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# Piece: https://leibnizlegible.com/api/records/475/text
# Catalogue record: 475 — Hac ocasione affulget nova Methodus solvendi radices aequationum irrationales
# Incipit: sit x aeqv + vy + z + ω substituatur in aequatione data x5 + px³ etc.
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=475&katnr=58052
# Work: https://leibnizlegible.com/work/00068124
# Title: Leibniz-Handschriften zur Mathematik LH 35, 8, 30
# Folios: Bl. 33–34 (from the shelfmark LH 35, 8, 30 Bl. 33-34): 4 pages, canvases 65–68, page ids 00068124:0065 to 00068124:0068
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068124&page=65
# Lines: 190 recognised on 3 of 4 pages, mean confidence 0.80
# 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 33r — 00068124:0065
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 34v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068124/jpgs/default/00000065.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 82 lines, status machine
antur 8+ d equa d+c  1s eed pponatu equ
et puendo d+c+ao d0c rd seustacer c 
sed pamo
non licet au
d’aequ nis drC +2de aequ. m3 17 + 1- 2cd. seu daeaqu.  m
c-C- 2de - cd equ il+cd. faciamus d aequ fcC
fiet: cte-2fe ef d n ret
euesqu
1c
Datum cubum oblitri partiri in duos
d’+C3 aequ. hi3.
seu d-
seu d.
cubos.
haequ. m d+c. Ergo d + c- cd aequ m, d +cq ud
3- 2cm
1- m3 
4
elcet
=c - 2c m aequ. f
E 1- m3
Equ. 4 (2)  c. Eem + m31C
etrcm aqu. sC. m4 +C m) m 2C m + m
21 - m
4 = B M + 4 mo
tc + 2Cm aqu E Y () 1- Dem - 4(3) , - m
2 - dre
sed difficili. paulo est efficere ut -7 +16m B8m+  4m sit quadratque
la problema aliter aggrediemur, unde apparet multum
interm
2)
d+ C’ aequ. h. Ergo d’+C + 3dCd Hc aequ.  h2 + 3 de, dc
ponatur dac aequ. l. fiet. l’aequ h3+3 de dc. sive l - k aequ. 2 de dtc
fiat: d+C aequ. m 1- h eritue P+ h + hl aequ. 3dc m Nam.
nimirum in aequ. 5. pro d +c ponendo valorem ex aequ. 6. et utrobique
dividendo per l- h.. jam. per aequ. 6. est a aequ. m l - mh -c
quem valorem substituendo in aequ 7. fiet:
l+ h + hl aequ. 3m d - 3nch = 3mc, et ordinando.
(10)
P+ ti hC
s- (h -13 - el
N+I +lh
seul = h3
h +I R + lnsmetgnie aqu. l’ +olmer guic, +, 2ni le - 12 me
+ 3uc
(11)
quod rursus ponatur aequ. l + lcn + c21.
aequationes, una: ha lorvis aqu L + ca
Tende oriuntur duae
13)
L + 1h + 1 46mic + quie laqu.  + len + eri, seu h + mC aqu (1+ cn
F3nC
(14)
(12)2 12
et: (17 +6 lm C +9m C (4 12 m. 1d = 12 mC aequ. (17 + 4 len + 4c
divisisque omnibus per C, fiet: 18lm +9mtc - 12mc aego 4 ln + 4cn
supra autem habuimus: d+c aequ. ml - mhr. Ergo pro tribus
literis. d. c. C inveniendis, habemus tres aequationes.
Nempe per aequ. 13
sit: C aequ. het 2h. per 13. et C aequ. 4ln = 181m ex aequ. 1
n - 3m
et duos ipsius Cvalores aequando fit:
18)
9m + - 12m = 4n
laequ 2h, 9 m t - 12m - 4n
denique. d aequ. Emh, 9m - nm - and = mh 2h n - 3m. 4 - 18m2
n - 3m, 4n-j m
dere
n- 3m
(19)
Unde jum d’ye  esse aequ. h3, substitutis valoribus ipsarum d et C.
verum ad huc debet esse aequ. d+c per unde ex aequ. 6. fit:  aequ. ml- inh
q nh qui vator aequandus est valori reperto in aequ. 18. ubi
m- 1
patet primum destini h. datam quantitatem. ubi sx poneremus
2+ m - 1 vqu0 fieut, n - 3, 4n - 18 aequ. 0
35
id est.
(25)
ponendo e in aequ. 1 et n aequ. 3. vel n aequ. 9 35
erit problema solutum. Tantum videndum an numeri, d. c. fiant atigmatir
ed vndeo rem non procedere feres enim per 20. Claro ip 1g0. 0

## Folio 33v — 00068124:0066
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 34r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068124/jpgs/default/00000066.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 64 lines, status machine
An sic:
inXau
1
s
tenx +c
pbuteus alag +bage
inA
amap.  +lm -b
b5 qaquot aequ 6) 2Y
(15
baequ. - - baequ. - 2r2
+bs 1Z
+anx i larX7
+b9
eq 35) Bp
trn 
(10)
per 12.
per 1.
qup on pa saequpn
(16)
(17)
ex aequ. 15 et 16 fit: Y aequ. 
a X = 8y
m mix
imX + n y + pZ +9Xy arX Z+ 5y Zt XAvy. 1Z + a
dmX tdn V top Z +dqH) Acr X L ilsyZ +dt X +edhijs 4 duo Z t duo
+aWZ
tetau reda d ica
mX in G + 1Z +gX) + rXZ + 5yZ
+end ten )+73
+bm X) +bng +bgu
+amX taqus
-ubru:
maequ. ru
superesse efficiendum:
am ee
a X + by  cZ
GXyZ + erXZ +C yZa
+br Ay +bp yZ2
tasayze tapxs.
+brin
quod est impossibile in numeris
-ber
+bY 19ue) c aqu. -bp
be
aequs
 aequ. rn seu
2Yn
aequ.
Sen. naego
pap
per. 1
ausu br
Nimirum fit:
Demedium hoc esto: ubi ad aequationem 13 pervenimus,
scu bermus repertas aequationes: caequ. - br. per 1.9. a aequ. br
saglreae. Maqu. n aequ. 
9 aequ. + m aego - 
Y
saequ -  per 12. 13. paequ. 12
nX in V ret
aa 2b9762
+ ak 4by +Z + d

## Folio 34r — 00068124:0067
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 33v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068124/jpgs/default/00000067.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 44 lines, status machine
fac occasione ciffulget vera nova
ille hotus solvendi ratiees
aequationum ninationales.
sit Xlago a-
a sit X aequv. V + Z +co assumantur praeterea requationes
tes aliae, ut sint in summa a aequationes. Narum
quatuae equationum artitrariis suis instructarum unaquaeque multipliae
tur substituatur in aequatiane data X +pX et aequ. 0.
sunantur adhuc tres
pro his quatuor V. Y. Z. co sarmantur quatror aequationes
quae attolantur ad uigesimum gradum, ad eundemet addantur
in unum cod eundem extollatur Et aqualio data per
novas arbitiarias. com comparetur lum summa
pittrum. Sed video jam cum quaelibet litera peculiam
modo tractanda sit, non sefficere numerum arbitrariam
santus uique forte sit aut si sufficit calculus alte assurge
et eut horribiliter prolixus
+ yt aequ. 23.
Hl
 Z egal a X + y + v
celet
+
X + y + 2xy (X +y+ 3V (Xr9) NapraV + 
XB
2u
sille..
(3)
3 +_ Xy aequ b. X 1 aequ. C.
hac
ty Fry Xy
fiei. 3 b3c + 3vc + 3VC + V3 aqu. 0. fiat.
fiet fl3+ Yvcao fial. D erit 2c + v aequo.
Ergo vapqu. b2 - 3C seu (Xy) aqu. EX + BX y + 3y
ergo X +5y N + 25y aequ. 25 y - 121 aequ. 13 y
fiet:
X + Ey aequ. f13 y et:
X aequ. y FEF:Y13
vel X anqu. Y Y - 20 +  13 Xequ. 3 Y- 10+ 108
b 16
08
8
- 20 + 1208
-900 4 3, +00 F 3,20, 208
+208 1208

## Folio 34v — 00068124:0068
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 33r); this page is its left half.
# No recognised text on this page (recognized).