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# Catalogue record: 58009 — Inquisitiones de [resolutionibus aequationum] cubicarum intractabilium [II]
# Incipit: Sit aequatio y³ * ahy + a²l ⊓ o. multiplicetur per
# Date: 1675-03
# Akademie-Ausgabe: AA VII,2 N. 018
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=58009&katnr=Cc2+1004%2FA
# 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. 111–112 (from the shelfmark LH 35, 14, 1 Bl. 111-112.227-228): 4 pages, canvases 221–224, page ids 00068201:0221 to 00068201:0224
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068201&page=221
# Lines: 382 recognised on 4 of 4 pages, mean confidence 0.77
# 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.
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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 111r — 00068201:0221
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 112v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000221.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 109 lines, status machine
in quisitione, de Resolutione Tubicarum intractabilium
sit aequatio  X alieus a ael 10 multiplicetur per y + P fiet: V + ab y + ae l 
Trast. + pal 5s
Pro y ponendo m+n. et dividendo aequationem per 3, vel minennari, et taglocando,  atque o addendo simul atque adimendo mo
fier: mermueu 4P men +ah , - al
Cunque alterutra exipsis m aut n, sumi possit m n
Fal
+ p’al
m+ rmn +n
pro artitio, ideo valoris ejus explicandi carsa
poterit aequatio divelli induas, ex quibus una sit narmn+n  + p al. sve m =n  (+) P s + al sive
in armuen
erit m = n e (+) ps + at. et pro Y = al ponendo 1 compendii causa, fiet: m -n  C+) p7 +gL. sed
ponendo literas ipsas p et L. ambigni valoris, omissis compendii causa signis, fiet m- no ) pL
Residua sive altera equationis pars est: 4mn +p Emtn + at  s ael
. ubi pro -a l + p h ponendo da
aequationemque to multilicando per m -ni, et in hujus locum, ubi remanet, subtituendo m ejus valorem pL. fiet
+mnpL + pL men + pLat   ad m.n sive Me +p L
=aea n - p L at ponc m -a d + p1
compendi carsa scribatur: m Fs D n + H Et A poundo au m p . fuei te  apL n + P
vn +7
hic 2DF,  = sive -q L hp - P + (at d) a 1 ret p B + P L
+)e
Ponendo b +c B ex. fict:
videndum an tirens cubus a
a2 14
= 2a L2p h
aL + p h. ne p p +D .p - aet po vnc in ponndo: = pes: R 4+D -hadi ((+ B
X m-  agnm m n
ex aequ. 5. X
sive X b2,c +2bC.
Aequatio data _
b + 36
+abb
 pra   a L q0. mulicima pe P etdinbendo pe aL fiet: R   ha R 
h37 27a8. Ergo Eh2 p 27 L. Nihil ergo hinc novi.
sit bs d + sac
et cri f +rag
ert bcri df +d sag + saeeus + fhac
et b r d2 + id sac (ac
et ct B s2+ 1f sag C ag
detet esse 0
et bc p d’f +  2df sac  d sag 2d saecus
et  sd 2fd sag +s sac + 2f saecg
+ag ubi rursus 2fd tag f tae
sed ne ninis multae eveniant
2df sae e. rag  eg
explicationis satius erit ponere: b d + h sac
et cH f + d sac.
ent B e d + adhsac +h ae
cF f2 + 2fg sac +gt ae
bc r) fd +fhtae
+gd vac +gh de
Unde ut tollantur quantitates irrationales,
ponemus: 2dh +2fg + fh + gd r.
et 2dhfa + dg’ a + heg + 2fgda + frha + geh 70.
erim dei s st
compendii causa rectius ita:
multiplicetur per.
sit jam aequatiodata Z H ab Z +a lo
pounio Z F +x fiet X + M dex + a laerae
Unde comparando orientur aequationes: f e 2bc: et a he e ada - ab - eae
et 4 ad b 2bC ac ergo ex profimi antecedente aeqoatione fier 1 a he p sa d ab  423
Restat aequatio le H -2bC - ed uti pro d. substituendo ejus valorem 2aC+ 2b
tiei al e p, bc = 2che (Ecb2)
Habemus aeqvati  duus: Bene notanda est daec operatio. Nam ponere bp fC.
non est determinare, et tann ita una incognitarum tollitur, itaque supernum eraria et nulla:
etsi videatur: sunt aequationes duae:
su
ram des ate + 26
sa et dr = 2eC. Ergorhea +2ba6c
vel X X
aced
X9+ bcX - bc
- b213 X bc boc = s
vel X X . l X - bc b Radices
4 rmn m
f sac
bC,, d7 F dhtaess
adda ae  ae
2tqd sac
A h sac +2fgh ac
 ae qe sac
=  c9 - frha =g en
Zhfa +gra + 2fga
+ad B
ac
+ adX  adb
+fsae. = fbsat
iberae - et ae
-c ac
X3 + X
 60 X
m’n X
X X -ah X + a l , 0.
necepario magor qu et quadiatie sent a abc b
26016
24b2c3
27  b 2
 ab. a  1d
72+ 1b + ub a 3 a s sit p
hve (+b +ac
26 (005)
vel acubus ab bc - LxC   27 1 .bc
BoC ep 20b
 ue iuice
et ponendo b fe. fiet 2ha + 2af  - bC
item 3al = Bsc - 2FH 1 o ubi pro 8 feponento  6C3 - Bhac
iod 2 a S. 13aC 

## Folio 111v — 00068201:0222
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 112r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000222.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 99 lines, status machine
Oidendum esst an radice cubica acedente aliquid fieri possit constat si bC4. c L b + bc multi-
plicentur per b -c fieri B - c2. Itaque sit:
aequatio multipluanda per X +hs Abatac
a3ce
+2sd sae  15) abracsae
-a dae
X s ab ressae X
+d tae
+ h Seab
=l 52 ac sae
-h d’ac sae
sit jan si sin duce radices cubica unia ex aliqua quantitate, altera ex ejus quadrato
tunc cubus earum radice cubica ceret Ubiue, quadracum ejus nusquam ine s +  By. ejus
cubus ent .aetey   qae sac Py), + 3ae sae ( (197 + tactac y Error 10
b +c p. 3 + 36c +3bc2 + c3
b7 4 ct.  u+ 7 13. c et L. +   3
erit 91 3 - . et u  9 - 6 +  ergo 69 - +  ) H
Eive.  6r = 5 Y + 1. sive x r - 5 Y + 25
quod in numeris fieri non potest.
4, 36
binonium iabenire ex radicibus, quae cubice multiplicatae carcant radice cubica, sint nomina
b+ b c7. fici:
multphieanda per:
YX - ssy X
fiei X _-sse X C sN)X
sin  X +35+7 N X +s’r
- 139377
+ + 7..
+s5
F par
+Sot 
13.) abedidu
sit iam aequatio data Z  ah Z + ael po
+8sV
+ Sxt 7
+ C256
F So +
+ S3t
(- + 7 (2514)
- + 3 73
1) a  n 2, a a + 3Y at
a a haer raarer
_C sap + 3 sap + 4
E ap sap +
(X +ab B X + Y. Ergo
(∞ ab  (4 + 2rX + v. et fiet:
X = fiet r2 ab - 2X -2X X ral
et repab - 2X - 2X 1X +ab. adeoque
Sxitab  ab 2X- X IXrab +X
C5 (+ 7s5FD urat
Fiat Z 7 2 6. fiet X  + v su aliX +   L B. erit 3s t Z e v h. punendo ses p va ves fiet n A  ar
fiet xco +Zppha.. et + 7 r vta. et collatis terminis ultimis, fiet: Co N- N h a  val N
sibe S - 27 a l a 5e + 27, 27a 12
adeoque sch = 27 a1
1 27, 27 ao l + 4c 27, +, at2
1f27, 27 a 13.54 h2a
adeoque s⊓1 B =Sia 1   h a3 + 47 a.
co s02
sed calculum cum tanti sit momenti, explicatis jam literis resumere operae pretium erit
Nempe
= c0 e
+aha.
co2 +
alti
+ o ox
(sit jam aequatio data Z3 X abiZ +ael B
or fiet Xa X ah Xa + ael 5 et musti-
plicata aequatione per  fiet. X X Nh X +,0. quam
aequationem conferendo priori -  X NH X + cor
habebimus aequationem collatitiam unicam
nempe
NN7
ati. adeoque
03
7294 4, 27 a h
Ubi ad determinationem scipionis Ferrei nihil tale suspi-
cantes redivimus. Videamus an liceat evitare,
2740
17a6 = a4, scilicet
er diveundo: XL n mra 216 n. adcom  S2 27 al 7 cos
altera aequationis paus residua: m  co A6
15 27a0
N Cos
ut i inie.
17a  a  sive fiet:
adeoque  729 a
+ a 
Me -rampa rH 27a  h3
Adesque M a + a
 
  ab rratetae
hd sae  16) ab + ac vac
Muctipuletur per
in
 aelioa
+o hcoo
= cose at
 nati
Nh X

## Folio 112r — 00068201:0223
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 111v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000223.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 88 lines, status machine
14 a) C a
+ 729a
C1
adeoque m-
( B L4oB a Vt + 7 29 a 
vel m  (108 a lh 4729a  + 24h3 + 27 a 
Aequatio ergo reddita quidem est planct sed iterum desideratur ut sit 108aPh + 729 at  quantitas affirmativa
vel 4 13 + 27 a l major nihilo. Adeoque cum h est quantitas negativa, debet cubus tientis ab h minor quadrato
semissis à l.
videamus an aliter divellere licet.
Residurm a - m  3 l pi
n _il 1 367 et nce a li
n B nC ans - at
Porro hoc interim ex colculo praesenti duciuus ut liceat curusque cubicae aequationis tres invenie radices
cum regulae cardam, dent non nisi unam
alter:
X - c 5 x + co a
+ aha
= cop a
ti
  a
+e i
Unde X 5. co NV X 5+ c aet
-coN X + e
Sac. Qui est uns valor,
t.copex k. Ergo X -idne  rah p f  C at ese haltrae h X
3c0a2
tco
Tres ergo habemus ipsius X autorus nempe:
2+3a2h 3 - 5 aeh co = 6aehao et 249 a t2 fier XB + co N han f V.3c03 , gati aliia
+ 9a h
4, 9co a
Habemus ergo tres valores ipsis X, nempe:
+Sec2r a 44, v7tia 127aet
si ponamus X q + Vap, tunc duae orientur
2, 3 cd a2
ubi ob signum ambiguum duo habentur valores ipsus
x qui sunt impossibiles ambo, quandocunque h. est
quantitas affirmativa
X co s hah 4 f - 1c2 A = - 2, 9 ali - 6 a4 
2, 3 co a2
X C, cos -hat - S a2s - 3, 9, a h = 5 a hcor 
unde ultim. catore erit: ran    hi firal fupra ratia
3ah - 10.
4, 9,0a
aequationes: una Co N (3ah)  3acoq + raco sap + 3at
2, 3, c0 a2
3ah = cor. ubi patet tertiam radicem esse duabus
3a00
prioribus aequalem,
sit X. sit aliunde Xe 9+Y sap.
vel 4  39457 2gu c + rraco sap , - co N.
et X 27ac aeg pagertaxr agiap  raptax -  N
H.27, v7ael 12, a7 h a (7 27a l Y 27,27 a l = 4, 273à 
hegt n rap e = 174 t. et
et cubardo: C N 1 27acX +27 aloIX Zah + 27 al Xa h  (A A)
(Y et CbN  co s ura t (+ 17a h3) neoque AB Z  X H7a CoX sah Y 27 aX a h  
3a3X + 3ah
3abXFahid er pauloante N p dae co X + 3ah
sed haedur Herae non nisi unius sunt arbitrariae, nam
si pro co ponas  vanc X ovanescit
Nunc ergo ad prera destinata redeundum est. Aequatio collatitia supra inventa erat unicas nempe, a s  a a
vel pro co s ponendo Z fiet p3 27a l3 + 27 a h. sit jum atC fac irit. st a rap esit rples
sive +2, 92  1p + 3a vac,  ne tac. et s B de  c
acvae sac
et 3 a nes hac raeptac trasare + bn e hapa bu cueae t ap act
actue ae p 
sit jam + _ B  13) aep tar sac et cut Z ap Fartac et Z7 ra s + rapr vae
et fict aequatio: ap + 2a er sae H 27 a a p + 27a lartae + 27 a t3.
quae commode divelletur
Quae aequatio commode divicetur
in duas, nempe naprsae  27 a larsad, sive p 1 27l.
Ninc Ergo altera aequatio erit: a 27l + a re ⊓ 27, 27 a  + 27 a h3. sed ita reditur ad diterminatio-
27a l a P + 27 a larsae + 27 a l 9 a + 27 ah,
e
yd
a tratpar  2apq, +arae + rataiqu qud est)
ar ae sae traraqae  27 a la pae +27 ae laraee +27 al qa ac sae 127 ah’ae sac qylnde
rursus faciendae aequationes duae nempe: p a2aprae + aeraer +9a ae  27 aelgaeae +27 a hae. Altera erit:
.Fiet  ex posterore, ae p 245p4-27 a2 1ap quae quantitas debet esse
aic
adaque ab e vane scente
arl prvoat
E+72941
adeoqe in .a
et con 1-
2, 2ha
X  + 36 5

## Folio 112v — 00068201:0224
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 111r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000224.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 86 lines, status machine
sit (0) dub ras ac. Quaritur ejus rodix. Sonatur esse: aP + Y sae + q. vel 
vel asae + rde sae +4e. unde cubando: ta ae tae 1hep ae qea ap sac qe a  que ea
Equante omnia:
+aep a 13apqae ac
+raxis
naprer
+ upr
ponendo ap +re e na fer: n + 3gen ebr0.
et et n + 4’e - ea 0 eritque er ng et e, nq
Cuumque valorem ipsius e inserendo aequationi praecedenti fiet:  +39ng - nt g+
et fiet nri a 2a, 44 + Dj a B
abq aba a9 196
Repetendus est calculus
a-2a9 4
nn ab rabest
gelibus 8
1) an sac + age.  a  a hge sac +randeage  an ae die + aqe  ab + a tae
et divellendo atque ordinando:
n + 2gC n -  Yo. itemque 3n ag + q et  ab
ex posteriore eio er) yriaq ct X 9a
et fier n
ab -43
. quos valores inserendo aequationi priori feb:
a+ b - 24b93 +q6
à b (2ab4 49
+9b4 
ponendo scilicet cea vel c sae esse cognitam,
TX + 9aq n -9aq n
1 a b 17469-390
9aq
ab-43
a +b- 2ab9 19
arbitraria. ideo, ut En fieri possit, sic
procedendum:
ab + detac et an sae ilge enaesaer inde qronasaeriqe raie
et divellendo ace   n’a + naq
tir. n ab  ent ae 34a aeqte
An ita ansue tagee  anaetae e na e  thnavaeaique a
Ta n ailae B
 ) s sit aequalio y + ah Y  a lro erit y B + E l + S: 12 =, 3 a 50 : 1 - +: p,  h
ergo y B i l  (d - Y) f 2 ( - 35, : l + S + 3 s0  1 - 34, + l - s
et fiet: v - =l p 3 s0 : 12- 0  s: 153 r s0 : l- 5 Eautem  (2. DB (= 1 - : 12 )  
Ergo fla V- 3 +e
Tronsponanus: y - 1B : t + D  s) Et - D cubando 3 - 2y : (+ D + 3y Se (: P. : 1)  1. . + LD
 - = l=D  a t 1 (D) siuis - 2γ Ergo 29t s0: 120 r2yS): 3 + 1D h
si sit g A ∞ +n. ponendo g. et n. esse radices cubicas, faciendo: GG. et Np 12. et Xp y 2. et vocundo N-GB H h. iet.
G⊓H +G2
fiet Y2 + 3HY - H Y + H. conferatuo cum aequatione enbica V + dV  d9B + 431 B. cujus rudices divispe per d. et quae angendo
jeas pro v. piucudo _ Y + e. fiet. 1ed +3C Y2 + 3C Y + e3 250.
erit He PC et HB  C+ dC  + 3C 4, 1
quem ipsius H vatorem comparando cumi e + de = 39C + 3l. tet : c + d  ge + d. vel
yae + ad s 279e +27d1. erisque e
it enim sunt termini penultimi; scilicet fiet:
3 Fir G2
C ea
1d + 17 , 324
42.
 144 4 543
13352 ad
5+872
FH 271 = a sive er d. Restat ullima aequatio, conferendi
ga = 279
+ + H G + 17 8 )
 ubi patet d esse imutilem. Nemus
eo redire ut constet an Radix extrahi
possit, et vereor ne res redeat ad deter-
minationem scipionis Ferrei speciem tamen habet diversi
Plane evensus: vel. Ros in numeris tentanda sit aequatio Albessi
t inZ + 1o ponc ZH v + 3 a friX a2da
seu y . y 
Ponendo ergo compendii cansa in praeserati
adet; +ned rnd
exemplo O20. et 91 - 13. quo plus quam opus est
FA) contra not grcedimus, fer. B00 + 180 =13- 10
 27, 272100 4 17 27 210+ 1702
y12c  ero d  4 uC 9724 e
d + 38 1 65
 es ponendo = H+. fiei D p 1317
Sed repetendus est calculus, cum videatur habere nescio quid
Si get 1126 foret 102
ves as e 
d 351
Singulare; illud certe, quod ea ratione videtur venitur ad radicem  aequationis cubicae 
20 + = - 3
secundo termino non sublato-
Nemmur