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# Piece: https://leibnizlegible.com/api/records/23621/text
# Catalogue record: 23621 — [De divisore communi duarum aequationum]
# Incipit: Si aequationes 10x_e + 11x_e-1 ...
# Date: 1684
# Other printings: Knobloch: Beginn der Determinantentheorie, 1980, S. 229-232
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=23621&katnr=56687
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 65 (from the shelfmark LH 35, 13, 1 Bl. 65): 2 pages, canvases 129–130, page ids 00068196:0129 to 00068196:0130
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=129
# Lines: 150 recognised on 2 of 2 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.
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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 65r — 00068196:0129
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000129.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 128 lines, status machine
si aequationes X + AX +aX etc. itemque
2 Si aequationes 10X +uX + 12X + etc = 0
et ) + 2X
rem communem X +. _X .  + . X n.  + etc. = 0
ubi n non potest esse major quam m vel e;
tunc ope aequ. 1 et 2 tollentur quaeratur nova aequatio
ia qualis  quae  assurus at ad X non ultra. Et
hoc fiat bis. Et  notari commodum esse modum deprimendo
si tollatitur modo summi termini modo infimi:
Hae jam duor
aquationes.. X+xn + etc debent coincidere
inter se, si scilicet aequationes datae habent hunc divisorem
communem equationem, ut  o
10 X 10X + 1 = 0 20X + 21 = 8 to fit (+ 10.20X + 10. 21 fit
10. 21 - 11. 20 =0 Aliter + 10. 21X (4 11. 21
necesse est idem prodiri sive
10. 20X 11. 20
Ut aute. nempe in finc
- 1. 21 ultima sive per prima procedas
per
Unde alia fint, ubi solum per mulabuntur
et Q ilan Xet  - 10, 20.. - 10. 21X = 10. 2
et fiet 1.22. Nempe si procedendo prodit calculo
+10. 22XX + 11. 22X (+ 12. 222=0 seu 10+10. 22X + 12 21 =0
=12. 20XX = 12. 21X = 13. 22)
- 12. 20
II. 22
quae duae aequationes coincidu
duae 6eq
nem, et fiet 1 X = 10. 9E 1220 0
qua si identicae sunt si 
ex 0. 21 = 0 et W ex posterior
1l
2. 2 - 11. 22 =
una aequatio utrobiue prodeant, habemus aequationes tre
12. 20 - 10. 22 -
pro indivi inveniendo divisore communi. Sed si nullus
eex du Et cum
sit divifor communis seu si tolli earum ope possit X
quo
si
X = 16. 22 - 12. 20 = 11. 22=-12. 21 aequalioidentuca
haec aequatio per se verificetur
aias teriper prodit abla
et 10X+ UXX + 1X + 13, et ita porro.
nobvidete
12. 2
inelientur
10X+U et 1O9XXLIIXF12
Et similiter 20X + 21X + 22,
ducere per omnes 1OKX + UX +1 Et 10X3 + uXX + 12 X. + 13, et 1oX  N XL 12XX+ BXt C
Et similite
20X3 +2 X + 22X + 23 per param et altiores. Donec regula
tellendi incognitam  habeatur
ubi eadem fera se per in destigabitur cgneneralis
iserior aequatio conjuagatur superiori ejs ope inferioris exallala
modus ex hibendi communem divisrem
ec tilletur terminus summus, suide ope nit etiam inferioris telletur
alsumus verb. gr. 10X L UKX + CX + 1320 20XX + 21X + 22 ()
iet (416. 20x )+ R 1. 20XX + 1. 20X + 13 2020 et 40. 22X tu exp 2n. 22X m2
- 10. 2073) = 10. 21 XX = 10. 22X
(4
3. 2X 1322
0=-B lo 48. Z et
prior adibectiant
us
seu 10.2XXLMAIXAM 0 sera  a er sutu
13. 20 - - 13. 21
C + 21X + 22
deberet esse tip. ∞ = 10. 22 = 20f et . 2. 20 = 1
12. 20 - 10. 22 = 11. 22 - 13. 20 = 21 et 13. 20 = 12. 22- 13. 21 - 22
Neum prb aeque
10. 22
1. 20= 10. 21
20
+20. 31 XX +4 31X + 22. 31 5-
hanc combinando cum aeg. 20XX+ 2XF22, ex quaprodit
12 = 22. 30 +21. 31 13= 22. 31
mult plicando hanc per 30k +31. fir
10 = 20. 30 I1 = 21. 30 +20.31
inde (20. 22. 70, + 20. 21. 31 = 20. 2230) =21 q s verum
(20. 21. 39 +20. 2031 (- 20. 21. 307 20
Iem 21. 22. 30 (20. 22. 31 (20. 22. 3 =21, quodes est erum
20. 2. 30
22 22. 22. 30 4 21. 22. 3) ( 21. 22. 3) - 22
20
20. 2217
uae omnia succedunt,  recte deter-
2022.31 (20. 2130)
minavimus modum quo invenitur divisor communis
(20. 21. 30 + 20 20. 31
duarum aequationum sublatione altiorum graduum
in per 22
+11. 20, 22X + 12.20. 22
fiet +11. 20. 22Xst 12. 20. 22 X (F 13. 20. 22G = 0 seu - 10. 21. 32. = 10. 22. 2
- 10, 22. 22
= 13, 20. 20XX - 13. 20. 21X = 13, 20. 22)
-13, 20. 20. - 13: 20.21
similiter conjungantur aeq. 4 et 2.  multiplicando per termnos summo
- 10. 21. 22
C010. 20. 22XX) + 11.
13, 20. 20X
- 10, 20. 22 XX) - 10. 21. 22
sed si aequ. 7 et
conjungrum
3. 20. 2
0. 22. 2
d prodit
(+ 11. 20. 22
=10. 21. 21X - 10, 21. 22 =0
= 12. 20. 20 X = 13. 20. 20
+ 10. 20. 22 Y
1
amus per terminos imos, fit
21. 22
13. 21. 22
= 12.22. 22
13. 21. 22
E
k11. 20. 2BXX) +
= 10, 20. 21
= 11. 20. 20
et 10, 20. 21
Et si aequ
410. 22. 22X
1.10  
1. 21.2
3. 21, 21

## Folio 65v — 00068196:0130
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000130.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 22 lines, status machine
fomula 20  = N1XX +EX+  multiplicanda perde x 3, dat
dala 1. Erit A quetens
30. 20X7+ 21 X + 30. 2IXX +023X
11, 20X H. 2 ex A 22p
Unda datae
logoX + 10. H X + 10. 12XY + 10. 15X +10.14
similiter
C10N et per 30KXF3IX + SL
A dividendus B
B divisor
quotiens
10.44 residuus
firt 30. 10X 130. 2X + 30. 21 X + 30. 23KX +
+31. 20 31. 21 31. 22 431. 23X
32. 20 32. 21
iddatae
32. 22 +32. 27
+ 10. +4X +10. 45
0
A
10. 10 k 4 1o. X  2x Y 10. 9XX + 10.14X + 10. 15
patet candem semper esse lojem moemendi 20, 2, 227 23, etc