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# Piece: https://leibnizlegible.com/api/records/957/text
# Catalogue record: 957 — [Aufz. zur Umformung rationaler Ausdrücke]
# Incipit: [x+a], [x+b], [x+c] aequanda quadrato; est triplex aequalitas
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=957&katnr=58538
# Work: https://leibnizlegible.com/work/00068049
# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 4
# Folios: Bl. 81 (from the shelfmark LH 35, 6, 4 Bl. 81): 2 pages, canvases 161–162, page ids 00068049:0161 to 00068049:0162
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068049&page=161
# Lines: 137 recognised on 2 of 2 pages, mean confidence 0.74
# 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 81r — 00068049:0161
# Source image: https://digitale-sammlungen.gwlb.de/content/00068049/jpgs/default/00000161.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 68 lines, status machine
atque ita invenre m, et n, et aequavemulbimam resi duam
cognitis constantem. Hac arte autem haoebitur rationalit
atam vi, quam n ita omnia demum reducuntur ad oquationem
in ipsis cognitis
Sed potissemus tamen bilasse  obtinere nuncrum arbitrariarm
Y + my+n: y +vy+ vD. Ja non
es apicieutem, faciendo, X= S
 despero pranc, rem posse prasari ope detemi in in determinalis
liceret ponere ar, ret radices prian
c
a
Sed a hoc possibile esset engforet et
X ta quad. xtb quatr. X+C quad sit X-
ubi demontimi videta reuet, nontes semper possibilem
104 114r12y 514V el
20,730. 21V. 12g.V+1239
+ eu 1
e e
si scelsemus si deberet aequari quo h X + aX + bx + CX + CLB
2t
fieretque X=.
+m
sg
3+
et pou ygit 2t vo, habeam
V.
explicando X iser aeg.2 in aey. 3. spet _ ag. Effurgeus ad Xerenius
temi formula terminorum  4, in qua duo arbilariae, m et n
hunc oparando cum quadrato excrlato ab y+ py+gq habemus omnos
t.
oeamtiones 4 et literas 4 nempe fit hyF + am y3 +6h m y + 4bny+hm
 +3h3ms +3m y + M3
y+n
yt 13my +3myy + n y
tgr +3mi tnn. tren
 +2 my + m
y+2ny +N
+ m y + mn
+2m i5 + 2mnys
u
+m.. +2ma mars +nin
1
n
aist +3am +3ani + am tam
+an +3amn T3amn
2b mk
+Cy + Cm + 3CNn +ZCmN + CMN
eytaen +sen + 4en + en =0
Y tgny+Suy + quy+ ut
Cujus fractionis C numerator divisus per haaabtete, devel aequari
cito  i ab y + py r4q, quod est y +2p 1 +299 C
e
4 C
+pp yy +2pqg +q
sa velinus X assurgere ad eb haberum de sultus
quacta
tribus aequivalere tantum
rescioan aliquid alguieruur faciendo v curm. ytn se 
ne
by + abm + 2bma +2bmn +Bmn 
a + b
+61V
+3CM +3CN +Cn
sed video revera has quatuor, albitrarias
unum quatu et qua formlie eplicata assum ad quo adacu habema
ostamm gradum possumu conganele qacm cto excilaso a
formula oti gradus in quo parbiharia Habernus ergo arbitirias  et comparititia
aequaves etiam 8

## Folio 81v — 00068049:0162
# Source image: https://digitale-sammlungen.gwlb.de/content/00068049/jpgs/default/00000162.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 69 lines, status machine
(ubi X quaesita, a, b, c. da 6)
∞+a X+b, XtC aequanda quadrato; est trisitc
aequalitas fit X + a X + ab X + abc equ t
Xscub prox sribatur + my+ nn et fiet
( = ( y 4 3my + 3nn y4
ut
42
+ 3mmy +_ +3Ry
+mu +3m uny +3mnty +
+a V + 2amy + 2a un y
+ammy +ramny +an
Fab y + ab m y + ab A
equandale quadrato ab
quodent
y + 2pV + 299y +273
+ pB y+ 2pgy + 2r y
q
+ abc
+ g y + 197)+
jatepio
y + pu +9y+
arta colls
Habemus terminos aequandos sexo sed literas equaas imemenias
et per () 6
o
1)
4
tantum quinque giet m pi p 3M. 2Yyntantabn t abe
et pr 9 sit 94= Zun s+ 3nm + aL et per 5 fit 49 = 3mu +2a ma +abm,: 21
giqsum
e
et exen 3etq sis m n+ 2a mp S- 3M = = M + ab.
4
 (+16
1n
etu comnpendio quantilas aequanda quadtato scrbatur V + 1)V + 1Y +. B9 11 4) r
fEt PL HE fet ex Y, P = 11: 2 ex y fit 99 = 12: 2 - 1: 8. EIGui per 2
s (2)
fiet ex Y, r3 = 15: 12- n: 4 = (per Y Sib, adeoque per ex 4 fet
+
3
15. _ (: 16 17- 11. 12:2 +157. (16). Restat yh dunde fcet ex y
l
3 = 2. 15:, 12- 15:4 + 11. 12:E - 11~8, seu 13 = 2. 14 + 11 () 12- 11: 4, :, 12- 17: 9
Hae tres equave
1
=14.
superest Ex Y, 11. 15: 12 - 15: 9. + 12_ -11: 4(2,: 4
Y, e es sunt vsolvendae per m et _ et datas. pro compendio rqt= c
Let  s 416  et 19, 4.  1 H eC et ex 7 fiet 13=4 15+ 11C,: CC1+)
+
et
et
2 c
-TH- C
seu ( exirtiot ie e+ -, 11 e + 4. 14 24 )0. Ex 19 fic 4. H 15 = p. 13C- r IBCriiC
1. 13(10 
Eurgo ex 1 6, 17 fit N. 13 - 2. 1(. 13e
 et = 4. 15. 14 + 13.,
et ex 1o...-.2a, (, 1604
11. 16.13
10) 16 + 211. 13e. 1it. la opus jam in gna–
a9y. 10, 19, 21, explicare un ee, per quel 2, tum itl ps 18 b, 14, 1, 12, 13
+abX =
Tabe=
1 2
e
i
e