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# Piece: https://leibnizlegible.com/api/records/36611/text
# Catalogue record: 36611 — De aequatione cubica y³ - 12y - 12 = 0
# Incipit: Sit aequatio: y³ * - 12y - 12
# Date: 1675-07
# Akademie-Ausgabe: AA VII,2 N. 040
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=36611&katnr=Cc2+1017
# 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. 277–278 (from the shelfmark LH 35, 14, 1 Bl. 277-278): 4 pages, canvases 553–556, page ids 00068201:0553 to 00068201:0556
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068201&page=553
# Lines: 310 recognised on 4 of 4 pages, mean confidence 0.79
# 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 277r — 00068201:0553
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 278v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000553.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 117 lines, status machine
 Ahaf6t1a
536
7d
cirqu
16d
409b
sit aequatio
Multiplicetur per
fet
V X - 1V
=, 12 H
+m
 X =1) =RV
im X - nm.
Jam ex regula Huddenii, si data sit aequatio rbica,
reduci poterit ad Cubicam hanc:
Nostra ergo num. 2. dabit cubicamihan:
Tollatur hujs aequationis productae Terminus
u
secundus ponendo Z H e = 4, et fiet
3
=3, 4 C +3, 16 C - 64
+12 . 412 2, + +12, 16
+ 36n  +36m- 4 (6)
= 12 m. -12m.
sive per
+2m3
(4 144m)
= 144
+D, 144m
0 compendium
= 14 4m
C+ -48 C
= 16
+2bm. +144m 0
-12m.
=96M
+ 12m
m Jam ut videamus an ope arbitrariae
144 42
+2i4am M aequatio proposita reddi possit
144m es tractabilis, cubus a triente termini
penultium et quadratum a semisse ultimi jungi debent, ut appareat;
an summa possit redii quantitas affirmativa.
Am + 12m - 16, 64m +3, 16m Enm 13, ∞4m 144m + 12, 12, 1L m
-16
2, 1216m +3, 144m, -16
+256 +3, 12M, 256
= 16, 16, 16
2 6m. a8m +m2M - 8,, 36m + 2, 6m  = a8m +48, 18M +2, +8 Ni 272m + 72, 12 M
-28m
+y2m
Et horum summam ordinando
=64 mo +3, 12, 16 m
+36
-----
+2, 72M, -8
+64
+
-(31212 M +2, 16, 12, 12 M - 12+ 25bm +3, 16, 12bm
+6).
C35 16, 16
9, 1..
+ 127 1   -27,..... - 6 
= 2,4,b 1 . +37....
= 8, 12.. +9,36,6..
Et sm subducta ratione;
=16, 16, 16
64, e
Hom +6, 16 7 M - 5b, 12 M - 36, 16, 7 M + 42, 12, 16M L
ubi notabile est omna singulos terminos dividi posse per 28, coefficientem
28, 144
altissimae potestatis. Quaeritur jam an aliqua valorem, ipsius m
arbitrarium, ita eligere possimus, ut tota haec summa sive formula
quantitatem constituat affirmativam vil sive nihilo majorem. 42, 1216
Quod ita  commode tentari posse videtur, pona
Ponatur: 6 (10 () m
syl
4(50 (16)
et mr f 1(46
62t 12, 9, 4
divisa per 28 
et summa ipsa destructis  destruendis, sive exenitis inter quae aequationem supposumus erit.
616 -2416, + 116 - 144 16 +288 + 146
et rursus dividendo per 6, habebimus:  16 446, s6. - 24 16 + 48 116
ubi necie est fucere fH, dividendo per 16, fiet 25 - 4. s6 (4 48 (6), 48 s 3615
+
i est uti formta proxine anteceitute
3
64
-+
4039
Quadeato quadusi 
1
+P, +9V+r) +s1)
-45Z
+ pr.
tit
12m.
=12m.
S
149)
144
7
t
72
JL23
1629226
3621646
73
+12Z
+3bm L
um.
36, 16, 49
F1M L144
424
L4M

## Folio 277v — 00068201:0554
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 278r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000554.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 86 lines, status machine
pt 3
o
quia ibe nono ilius radix occurrit quam s6 debetergo esse 48 1 p 4 146 +25
0, 4 4. 
Erit ergo
s B (6 44
minor
157
Y(46 major quam
100
adeoque majos quam
Si poneremus esse 2, fieret:
48, = H 423 +25
Sive 32 H 6 + 25 pr 31. Ca an
Sed ut exactis appo securius agamus, ponamus
esse
4815
1670
p
1010
 debet esse
Sisi
8 N 4C 8 + 25
sive 30 p 6= + 25. quod est falsum
ponamus ergo 48100  4 157 +25
157
100
 +4800,00 H 4, 157, 157 +25,15700
15 7 quod est falsum
2
785
s
Utarum ergo alia Hypothesi et
514 faciamus:
3925)6(16, 2) m t - 36, (16, 7) m - 28, 144 570
et 
suc m = 6m + 36
9 + 6
41 5
4() Codu
sa et m e _ 3 + Sis, et m 3
et mpi 3 t = 13 + sis
restat:
=m = 24 n + 288 M  quae summa deber
esse quantitas affirmatira, ponendo ergo n isse quantitatem
uum, seu , dividendo omnia per m, fiet:
m - 24m + 288, sive numerus 288
sive 2889  671,   Quod succedit, nam
S. si restitaemus m. fiet:
si
7252
288 +(3+M = A82e5
+(2
+ 2 64
424. 243f
+Nis
=9 +es
 +288
+ 288
16
(4,2) E2
sed nec hoc succedit
Sed quoniam reatumenta An forte licebid facene +28 m 1) 6, 16. E) m et m 24
e et m r + V24., fret: det et
E
=m - 6m +24m = 6 quantitas attirmativa adeoque ponendo: m - 124
fiet: 2424 - 144 - 24 (A 24 = 6. nec hoc sucedit.
et
24
a
s
ne
F9
96
56)
s
tive 2887
4
1t  3+ Sis (- 9
6515
+9 f6d
3
sed
major
mino
+3,4

## Folio 278r — 00068201:0555
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 277v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000555.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 54 lines, status machine
sed ne tentamentis inutilibus tempus teramus, generali inquisitione videndum
est, an id de quo agitur sit possibile; immirum mutata determinatione
in aequationem videndum an possibilis sit haec aequatio:
Mo X +24m - 24 m - ∞ 1C4 M, + 288m = 14 4- n
et ponendo m, et n. pro incognitis, videndum est an curva, ad
quam sit haec aequatio describi possit. Quod  item an  ad pan
tangens possit duci, sed Sed directius credo ad rem perveniemus si
in quiremus an haeq aequatio producta intelligi possit, ex duabus cubicis
vel ex una cubica, et a Quia omnis aequatio cubica est possibilis
ita autem sa stat aequatio: M X ∞24m +24m + 144m 288M + 144
sint duae aequationes Cubicae
in se invicem ducendae
Institutis comparationibus
fiet:
aq  P - ab - 24. extermins htius
exquatis -2
m + em tagm + ar 7.
mZ = p m tab M + aC B
m (+pn) +aqm + a M
arf paqe abp- ac + 24 ex quintus
p pad . pra
-pm - pt m. = pag... - paer M
a codonie
Habaq + acp- 144
+ n2 1BO
+ac... tacp.. + a caq. + ac ar
Oex quartis e quentis
adeoque  p - 24 p 2ab p - a cp + rp  ab p - a b- 24 ab + a c p - 144
aaq ab - 24. aer p ac + 24.
2sque ex terminis quintis
et ab + 24 ab + 144 10
ex terminis sextis
sive ab  = 12. felciter et (F 1249) - 12, 24  - ac ab -ac 2,, - 288
-12
ubi rursus evenit feliciter, ut sit utrobius -0D, a et sine hoc quidem
estat ex terminis
ultimis aequatio 1+4 + n  - ac + 24ac. quae aequatio
est impossibilis fieret enim c - 12 p ns - 2 Quod est impossibile.
Aequatio ergo proposita est impossibilis. adeoque nec quantitas supradicta
affirmativa reddi potest, quomodocunque explicetur quantitas m.  si
Nisi forte hoc non evenisset non destructo  p. et certe non
potest fieri ut aequatio data reapse sic produci nequeat, ex duabus aequationibus
cubicis secundo termino carentibus, restituta ergo p. resumtoque calcaalo supra
ubi signum O erit aC p, p4 - 2+ p - 2 ab p  +24 p - ab p  a b +  b
quo
t
et poncu
ex teminii 3tris
2 ex suartis
exquintis
- 2p +2, 24P = aC + 24
Et  aC + - 2p + 2, 24 p2  +24p +144 et fier ex sextis:
et ar r - 2p2 +2, 24 p +24 P. sed ita rursus o ex terminis sextis
madetur in aequationem sexti gradus, unde nihil discimus

## Folio 278v — 00068201:0556
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 277r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000556.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 53 lines, status machine
 -48e erc, aliqua ratione utiliter divid
possit in duas, ut an terminus ultimus fieri possit aequalis nihilo:
O sublato secundo ternus ino ponendo
in H n + 8. fiet
Videndum n aequalio
erit
I3 - 8m + um -  4
23, 8,5)8
+ 9, 121,8
A
+ 3
264
8 n=
n
010,0
64
G. M
p
08-100
1E,
F 12 n
1028
+ 12
B
6
N n X = 2d R1
81
4
s
222
2
Orvnetur tota aequatio terundum M, fiet: N 12M
e
B (e o C, = 8N
Ruusus attollatur aequatio: M
28
mutile
- 96 M5) + 144 MS + e25
=eis) +36CN 5 - 48 es 5)
16 N3 5t
-96m + 144 m + C
e.. +360.
248C
6
12m F = 96 m3 +144m +e in
re.
2360 43e.
d, 12.1 +d 296.
ne.
d, +144.
36C.
de
486