 11.31 .. 12. 31-. 13. 31. 14. 31.. 15. 31.. 5
2agX +. 4) X 42o qat2o. qs X t20. +bX 120. 3 X) - 0
21. g1. 21. 42. 21. 47. 21. 44.. 21. 45. 21- 4
22. 40. 22.41. 22. 42. 22. 43. 22. 44. 22. 45.)
ubi 10, et 16 et 40 et 41, et 4b et 47  quivalent nihilo, tantum tamen
tantumque adhibentur ad servandam uniformitatem calculi
Fortasse tamen hoc supplenento non erit opus, itaque consideremus solas aequa
tones dextras, quia in illis incognitarum numerus crescit, qui in sinistris manet,
et sit 20.40 X + 20. 4) X + 21. 41 (   50a X + 51 a X + sea
21. 40
et siet 40 = 50a :20 41 = 52a: 21
unde _ 20. 20. 52 + 21. 21.50 = 20. 21. 51
20. 407 + 2t. 40X + 22. 90X
E  ~ solx + siX + 52 X+ 53
+ 20. 4 X + 21. 91X +22.41 b
30 = 50: 20 4t = 20. 40  f 20. 4) X + 20,q2X. 
21.40
+ 212X
2 soR Ns X rIXESIX +54
22. 40 5. 41. +22. 42
40 = 50: 20 42 = 54: 22 42. 20. 20. 41 + 21. 50 = 20. 51 21. 12. 41+ 21. 54 = 72. 53
fit 41 = 20.51 - 21.50, : 20.20 = 22. 33- 21.54,: 22. 22
202.G4 +,22. 50 +20. 21. 22. 51 - 21. 21. 22.50 = 20.22.52
20. 54 + 22. 50 + f 20. 51 -2150 +- 22. 53 - 21. 54 = 52
2.20.20
sad in his non satis apparet ordo.
2- 22. 22
20. 42 + 21. 41 + 22. 40 = 32 20. 43+ 21. 42 + 22. 41 = 53
41 = 5i = 20. 42 - 22. 40, : 21 = 53 -=20. 43- 21. 42, : 22
fil 22. 52 - 20. 22, 42. - 22. 22 40 = 21. 53 - 20. 21. 43 - 21. 21, 42
seu 42 = 22. 528 = 21. 53 - 22. 22. 40 + 20. 21. 43,:, 20. 22 - 21. 2)
20 ve (2091 X + 21. 41 X
20. 40 X + 21. 40X + 22. 40X
+ 20. 41. . + 3. 41.. + 22.41.
20X + 4X +  40X + 41X142
20X+21X +2
+ el.40 221.tes
+22. 40. frc. 41

## Folio 155r — 00068201:0309
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 154v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000309.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 127 lines, status machine
20X +21 = 0 fit 10. 21. 21 - 11. 20. 21 + 12. 20.26 =0
20 X + V X + 22
1o 2 411. 20X + 1220 =0 + 11. 22  + 10. 22 X
- 10.21 = 10. 22
1t. 20 + 11. 11. 20.22
- 11. 12, 20.21
- 10. 11. 21. 22
4 10. 12. 21. 21
= 12. 21
 (10. 10, 22. 22
270. 12. 20.22
(+ 12. 12. 20. 20
= 12. 20
410. 20X f. V0X F 12. 29F 10.20
= 20. 20 X - 10. 20.79X
10 X3+ uX +12X + 13
=1. 20X - p. 20.21
+10. 2v3In 410. 20. 21
20X + 21X +
+11. 20 X + 12. 20X + 13. 20 =0
= 10. 27 . - 10. 22..
unde
+ 117 + 12. 20. 20 X + 13, 20. 207
- 10, 20. 22 n
= 11. 20. 21.
+ 10. 21. 21..
= 11. 20. 22
+ 10. 21. 22
(+ 10. 22. 22) (H3, 20. 20
 - 12. 20. 22 f7 - 11. 20. 22
+ 13. 20. 2
+ 10. 21. 22
fHI. 22, 227 (412. 20. 20
- L- 13, 20. 22
padratum
A b ut una p n
rer ab illa ab dena  in
cubicae assurgunt ad quadratum, et coefficientes aequationis quadcaticae
fic ni
affuus unt ad cubum
+ 10. 13
+ 13, 13, 20, 20. 2
0. 20. 22
(2) iea
itata
4(2)  10. 13. 20. 21. 22
= 10, 10. 22. 22. 22
= 10. 12. 20- 22. 22
+()0. 13, 20. 21. 22
+11. 11. 20, 22. 22
= 10. 11. 21. 22. 22
D1. 13, 20. 20. 22)
+10. 13, 20, 21. 22
f 11. 12. 20. 21. 22
= 10. 12. 21. 21. 22
= 18. 13, 20. 21. 21
+ 10, 13. 21, 21. 21
adimando secundum 21. potest tolli 21 faciendo
- 11. 13. 20 - 10, 12. 22 =0
et potest tolli 21. faciendo + 3. 10. 13. 20. 22 FL OB  - 10. 11. 22. 22 + 11.12. 20.2=
sed ita non potest tolli uterque terminus quia aliobique vabor 22: 20
tun
seu saciendo 3. 10. 13. 20 = 10. 11. 22 + 1. 12. 20 =0
sin ollamus 22 secundum _o 22 ordinemqus
 ad lollendum 22 firit - 10. 12. 20 - 10. 11. 21 + U. 11. 20 =0
et ad lolendu 22 firit 3. 10. 13, 20.77f11. 12.20.3= 10. 12. 21.7 =0
taque verendum ne du. de sschiruhans in calculo suo Maverit
quid si ergo opus sit ad hiber aequalionem assuntitam epsdem cum prirre gradus
ex in aeq, Eecta ex iox etrox fet 11. 11. 2D - 10.11.21 = (2)1072.20 = 0 uet au
Et pro tolendis
v
terniuum secundum seu fit 2D: 2D = 11. 11 - 2. 10. 12, : 10. 11
fortasse tamen  du. Esch. rem sic peregis ut ad tollendum demmiam primus pro se
atque ita in aequatone
fit 2 4+ 21 =y. ita pro tollendis duobus faciendm XX ∞2IX + 22 = y
que  ite uit au qu n festetu rs
102 +y
2020X L 2X + =+2t
10X + L 2 = ( a
S. ptro et  fuit x = Zos..
1X- 10. 21, + 3. 3
Et pari jure
+20u
+ 10. 22. 22 X + 11. 22.22
- 12. 20.22..
+ 13. 20. 21
- 13. 20. 22
=(12. 21. 22
F 13. 21. 21
+207)
useuu
 20 2
10, 12: i
= 12. 21. 22)
1o X + uX + 12
1BX +HX F IE 10X +UIX + 12
 1030 x 1
XIOIIZD
12. 13. 20. 22
Hie nulelidita
13 de ascepdit 20 et 22, sed _ m
colfficientes aequationis,
seu medio
efficientes
cubicae in una
extremos inaltu
membrissuntiam yposte
Elidendis nemp
-10. 11. 20. 20
Generaliter eruo dici potet
illic  Nunquam cum effecti
ab una parte ad quadratum
ascendere 10, 12 in una; 10, 13 un
altera et nunquam cum
effectu ad cubum ascendere
21 in rna, 20, 22 inti
seu medios coefficientes
quadraticae in una
 illa, extremos
video ad hac amplius
elidi nempe debent omnia elidi
membra, in quibus abest 22
2 nam quia 21 assurgit ad
gytum, oporeet fieri depressione
ubi eadem diffi
cultas
X. et X. Ct

## Folio 155v — 00068201:0310
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 154r); this page is its left half.
# No recognised text on this page (recognized).