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# Piece: https://leibnizlegible.com/api/records/60633/text
# Catalogue record: 60633 — [Aufz. zur Reduktion eines Polynoms]
# Incipit: Desideramus productum ex x + b, x + c, x + d, aequetur quadrato
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=60633&katnr=57621
# Work: https://leibnizlegible.com/work/00068197
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folios: Bl. 51 (from the shelfmark LH 35, 13, 2a Bl. 51): 2 pages, canvases 101–102, page ids 00068197:0101 to 00068197:0102
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068197&page=101
# Lines: 178 recognised on 2 of 2 pages, mean confidence 0.76
# 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 51r — 00068197:0101
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000101.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 151 lines, status machine
Disideramus, ut  e productum ext X +b, X+C, x+d aequetur quidrato. Alumo
oe et esse LLMm (x+ b) = ABCd + b LLMM = EEFGEt LL MM(XIC) = ABCd + Cf LmM. Et A B. Sa B au Mm =rGaa na
Enim productum ex X+b, X+C, X +d erit quadratus EEFFGGHH IIKK: L M. Ubi quaevis sitera signilicet formulam simplitem
Fiet A qualis est. H +.. a
Omnes autem aequationes quae sequuntur sunt coinidentiae ideunficationes
Eliu medium, ABCd = IIKK - CLLMM. itaque seponi potest _ ABCd, donec I, K, L M, sint inventae;, inde cum statiae dabitur
Sed quia etiam ABCd = EEFG- b LLMM et ABCd = FGHH - dLLMM. Hinc fit idKK = EEFG + (C- b) LLMM, et iIKK = FGHH H (C- d) LLMy
At c et sit C b = β et c- d = d fiet EEFG = IIKR - 3LMM et FGHH = IiKK- t LLMM. Hinifiat
HH su (X + LM 1B K Im tC) EEFG A, IK + LM Vo  HH LM Ad) FGHH Ex his quatuor sinistris hacum duarum aequation
B(IK + LM +B) in f (iK. LM sB) = EEFG et A (IK+ LMS) in Z (IK- LM YS) = FGHH. Ex his quatuor sinistris harum duarum
aequationum binas formulas, aquali quadratis, vol ad quadrata EE, HH, rationes constant a habentibus, non licet;; aliequi enim binae altere
aeuate qui
videa deberent habere rationem constantem inter se, quod  commodum non erit  Hac ergo resolutione omissa ali tentanda erit,
p
EP
tuc tame
Ex IK + LMIS =HF. K: d IK + LMd = HF L: K
vil HF. 9
Vet H∞. 1.9
iK- LM+B= EED: BC
s in piori pari, nihil resent cui attriouss F vel G; in poseriore, quia refert post electam jam in priore F vel G) adhibetur signum ambiguum, ita
nin
pursus nis refert: Ex his quatuor aequationibus fit ik = Ep. EF + E EG = Z9. HF + = HG. addende inuilen  po runus moscem
ubi sufici ik iton 2
hi coincdentiantur
Hinc E=
dincidentiari alterubi suerum valorum, quia
er
F+ G
et
PF+ G
=
1B, 9F - 4G
os
 ip. EF -  EG = 2q. HF ig HG subtutento a primasocurdum, ta tetia quantum
LM= +B
418, pF-  G ubi rursus sufficit E coincidentiari alterutii valorum, quia hi coinsat atantur ater se
Multiplicatione ergo per ceucem instituta, fit +8, pgeF G(K 2) FG = YB, p9FF -  GG + Cq - P) FG. jam com–
pendii causa paFF.  F GG vocetur R, et (Y-4) ve FG vocetur S, fint +d, R+5 = +B, R- 5, seu
R (1β +8) = S(1B 488) sou (1B. f 18) CpgFF- 1 GG) = (+β + 15) (L. 2) FG dividatur haec aequatio
1B + ss (f -Z) FG seu compendio FF- rrGG = at FG. Htoc jam
per (AB = 15) p9, et fict FF - poqu GG = YB+ is
tandem aequatione identificatoria praestandum est.
fiet +10. 10aa yy + 2. 10. 11 a3y + 11.1 a
Sit F = 10y , 11a, et G= 20y +21 a
=20 orrgy - 2. 20.21 arrys - 21. Maart ~ 10. 20 at yy + 10. 21 auty + 11. 21 cet
Is
11. 20 aaty
identificando teocoefficientes ipsius yy, fit 10. 10 aCD - 10 at = W, Au 10. 10
8.2C
Eodem
2c
10
20.20 20 a
12
-
+ na
Et + 4rr
Aaa
20= ta+CAtarr. pari jure acusi Eodem modo coincidentinadidentificando voessicientes ipsius y
seu 10
Eodem modo identificando coeficientes ipsius y, fit 1P. 11a0- 11 at = Y Y, unda ad discrimen prisrisfiet,:
21.2
= F- HCH+ arr)
la
Tandem identificando coefficientes ipsius y fit 10. UIaa = 20. 21 EY = (10. 21 +. 11.20) at, seu dividendo per 20. 21,
S.aa - rr = (180 + 1) at, et fint Et - H - a Iaa - rY = L + Y (H + arr) + L - s (H + arr) at
fit 10.
et fipt -Err = FE et fit to
4aa
+
 a
t 2 
Tra
A ser Arp =- 2tt et (H +ari) = (H - 2H) = H tat F f-
10 = E
1 1 + Y-1
Ea
et IR = F, 1 - -1
et
a
t:
2D
1β+
jam quia -2rr = H, siet 1a = Btd +21Bd pp 44
c
ut  24
A+S FZ +Bd
PtS L2+
peisa B+d + 2 1βd
F
seu  = YI- 2 = F fiet
unde st=
a
l-2
L 10 = aa l- 2 - 12
(p 2 yu  = 1ay Zu2 aa
Ergo fiet Ik = _ LM
a 2a
13+ 18 -  seu
et i - aa, L.2 + 2 seu 10 aa
jt fiab F et hut asensur, et  
et hiIK- LMS) fier 
et A(IK + LM S) fieret = FG =
 +F 30yy + 31 ay + F 32aa
+ F d 4oyy + F 1S. H ay L F 18 42aa
le
et
1- 2 = pp 
YB + 1s
L-1 + 1 et 11 = ea
2 AH= pE t G. m
h.d
Tentemus ante omnia
postremum; sit ik = 30y9 + 31a 1+ 32 aa et. LM = 40yy + 41ay +42 aa
+p2R yy + (2a. y +32 aa
X
Epet
+ t-Epp4q, seu eaa r- 1 =
ao Y. 2 =
1β+ 1s
et E Y. - 2 = a l- 2 fit Y = a
veluti si B(IK + LMB) fioret = EEm
ex  (1KC L M feret = FG+,
94
pP
4
10
fiet
per has tres identificationes qu
dabitur raho constans 20 ad A,
 40 ad 30, et A ad 31, 42 ad 3 d
et quidem semper eadem, et fiet
40:20 L 41 : 31 = 42: ZL
et fiet DitE + Ba t BV LM + LM+ B - EE  et  LM - YS LM = H H 1 
ergo etiam E et H erunt in ratiore constante. Ergo etiam EEFG et FGHH erunt in ratione 2 constante, (labe
sed ita etiam X + b, et x+ d erunt in ratione constante, quod non licet
procedit,
et  (iK DLM =  FG
itaque seposita iniErethidus non ps
possemus ad huc  unam tlatare sepositam, verb. gr. faciendo B(IK + LM YB= Et sur
et  A(IK + LMId) = FHX et ZCIK- LmB) = lC H = hac methodus 
videtur extum bossehabere, et absolvenda est, it vid atur quae exduabus possibilibus sit commodedes
ita vin
1- 1 - 1
pa
ideu
Pa12
i
ito
x

## Folio 51v — 00068197:0102
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000102.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 27 lines, status machine
E= 9F + G, A = qF - 2G, B. n
H= PF +  G, A = pF - fG9 +, n
Ag. 10y + G. 11a)
120y+4 21a
sid ila fieret 10 - L
c
F = 10G + 1a G= 20y +21 a
+nqV B10y + Mqu B1 14
= 1β 209 - 1β. 21a
quid ergo si faciamas F = 109y+ Hay + 12 aa
AZM + B3. : q
MY B = sl, 9
A+ B,:q
MYB- A,
et G= 20y + 2 ay + 22 ca
et da facenemus qFFt G, et PF+ G habere dvisorem communem L
et qF- G et pF- G, etiam habere communem  et hoc sublato
esse
1F + 
4
G
 M
it PF+
5
eo
eF. G
N