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# Piece: https://leibnizlegible.com/api/records/29064/text
# Catalogue record: 29064 — De extractionibus Radicum ex Aequat.
# Incipit: x aequ. a + b + c etc.
# Date: 1678-05-06 (78-04-26)
# Other printings: Knobloch, Kombinatorik 1976, S. 124-128
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=29064&katnr=56114
# Work: https://leibnizlegible.com/work/00068201
# Title: Leibniz-Handschriften zur Mathematik LH 35, 14, 1
# Folios: Bl. 37–38 (from the shelfmark LH 35, 14, 1 Bl. 37-38): 4 pages, canvases 73–76, page ids 00068201:0073 to 00068201:0076
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068201&page=73
# Lines: 159 recognised on 3 of 4 pages, mean confidence 0.83
# 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 37r — 00068201:0073
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 38v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000073.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 89 lines, status machine
de extractionibus Radicum ex Aequat.
X aequ. a + b + Ce M. L aoqu. ab +act be dc. L. aequ. ab
2
X’aqu. a + rab.. Ergo ab aequ. X  _ ab aqu. X- 2a X + a t 146)
et
5
4
4
et a b aeq. X- 4a X + 2a X +  4 a X
qua
e
iga
+a8bX
 16
vel quia X aequ. a erab.
+ 28
4 aX +2ae b+)
Faab
+4
fet X aequ. a +  + 4 a b + 4abc + 4 ab +8 abc
seu X aequ. a + 4ab + +6 a b + abc. ut alionde
Ergo possumus statum scula X8
est autem ab aequ. a b + 4 a bc + abC + 1 a b2
4
Inde aequatio 14b aqu. X 4a X +6a t - 4a, ∞
X aequ a + b + c.
a + b+ Ct aequ. 54.
+gab 12aY2
a+ C+
c4
+ a
+4aV
1E 24 at bCX = 4a bC
idu
- 24467
=12(1bC
X aequ . qX + rX + (+.
abc aqu. 93
quaeritur quantitas a+ ex datis q. r3 54
ro X subslituatur esu intr in aequ. 2. ejus valor ex aequ. 1.
(0) Y12 ex aequ. 4
utique a aqu. s4 b  ex aeq. 3 aequ et a aequ.
Ergo 5+ b
+ b C8 + r 12
144
locum In aequatione 2 substituatur valor ipsius X fiet.
(a + + ab +6 a b + (12 abe 2 a aqu. 9a + 24 ab (.  at s+)
ubi pro a substituatur ejus valor,  ex aequ. 4. fiet:
42 r92b +ab b +6 n 16L
2
Aqu. 9
12 Y
e
bc
b c + 6b
4, 12, Y,  + 4b b +6, 1 Y, bc by +6bc aequ.
o2
aequ. r2, qr bc +2, 12, 972b  2b + 29b=
X  X + lae X + n _ X + pa
Aursus:
d
c
+
+mab + p aeb
+ A abc
in X
+
a
3X3 +7qX HIa X + 1 a
43 ab + 1g ab in X2 - a
ab
+ 23 abc
Sed posterius inutile esse puto quae cujus posterioris multiplicationis productum addi posset priori;
sed jam re recta expensa, puto nos nihil  lucrari per posteriorem multplicationem, nam
multiplicare C in Ft -a, est multiplicare C, x- a in A -g. jum O,X-q potest
considerari ut species formulae D perinde ergo foret ac  si ac si loco D aliam formulam
suntem fiigeremus, mutatis tantum literis (. l. m.  etc. et productum puori producto
adderemus ita enim, nihil lucrabimus, nam si quid lucraremur, nimium mcraremur, possumq
enim similes alias formutas fingere infinitas (N. eae notabilesi ratiocinandi in tabib,
exemplum 3). Et perinde esset si ponamus loco in secunda formula loco D sumta) nos
sumsisse I. utique poterit C+ A assumi pro una cnuantitate. Sed aagendi
tamen arbitrarias uno succurrit pulcherrima gatio si ducamus  in X- g. et
productum addamm ad productum ex D in X-a. Et nusus aliam fingamue formulaque
39X +29 4X + B0 e in X - et denique 41X + 43a in x- aHis omnibus simil
aditis a habebinus utue ad 23 inbinarias idest multo plures quam sit opus e
20 fals 1678
2
abab
aequ.

## Folio 37v — 00068201:0074
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 38r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000074.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 57 lines, status machine
onamus ergo nos habere aequationem omnium, maxime arbitrariam
(+ N ex 24 X + 5 a X + 13 a+ X + (31 a) aquo
3 ab
+ 7 ab.. + 17 ab + 37 ab
+ n abe F 19 aeb
addatur. (- X) + + 92 X + r3X
23 a bc
q
+ 19 abed)
(+
pro a ponatur
+4 a35 (esqu. 0
+ 43 a
+ 47 a bed
+3
aqu. 0
Explicentur jam X. X. X. substituto earum valore, seu atbtCd eorumque potestatibus
Orietur inde fola aequatio ex formis ab a+ b+etd. Omnes autem
formae provenientes erunt tantum hae a. ab. ab. abc. aa a. a b. abc. a. ab. a.
Dubi illud malum animadverto quod priores habent arbitrariarum, plus satis: at
posteriores habent arbitrarias plane nullas. Unde dubitari potest, an postea
sibis mulliplicatio instituetur, ut toassurgatur ad ab et  ad ab
satis sit arbitrariarum illis tollendis. In lasu defectus remedium in prontu est
sadendo in D paginae superioris sic senbendo: X Ba X3 + la X etc.
+ mab
fa
praecedenlis,
in utoquoque termino repetendo formas praecedentium
nullas habeant nunc arbitrarias in mulliplicando tamen accipient producta
ab iis, quia producta ab aliis ope cognitarum a et abcd. ad producta, ab
pono quia poteriores priores plus satis habent
arbitrariacum, hirc illa additione  ad esD 
his deprimentur;
M. opus non puto, et sufficere solam formam (
Quoniam verojam in ipsa multiplicanda  formula
santum, numerum abitrariarum habemus, hinc nullis habemus opus
in multiplicante forma, ideo eam assumere poterimus simplicissimo,
ut scilicet tantum assurgamus ad ab et ad abc. reliquaque tollanus.
Fingamu curliba ix formis multiplicandis praefiram esse ubitra
tiam, id enim obtineri potest. Et multiplicentur omnia prtmum per
ab ab abc. deinde per (2 VEm
a 4bC. at abe a b. a bc. ab. ab a abc abc. a
Porus sir procedemuss abuna parte
forma (ex his tubus  ex tibus prioribus  ut atb vel a b. vel abc 
poterimus, per a Ebitrarias
N
ve
ullum alhum allatae hactenus aibttrariae
usum habere possunt ind possunt de quo
reliquis destructis
c a b + q ae +3 9 abf 69t abc + r a + 2r ab aequ. 0
Multiplranda per  a ab + abE sed deprehendo hoc modo nullas
susterfore arbitrainis sufficcentes ad tollendas sormas imitiles provenientes: ideo
necessaio  coginiuo helinquere radhuae arbitrarias superiores. et cogemur,
per multas cormulas simil cultiplicare, quod prolixum quidem facit calculu
non puto laduen ceritari posse.
agu. 0

## Folio 38r — 00068201:0075
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 37v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000075.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 13 lines, status machine
ab  a +6 X+ y aequ. a. Xy H B. Xe a - y. X aequ. b
Ergo ay - y aequb x X 1y +Z aequ. a. Xy + xZ + yZ aequ. b2. XyZ aeque
X aequ. a - y = Z,  X aequ. B =Y2. aequ. C
scibatur compendii causa, ytZu aequ. (). et yz aequ. b) fier N
a - (a) aequ. B  (7 equ.  seu a ()6) - (a)By aequ. b (6) = 12) aequ. C19)
Ex prima fiet: (b7 aequ a (a) + ) - b. ex secunda (b) aequ
adeoque habetur aequatio, (ubica
Xy + XZ + yZ aqu. b. et xyZ aequ. 8.
a -(a)
xy ag. V XZ, gZ. Xy aequ. b - xZ - yZ aequ. 3. Ergo 2
Ergo Xy aqu. bc - xC - yC vel x yt aqu.
Ergo Z aequ. bc3
Xy+ xenyC

## Folio 38v — 00068201:0076
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 37r); this page is its left half.
# No recognised text on this page (recognized).