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# Piece: https://leibnizlegible.com/api/records/57004/text
# Catalogue record: 57004 — De aequatione cubica resolvenda ope comparationum
# Incipit: x ⊓ √③√b + √③b.
# Date: 1675-08
# Akademie-Ausgabe: AA VII,2 N. 041
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=57004&katnr=Cc2+1018
# AA VII,2: its series at leibnizedition.de: https://www.leibnizedition.de/de/reihen/reihe-vii/
# Work: https://leibnizlegible.com/work/00068201
# Title: Leibniz-Handschriften zur Mathematik LH 35, 14, 1
# Folios: Bl. 107–108 (from the shelfmark LH 35, 14, 1 Bl. 107-108): 4 pages, canvases 213–216, page ids 00068201:0213 to 00068201:0216
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068201&page=213
# Lines: 277 recognised on 3 of 4 pages, mean confidence 0.81
# 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.
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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 107r — 00068201:0213
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 108v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000213.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 110 lines, status machine
X   10 arb + (3 b
XB +b + 3b X. adeoque X arb X - b e sb. et quadrando utrobique; fiet.
N - 6b X  - 2b X, +9b2 X + 6( X +b Y est autem b. quantitas
affirmativa:
Haec aequatio duas minimum habet radices reales, ob sb Sed matum in eo consistit
quod non nisi una habetur arbitiaria b.
An c An rectius ita C
XN 1O. LY + Sir + 4 + 10 + 1r + sirrus
XH ZS r + q 4q X sed adit determinatio cipionis Ferrei. Aliter.
X N 3 r
X r
: rq t
ult +......
X +ZY (-  =r + 90
+ r + t...
221 L 6 7 13
7
0.
i
4Sr 14, 2tir
1li 19
summae Ergo bE HC
Eoce notabilem formam duarum radicum d
quadatranum, quarum quadratum est numerus
Pro 24. compendi causo ponanus v. fiet aequatio
X2 - Y A0
Rectius ab initio ponemus: X + v B ..... et fiet X + 2XV +V A..
Serve adhuo rectius faciemus X ponenda, sed malum in eo praevideo quod ita duae radices fiunt
aequalis. Vi d eamus tamen pro V. = Y ponamus f. fiet aequatio: X - f 1.
Cx pro _ X ponendo Y + n. fier: _ + 2n y + nt B0
comparetur haec aequatio cum
aliqua cubica data,
+ m
+ ings + n y
m. rmm. m
mas
y + p y + aqy 4 aer pjo. fiet 2n +m pP unde paras aq. sue np P n
sed manifestum est eam comparationem esse impossibilem, nisi alteretur datu, et ejus loco
scribatur: y + 3d72 + 3Y
+d3
+apd. =pA2 B unde comparando:
Sed haec inutilia arbitror.
m 3d +p - 2n.
tar afp n + hdu + 2pn = 4n - 3d-2pd - ag.
as 3du +pu - in  - d - p d - aqd - ar
3d + p - in
Sed si abinitio posuissemus X +G Y3.... + Y6)...
habuissemus aequationem X 4 2g X + s  f. sive: X + 2gX + g ppo.
si a Ergo aequatio inter duos ipsius af valores,
Notanda haec
comprehensio
forma:
(343 ut 43
d
+
malliplicetur per X +m
= pd XE + eg N +g X
- aqd (
er
Am..
fa.
 id - 3d 3d = 2 pd 3d - aq 3d, + 9d 6 pu p (n 3d +. d 
n)
Lin
-2n
in
sed ita ascenditur ad aequationem Culicam
in 9
+4
non minus difficilem quam est datu olsi mas habeat in specrem
abitrarias:
videndum quid fiat si radix superior culetur: tiet: Quad retur potus X - 3g X . L Y +96. fet
12F
+ r
+ 29
+13
14fr
-39
compre
X = 6gX + 9g X2 - p Y Corparetur cum X 4. 3n X 4 + 3u V X - 3
-Eg data
= p . + 2p - pn2
fiet. q 7 3nat pa. et 27+ Epa 2n + 2pn +aq et fer n + : pn + Ap e taque agu
et fiet n et n  - 
post 5.4 uta q _ na  Pa + npae n
dug
216
+ et
+ 33
H
+ C- ag
21.
e
73a.
=ar
72 - 2pa (7a) (2p a) + L pa
+9a +ga aq
16
14
ai gueltis qudi sa raa guanus 5. si ete pe s. e age us pui  a e , ver 1 n i1 
scilicet: , + 3d  2 - 3d
d
+ag. +aqd
3d
4bc H 4
6dp
-12 dn,
C2n) ragd
en
-Fu

## Folio 107v — 00068201:0214
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 108r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000214.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 78 lines, status machine
si ponamus XB SB +Y + s  Y + 4 +  1B ex - l2  + 4
fier x Y 4 4 X, et X + q X H r. et X + 29 X  + qX - r
sit data aequalio X y  -12Y - 12 7. fac y e 7 4- 10. fier: Z 43oZ +200Z +- 1000
25 sed tunc aequatio data ad eam formam revocanda est
ut ocune, termini proter altimum
sit, an equatio data qualibet. sint affinuatibi, quod generaliter
y
difficile est.
= 12 Z +120
73 - 30Z 100Z. ogr
Crbicom
Melior ergo est ratio prior. Si sumamus tantum aequationem quadraticam
NA- bq X + = Y + 9 o. cujus utique habemus constractionem,
sit data N X - qu X
+ a h r.
reditur ad eande determi nationem nec alio opus ealenlo
Rectius: sumta aequatione X Y. 24 X +4 X - _ r
tollatur trminus secundus, ponendo X e Z. + 24 fiet:
(+319
22) 130297 et q
14 27
- 243 39
+q..
-39, 491
12q
1012
P92
 B. eit debet esse a h p 46P13 et quod si
verum est, mutata determinatio est, hoc enim semper fieri potest
en  6m a 15 m nt  15 m n 16
n + b mn + 15  n + 20 m3n2 + 15 n n
m 6 men + 15  n +2omn + 15 m nt +bmu + n6
e
+15 mn + Cit
erit
in
57 9. 1t 90
H
M+n., Z P
ns
6mt  9 mt + 15m n + 20
110
eur scipsi  pote
7 6mn 4 mY + 16 mn + Q n. vel
s neque 
sire
6mn X-gnen X
maner a quantitas sit jam equatio. X  
vel generaliter: X B ida X+
Apa..
-+aq X
3 d’a X=
-2pd a 
aq
Quae aequatio D. conferenda est cum aequatione
+
+ aY
+ 4343
2
0
quod ut fiac tantum opus
est ut sexta pars termini secundi quadrata, aequetur nona termini tertii
3da +  (, sive Dd + 6dp +P  3d + 2pd  + at q
5) 4
sive. 3d2 + 2pd + 4aq = p   sive d +2p d + +p  10 p - aq
Alterutrum ergo necessarium, est, vel ut sem tertius terminus aequationis cubicae propositae
ecima octavt
completae sit regatious, vel ut quana sais te quadrati a secundo sit major quinta
() m n 299, n rn + 16es mn
162
multiplicata una
66 comprehensionis
parte, et divisa altera
Nota character
Sive distinctus ab
aequatione, quando
scilicet aequatio iden
tica est.

## Folio 108r — 00068201:0215
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 107v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000215.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 89 lines, status machine
parte termini tertiis.
Ante omnia ergo hinc ducimus, omnes aequationes
Cubicas completas,  tertii sive penultimi termini negativi, hac methodo esse tractabiles
Item
l altim Dende hinc omnes in quibus decima  octava pars quadrati
a secundo est majoi  quinta termini tertii . Ergo tam si decima
octava pars major est, etiam decima quinta pars major erit. Tam decimam
quintam uniqus esse majorem quinta atterius est, sertiam unius esse attera majorem.
cubicam secundo termino carentem secundo term
Quo Ponamus jam datam esse aequationem (Regulis Cardani intractabilem, erit
ex ejus penultimus terminus negativus. Angeatur rvel padix minuatur radix parva
admodum quantitate, habesitur aequatio Cubica completa, in qua penultimus terminus
vadratum quantitatise
terminus manebit negativus, modo scilicet quantita qua aucta vel minuta est radix sit
tertia pux
minor quam quantitasis cognita penultimi termini
Sed est praeterea
opus, ut Terminus secundus et ultimus eodem sint affecti signo, qnec
refert affirmativo an negition quia no lebitu mutari possunt. Id autem in aequatione
cubica petimo termino carente, penultimum autem habente negativum,
secundo
qualis est omnis intractabilis, nullo negotio efficitur; si scilicet fint et itimus
negatious; et postea radices  ang non minuantur, sed augeantur aliqua
quantitate mpra cujus scilicet quadratum minus tirente x quadra) triente
termini penultimi aequationis datae. Itaque semper aequatio (ubica intractabilis
ad hanc redigi formam potest:
-V - y -a 1 quae
formu vel y +1 - y + a. qualis formula semper hac methodo tractari potest
His itaque positis sit aequalio Praeparata:
sit jam aequatio praepmntu  N  +X-  X +  ar
Compurundacum hac: X mnX. 9 enea X ( 
In altia loco Z jonumus (x et pro X e omn  - gniae +m
Ednn
n habebimus: X Lbm X - 9ME X (4) m
56 n n3
Unde comparando: fea  mn ex terminis primis secuntis. Et  omnu ap, sive p  ex terminorum
secundorum et tertiorum collationibus consentientibus. Porro hinc jam b6M H3 a
Superest novissima aequatio (+) m (+) n (+) 66 m n p faer
sive pro n, asubpi et nn sustituendo ratores erit:
sive m = (+) 1 a pr M6
32
(+) + - aer..
9
n + ae + (4) 1 a p 4 a
13136, m
36β), _
324
quae aequatio est semper possibilis, si modo signum literae p.
sit contrarium sigri literae Y aequationis praeparatae.
jam
ita enim nam 11, 11 a0
4, 327, 324,
-11 a pr +  ar- 
2, 324.
4 36, 36, 36
debet esse quantitas affirmativa, ut inde radiu inde extrahi possit, quae
dabit absolute valorem m. Quod semper fieri potest, si modo fieri possit,
ut in aequatione praeparata. Signa literarum p et r, seu terminorum
secundi et ultimi sint contraria.
Sed hoc semper videtur teri pessee salvis praeparationibus nostris.= sit
scilicet aequatio cubica, penultiro termino carcus, instructa secundo et quarto
signotum contrariorum, sit secundus negatious rtius af quartus attimu atioque
hoc enum emper fieri potest, si sit intractabilis.
cum absit g. it poterit
d semper haberi, et poterit fieri quantitas affirmativa, nempe 150 - 1 p. ( d.
Et 3d + pa  Sio p. Terminus aequationis praeparatae secundus.  ultimus ert
10 sio p3 + 10 p + a r. Y  rap - io Pd G hoc 10 5iop Atrop 4 aer
tivi 4100 4 10 sio P + sio a pr quantilas affirmativa sumtia serlicet literiu
p. a r. pro earum mote attirmativa. quod emper videtur fieri posse Ergo a100 s  10d1o P  sio de qugnttus afinutum
Het
eem n
et R, +(4) ac
()
929
42771 4
cubus a 36
1 problemata
ejusmodi difficilia
foren numerum
invenore, curo tres
pSurdesolidum,
quatu notar
cujus tres primae norae
coiicidant inter se
et unitate differant ab
ultima ) ut 7776
undesolidum a
Ergo 110. significat + ve
9, 36

## Folio 108v — 00068201:0216
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 107r); this page is its left half.
# No recognised text on this page (recognized).