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# Piece: https://leibnizlegible.com/api/records/40082/text
# Catalogue record: 40082 — De extractionibus radicum ex aeqvationibus, terminis discerptis
# Incipit: y³ ⊓ 2a³. sive y³-2a³ ⊓ 0. Pone y ⊓...
# Date: 1675-04 (75-04)
# Akademie-Ausgabe: AA VII,2 N. 021
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=40082&katnr=Cc2+938
# AA VII,2: its series at leibnizedition.de: https://www.leibnizedition.de/de/reihen/reihe-vii/
# Work: https://leibnizlegible.com/work/00068219
# Title: Leibniz-Handschriften zur Mathematik LH 35, 3 A 6
# Folios: Bl. 4–5 (from the shelfmark LH 35, 3 A 6 Bl. 4-5): 4 pages, canvases 7–10, page ids 00068219:0007 to 00068219:0010
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068219&page=7
# Lines: 273 recognised on 4 of 4 pages, mean confidence 0.78
# 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 4r — 00068219:0007
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 5v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068219/jpgs/default/00000007.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 78 lines, status machine
Hac methodo aequat
quadrato quadrati
a Ferrario
reducta
fieu votest
sneratior
n 4b
20
+3 lm
+ a Pm
G-2am
h
sit sequatio quadrato- quadratica generalis:
formula
+a e 
+af  unde extahatur q adix Quadratica:
P roductum
+ tatab- ea
eb
44 dbc = 1644b
v
+ cat 7 + 8a dbc
Residua debent poni aequalia nihilo, et fer
a
8b2 c
164 b6
8b a e- 4adbc + c3 a 
a
Iem 64b af = 16a d 6 + 8 a dbc- cta
er
errorlaul
7C
quae duae aequationes resolventur vel secundum a at prima,  et
plane,
tunc secunda necessario secundum f; vel prima secundum , secunda secundum fvel d
Itaque cum bisut infra patebit sint aliquando eaedem aequationes duae duobus
diversis modis resolvendae, hinc uno modo, prima resolvatur per  secunda per f.
Unde sequitur, ut
altera podo prima resolvetur per e, secunda per c.
ex Aequatione quadrato quadratica qualibet generaliter extrahi posse Radicem
mox ostendam
Quadraticam, methodo plana. Quod fateor non satis mirari me posse
nihil tamen habeo quod contradicam psab. Pro arbitrio sumi potest
sive (EI ram, W- p 1W =p aequ v - 2n P seu naego V um, 1.  p
2. P
sed quia daequ. d. hinc n aequ. (2) +- imph Ponamus jam W- p aequp
fiet et p datus
ve e fet C H .b n posito β esse munerum aliquem datum
ope aequationibus cubicae, cujus primus terminus pro arbitrio mutari potest
dalbitur P. et quidelu aequatione pli
ob B arbitrariam itat
trspetus
o
extractionibus Racdicum ex aequationibus, Terminis discerptis
April. 1675
y n2a. sive y - 2a370. Pone yZ + l. unde fiet:
h 73 + 3tel qa
a2
22
e
s
a
Do + l
- 20
+ 2al 71 4 3a1 q 5+ a
+
ti
-2ad
+ m
fiet: ad specam
Sed lhoc more
H 0. Multiplicetur hon aequatio per Z+ ms
74 + 34
m..
+32
+ 3alm..
+ db. ca
t

## Folio 4v — 00068219:0008
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 5r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068219/jpgs/default/00000008.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 78 lines, status machine
y4
2
dus exemplum, sit aequatio:. 173- sive ponerto c-g a) et f- m i 
+ ck + adZ + a eZ + af
Unde ex dutis 8aC +c3
sive 
Ae
4ac
d.
H ZF +9
V
Z
et 16d’a + et - 8ad
64 a2.
16 62a + 16ae4 +c fac
64 a
2
da+C
F+  64a 410 ec 3 + e6 + auc )64 8ap+0
12
18 a2CY 64a3
alculum ( resumamus. sit
2
+ e
ponamus ab initio d aequ. 0.
1263
Hb
p
+
aequ. 0
2
+
nt Z2 + 2H
sba
b
db
lsautum ergo opus fieri
Habemus en
summa
1Z
333
0
Z
ejus quadrabum
1692
L q
+ 16 2
eter
y +
86 2 64b
residuum: + esque +f6
sud eg4 3y3
+C L
8649 = c12
+C
64b
X + es, + fb
Habemus ergo aequationes. Ponamus aequationem 2 coincidere
Cum aequ. 3 et residua extractionis esse aequalia nihilo
ex aequ. 2. fiet haec: + + b 4 + c3 93
=i L = 2mn
onferenda cum aequatione 1. hinc b aequ. r+ m.
e aequ. 12n p. et f6aqu W + pE.
denique ob residuum, fiet.
10)
+e7 + 56(4)
= 2M p 392
Caequ. s+2mn. set 2rep - p
0
3(6) n+
16 (11)
e’aequ. - c. et F aequ.
36
qu. C
At- imp n
12
6466
Nqu. 0

## Folio 5r — 00068219:0009
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 4v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068219/jpgs/default/00000009.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 67 lines, status machine
Nunc ut valorem literarum radicem seu aequ. 3 ingredicubium
inveniamus, ita opinor commodissime procedemqu:
ope aequationum H1. et 11. tollamus C. fet. Nimorum
(6 (12) en
dividatur primum aequ. I1 per aequ. 1o. fiet:  aequ
seu
l
3
aequ C
(14) B
8b256 e5
Ergo c aequam
862
9 (16)
641)
1
5
s
8bI
( (15) 512(6f18
 et C aequ
C
15
17
512 bF18
sed per aequ. 10. est = C aequ. 8b e. Ergo per 15 erit 8b  aequ
20 (18)
15
vel e aequi b.f1d. jam loco b. e. f. substituantur ecrum
valores ex aequationibus: 5. 8. 9.  fiet.
8, 12 ((19)
(20)
16n p aequ r + m, (3) WW+p, vel _Yrn WAp, W + p aqu. 2n p
2 (21) 45
vel n aequ Vrim, m+ pe x W + p 
2β p3.
2l
L
Terravimus in praecedenti
folio cum initio hunc
calculum fageremu, ponendo
Y pro-
Faciamus W+ p β aequ 2p3. contrahendi caleuli causa, ipsa
litera B. numerum arbitrarium significante, et habebitur
valor ipsius p aequatione plana: et ex aequ. 2 1. fiet
2
13
n aequ
ex aequ 22.
restant adhuc duae aequationes
 (24)
Una: n aequ Y+ + 2mp3 per aequ. 7.
(16)
altera (3 aequ. ( + 2mn ex aequ 6. Jam per equ. et erat c’equ
e
5
et ex aqui pro b2 s et substituendo vatores, fit jam jam per aequo 14
erat C3 aequ. = 8 b256
e
, seu
=erm, uetp eqo Frmn.
2 y2 p3
sed ad hoc prestandum non satisnullae supersunt arbitrariaeqe nisi
omissis aequationibus 22. 23. redeamus ad aequ. 20. et 21.
sed facile conjicerelicet nihil
aliud ex hoc calculo
duci posse, quam quod jam notum est

## Folio 5v — 00068219:0010
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 4r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068219/jpgs/default/00000010.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 50 lines, status machine
e
dest
F2b22 +.
+
mZ + n Z + p
=2m (2
c
C
-Cd7 -q
+d
C
2
262 4b 86+ 64b
461 + cd
8b7
Hini si volumus succedere extractionem fiunt duae
aequationes. Una:
Hinc aqua ant residua sint nihilo aequalia, fiet:
4
8be- 4 d + c9 aequ. 0
ergo jam e aa
jam aliunde
b2aequ   N- m.
b 12146
et 64 b f- 16bd +8b dc-8 brg
chaequ s- 2mu d’aequ. t- 2mp n
13 aequ v - 2n p3. f aequ W = p6
sunt autem x. s. +. v. W. ter
cogniti aequationis datae:
Z + +s Z+ tZ1vZt1
thc in locum Compendii causa ponamus at aequ. 0.
aequi0
fiet: - C aequ. + 8b e, ( seu As4 mn aequ ir - m, ( v- 2r p3
et
C aequ 64 f
sgo (+2mn aequ
seu f-Emn. (4) aqu. 64 b6 fb
64, r- m 1, W- fo seu
Eg, r- m, v - 2n p3
ergo seu -C’e aequ 8f seu
F. Ergo a C aequ 512 be f2
Caequ. 8656
=5+2mN , v - 2N p aequ W - 
aequ 9bR), seu 9. ligu. 64 bf
seu bf aequ. e sive r  r-me, B W -p aequ ( al v - 2n p3
Ergo Eaquob
Z.
372
+ f aequ. t
et