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# Piece: https://leibnizlegible.com/api/records/42663/text
# Catalogue record: 42663 — [Aufz. zur Zahlentheorie]
# Incipit: m=a n=b | mn=aa mp=bb np=cc
# Date: 1680
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=42663&katnr=39559
# Work: https://leibnizlegible.com/work/00068195
# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folios: Bl. 11 (from the shelfmark LH 35, 12, 2 Bl. 11): 2 pages, canvases 21–22, page ids 00068195:0021 to 00068195:0022
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068195&page=21
# Lines: 75 recognised on 2 of 2 pages, mean confidence 0.85
# 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 11r — 00068195:0021
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000021.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 46 lines, status machine
ituot hit crvi vide firei Nitul hic m stt  a ompta
mnea m = a n = b. mn =aa
Ero  aa p = bon s seu p = bbon: aa. jam ex 3 est p = cc: n Ergo fiet exzero
denique nn =, aacc: bb seu n= + ac: b m = 14) ab: c et p bc: a
mmnupp  aabbec  (per 121), sin mn p= abc mnempenp= aatHt cC
+ n + p L6) ac: b 4 ab: c + bc: a L ac + ab + bc: abc Hinc si sint
ut mup, magmatup, et impurn, tiet abc= f aa + bbrCC = et denique
ac + ab + bC = fh et quia ex 2c est atic  et hinc jungendo 22, 23, 14
habentur a, vel b, vel E, saquatione cubica. sed nihil hinc novi
a Icidendum quid fiat si faciamus, nn + p mn + 12 = 0 et similiter faciamus
20 mp + 21 mp + 22 =0 ac denique 3eri p + 31 np + 32 =0, ac ita quaeramus
 valorem ipsius m. Tollatur ergu promum p, et ex 2 et 3 fiet Eoomnp L
230mh p + 2230n = (30 n LOmp) + 3on ZImp + 3on 22= (20m 3on ptroni opi 20 1 32
et fit p = 20. 32 m = 22. 30n, :, 21. 30MH = 20. 31 MM. Rursis
aliter ex 2 et 3,  32. 20m p +32. 2mp (432. 22) = 22. 30n p + 2231np 22. 32)
) et fiet p = 22. 31 n = 2. 32M,: 20 91 um 20. 32m = 22. 30n. quos
duos valores agy. 5 et 7. aequanto, fiet: 20. 32m - 2. 20. 22. 30. 32mn + 2. 30
= 21. 22.30. 3,Mn = 21. 30. 32 m n  + 20. 21. 31. 32m n seu fiet
=20203232M, - 20 21 31,  ipon L 20. 20. 32 32m, - 20. 21. 31 32 nn + 20. 22. 31131 m n 2
10. 22.
82. 22. 30 30 1 21 -2 3. 31 M 13 30. 30. 32. 221 30. 31. 21. 22 n M + 30. 32. 21. 21
-2. 20, 22. 30.52
d haec aequatio
per compendium scribatur 20 n + 4M eo
compendo 40m + 50n _ + 4 I m n + 51 _m + 45m pt (10
jungatur cum aeq. 1. ut tollatur n, primum tollendo non, fiet
90 m + 50 n +41 nn +S1n in _ - 45. H mn - 45. 12. = ruisus
multiplicetur haec, aequatio per mt, et siet 40 m +50m n L 4 m n + 5Im N- 4sumn
=45. 12. mt = 0 Ergo ex aeg. 12Expliata S H: o m N - 2. 11 2. 1. 1n +. 21
 LSIMNIL.1OmR. SIMMIZ.IO
fiet: 40m - 45. 12:10m ~ 5I. 1I12. 18 M  = 5C. M. 1d H. 10M - 50 1. 10 12. 10 = 51. 1 toM Mpi
 eAM + 50 (2.10 = 50 11. 12: 18
Esuito m t SL. LIOIA
=50. 11: 10 MN - 2. 50 11. 12. 10 Mn + 51. 11: 10 M mM - 51. 12. 0 nt ma = 45. P1. 10 m mn
quae aequatio 13 per compendium sic scribatur 
an +60 m  + Sim  462, mn ,  40 m + 63m +64n +65, posito
esse 60 = 45. 11: 10 61 = 51. 12: 10 = 51 11: 10 62 =50, 11: 10 - 250. 11. 12: 10
63 =∞45. 12:10 64 = 51. 11. 12:10. 65 275012:10 -50. 11. 12:103
fit Mn = 40 m8 + 63 m +64 ni +65,:760m 161m + 62
Eix aequil est mn = Si: 4 -  10. 12 = 1: 2: 10 Eergo tantum opus est
11: 4 = 10 12 aequari coto (quod facile est ne calculus ascendat ad ,i porro nd non eranescit
nec m iaque satis cquoteta prodit aequaliaa quae tamen facile solvi pba
i estet comn + 11 = 0 Eju ope tollendo. n ex aeq. li
fieiet 40m + 68 450. 1 1om 41 M. 11: 10 -51m 11: 10 n + 4515: 10 = 0 seu fiet
40 10 m  = a 1. 11. 10m + 45. 11. 10 M = 51. 1: 10 M + 5011: 10
den

## Folio 11v — 00068195:0022
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000022.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 29 lines, status machine
si simplicissime fnisset comn = umn + 12 =0
et
romp +21 = 0 et 30 np +31 =0 ex 2 et 3fiet p = - 21 : 20m
et  p = a8e - 31: 3on ergo 21. 30n = 31. 20m, seu fiet
H= M, EP 20. 31,: 530. 21, quo valore substituto in 164. 1. fier
10 m . 20. 31: 30. 21 + HIM 20. 31, : 20. 21 = 12
mup = a mng. b  m p9 = e npq = d fiet:
1 = b3: mn = C: mp = d np. Erbo  Et p jam p
hinc B. n = C. p et Hinc C: m = d3: n
Aursus.  p = b: m = d3: p seu p = d’m: b = a :mn ergo
fit n = ab: d’m = d’M: C fit denique abe: d = M
seu M = abc: d si quaesilsemus m +n = pendi esset X = abc: d + abd: C
aa  b   m + n = a m+n = b fiet m + n + 2Mn = 1
quod si fuisset
fietq 2mon = b - a et M= n = b - a,: 2n = b = m ergo fit
b - a = 2b M - 2 m. seu m = b M = 2 a - + b
n’+ n + p = a m + n + p= b mantp= c. ex 2a
fit in + imn = c seu 2mn = ce - bb misus
n 4. 3nn C + 3 mup seu  mmp = - a + cc - bb: 2C
x nidu quatuor rerum ternimes 4, biniones sex, ut
mn _mp _mq  ap Lnq , pq
mnam= aa Comisso ap Exopposilis is si
nu np nn+ np = bb
mp
N. hil hic novi repertum riem quod
quatu tres biniones trdarent lubicam baram, tsquatuor termones
dent quadruloquadraticam param. quiaque quaterniones; verum dent
surdesolidam purom jam noram
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