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# Catalogue record: 57005 — De aequatione cubica intractabili
# Incipit: Sit aequatio cubica intractabilis: v³ * - aqv + a²l ⊓ 0. Ponendo v ⊓
# Date: 1675-08
# Akademie-Ausgabe: AA VII,2 N. 042
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=57005&katnr=Cc2+1021
# 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. 281–282 (from the shelfmark LH 35, 14, 1 Bl. 281-282): 4 pages, canvases 561–564, page ids 00068201:0561 to 00068201:0564
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068201&page=561
# Lines: 328 recognised on 4 of 4 pages, mean confidence 0.84
# 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.
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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 281r — 00068201:0561
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 282v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000561.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 101 lines, status machine
Sit Aequatio cubica intractabilis: V -aq  + a l 1.
ea
Ponendo vH  =, fier: 
= ag ZX + a 10
scribemus: 
+pZ X - à l p0
quam compendum causa
Sin adhibeumus ab initio v = aqV +aeY. junmus V1Co +B
fier: 43 + 3b C0 + 3bC0 + 63
- aq. - aqb
hioc peropendum d + P Ed +aZ Cd. + a R
fier (6)
comparandus
cum
V + 3 CV + 3cV + e
+ P)
D
+2FCV + PC
+ag y + aqc
a a
y 46 muy +gmn y (- n
14qui utique major quam 279 - a F quadra Aebet ergo aequatiaii () termini
sive:
9
secundi quadrati pars quarta
aequari termino tertio:
= n
H-66 m En
Si velimus facea 34 Terminum etim
aq.
collatis terminis senudis
1 (2) C + 9 DC + 4 a 9
10P- 36 a2
P +Y 10 P - 36 a 2
9b agb e (B) agb ar
y7 +3CV + 3C y +C
+3b.. +6bc.
1312.
y(py 1. a
 hoc
+ N
vet (P) ( +a.. id est explicando (P et (2) fiet:
(9 664D + P
et F et 
Ergo jam explicando aequationem otam, inscrendo siticet in
locum ipsius C, ejus valorem inventum, fiet:
V. (3- P
+ SioF= 36a2
s + 10 V
- 36a2.
cujus terminum ultimum, i in summam colligamus,
habebimus - 
(220,
27  (10e)
=16) +3. a p.
3r10 -2
(19)11 2
Et loco aequationis 17vel6 ita scribemus:
Y + (10P - 360 QY (+ 10F y
2= 36a2.
Quam aequationem comparando cum
+7 P2 S..
-aq.. aequatione Y, erit
mM (2) 1- Sior- 36a 2 ex collutis na
fet 36+6bc +365 a9.
et pruo co ponendo V+C
nempe mn (Pa. ergo m n ()a
at idem m n et a (2). Emo (
aeZ
superest jam collatio  terminorum ultimorum, pro qua erit
66 mn
Cep
166  10 2F. 
3 6,6,6
(23) 11 2 107
3oae +Y..
9,6,6
C) d
noc: 13 + bs3, aq
sine e+t b, saq
+5bC
ubi notabile est
si fuisser 
tectius affinuatio
=ggc s7
id fieri
non potuisse
+ 3b5
 et haequidem ex
at idem in
3ptLV27
410
qoasa ciic
(4 PY 1)
id sou
acc~3
3t dicia
+ a R
, 10l- 36a Z
(25) ob H, mo
isp = 36a2

## Folio 281v — 00068201:0562
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 282r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000562.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 61 lines, status machine
Ipsa autem aequatio collatitia exterminis ultinis aequationum
nempt. 6. et et 7. vel 20 et T. erit:
no + n = 56 M13 - a (R) et pro n. et mn. substitiuendo
fiet, 12
valores inventos ex aequationibus 23. et 25
+ 112 10 p2
iba qu saer. n6 (7)
9, 6,6
P a (R)
- β3) 10  36a 2
quam per compendium ita enunti abimus
10 (28)
m. + ea M + eas A ea - 4 fa
et erit mo
=ea + ea - +favrace
29
Necesse est ergo ut C sit major quam 4fa, senes
vel ii ipsas seu habelitur determinatio
Habemus ergo determinationes duas, unam ex aequatione 16
2 (30)
quae est: 5P p 18 a 2.
alteram n e Bb) 4fa ex aequatione 29.
sive explicando e. et f. secundum aequationem 27.
C
et
pro 10 110 P - 36 a Q. ponendo (P) ut jam ante
feceramus quoniam scilicet (D) secundus terminus aequationis 6
coincide aequivalet ipsi ... secundo termino aequationis
coincidentis 20. ex determi 31. fiet.
117 11, (P)
4, 81, 36, 36
4, 11
9, 6, 6,
’a (R)
+a(2, (32)
+, R
1
131, 36,
ubi quidem quantitates T. AT. SI sunt affirmativae
sola quantitas 3 non est signi determinati. Patet
ante quantitatem Ante omnia autem videndum an
quantitas ) sit major quantitate TI. nempe sive an
sit  1 1L Y , hioc 11, 11 p 1, sive 11, 11 B 9.
6
est ergo T ad 1 ut 11, 11 ad 9. adeoque multo major
quod si ergo cstet B fieri possit quantitas affirmativa
(3q
(vel saltem 976, 6, 9, 18 aR
p3
haud dubie effecta erit determinatio 31. sive 32. Quorum
nentrum si  praestari possit, integra determinatio 32. in
calculum vocanda erit) et efficiendum ut 11, 110 - 8 (P5)
a (R) sit - 2, 11 (P3 R)
1, 6, 6 lt
9,36,36
Sed quoniam effectis
determinationibus 33, vel 34 habetur determinatio 32. ideo
brevissima
cum sit 33, quo ad calculum facillitua, ab ipsa incipiemus.
sive .h fie
vel (P F

## Folio 282r — 00068201:0563
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 281v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000563.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 72 lines, status machine
36)
Ut ergo sit (P)R) p 0. sive  (P(R) pr 0.
(nam
omnis determinatio per quantitatem affirmativam divisa non
un ideo mutatur:) binc tantum opus est, ut quantitates
P) et (A) eadem habeant signa ita ut una existente
affirmativa vel negativa, etiam altera sit ejusdem offectionis
Quod si ergo effici potest, ut aequatio 6 vel 20 habeat
omnes radices veras, et simul quadratum semissis secundi termini
aequale (Cartesius postulat id esse minus) Quantitati cognitae
tertii termini, nulla opus est alia determinatione, caeterae
enim omnes in hac includuntur. Nam si omnes rudiles verae
signa secundi et quarti termini coincidunt sunt enim-
ergo (8)(R) f1 0 unde sequitur determinatio 32. sive 31. Quantus
vero Tertius vero e penultimus terminus est affirmativus, ergo
10 P2 - 36 aZ quantitas affirmativa, ergo
quae est determinatio 30
5 P p 18aq.
Videndum primum en  sine calculo vi simplicis subtationis
secu aequalitatis inter semissem quadrati semudi, et quantitatem
terti in aequationibus intractabilibus id obtineatur:
y . 34y + 2ar. (0. ut sit jam intractabilis
debet esse: q  sive mutando determinationem
in aequationem: a q
38) 4
at
 e 95 ar + a +2
erq ar ar +a. et aequalio intractabilis generaliter
2 (40
ita enuntiari poterit: V X - a aY at V +2ar, po.
1)
=d4
vel quia aY  V q- a t aequatio generalis sic enuntiabitur:
C + - 3q W +Yq- a   tantum ponendo at minorem quam 4
quae aequatio ponatur esse eadem cum aequatione 4 erit  in
3
aequatione 4 H. P10
et 9 (44)
p-. 2, 4 Nam per quando autem
1 a
hoc signum. Q. semper designo ipsem motem affirmativam, adeoque
ffectilonio
per - designo signum reale cujusque quantitatis, quod ex signis
d
calculi non semper apparet. Ergo loco  ponendo y + c.
et explicando C per aequationem 16 quoniam Ppo fiet
r1  de
(45)
H1 +24
adeoqu loco aequationis
(+
siibemus.  f C1 . 16a 2 , S+ 1059  92913
atque ita explicando aequationem  habebimus loco aequationis 20
hanc:
9 16913 y + 279γ + 189 13 2 (46)
Ubi tertii termini signum tixum affirmatioum,
+ 2g-a 
secundi et quarti Terminorum signa arbitiaria suntcerte semper
io
o
eadem  idem tive diversum attribuamus signum ipsis duabus
radicibus 189 13. et 2Y 4. a quoniam ui si idem damus utrique
quod possumus, cessat jam tum difficultas; imo etsi diversum daremus
non ideo minus cessaret; cum 18, 9 +3. sit quantitas longe major
quam 2 Y a at, antoque erus cipum quod cum siguo emini scundinden ep, praevaleat
quantitas ipsa
est quadratica aut
alia parium
dimensionum,
ut cum est 34
loco ea, cessat
ambignitas)

## Folio 282v — 00068201:0564
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 281r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000564.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 94 lines, status machine
Jam ergo ut calculum de integro ordiamur,
data
Sit, aquatio
W X - 39 Cd +25 96 - at  0.
erit ea intractabilis per regulas Cardani, quoniam  a q
riente termini secundi 34  cTubus 4, major est quam qb - a+ +2
quadratus a q - a +, semisse termini uttimi, 2 Sq- aF
Ponatur jam Ce p Y + 29 13
-39
+3Sq. a F)
 =4N3, y +3, 4, 39V
y +6953
et fiet aequatio
+89.3 B3
- 34, 2953 (6 8)
+21q - a +
+189313 (49
+259 - aF +2S
ubi sed video, a me erratum esse, in calculo, debet treri
Aequatio data: Cd X - 9 q Cd  , 224 - a t H quae aequatio
est intractabilis per regulas Cardaniu nam cubus triens a 94
qole cognita termini ul penultim est 34 ejus cubus est
t2 surq aF)
Cd ei
(15)
y A - 2 9 13.
ut ante, fiet:
(3, 295, Y +3, 493, ) 892
99
+ 2794)
.92913
2 Jvig at
6qu 3
=ril 279. a
prtam sacisim potestates In qua aequatione si faciamus tantum, ut
duae radices, 13, et 1279- a  eodem signo sint
 ita iss
affectae, quod in nostra potestate est erit signum
termini secundi et ultrmi idem quod si ergo suam
molem exhibent, sie affirmativa intelliguntur habebimus
aequationem ex data factam, omnes radices habentem
veras, quia signa alternant; et talem ut a
3413. semisse termini secundi factum quadratum,
993 sive 279, aequetur ipsi 279 cognitae quanditati
terminipealtimi. Quod desi derabatur.
haec aequalio cum
fier ergo ex terminis ultimis:
m+ +(2243 ⊓ (6 qa 13) + 2aY 279- at
M12 + 160e9 s3 m + 64a962
2a279 - aFF2
et signa radicum s 
conferatur jam
hacl.  ay-omne +ominay -no 
C fiet mn B aq s3, et ne aq S3, et n6 2ra g
M 1 274606
mt
= 2796-a
(+ 16a4 r2rg- a F
1oZ abg
+279247
166
O3 aq() 13 nest
adeoque
=, 3, 16a9 (279- a F2
2449
quae aequatio est semper possibilis Ment C B3 4at79F sig 1 q6 - 1316,
et erit m. 6a 43. β3 = a Sirf. at. + sug
41 221
= 15, 16a9 527 96- a2 +. quae semper et postitili. 
tum a 274- a4+2 dimidio termini ultimi. Jam
faciamus
-0q0
L adeoque 9m n p 9a93 1 279 a
modoq sit masir gerim
quantitates
characterem portant
sunt formutae quae
affectionis ut
sunt
quae characterem.
portant ambiguitatis
et quarum signa pro
arbitrio, aut rei usu
atque exigentia variare
dicet, ut radices
quad raticae, vel
aliae parium dimensionum
Cterae omnes quantitates
sunt o in differentes
non portant affectionis
characterem.
ex terminis
secundis et
tertiis