# Leibniz Legible — https://leibnizlegible.com
# Piece: https://leibnizlegible.com/api/records/38650/text
# Catalogue record: 38650 — [De aequationum resolutione: Aequationum resolutio generalis]
# Incipit: Aequationum resolutio generalis in eo consistit, ut inveniantur earum
# Date: 1675-07
# Akademie-Ausgabe: AA VII,2 N. 034
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=38650&katnr=Cc2+1004%2FE+%28tlw.%29
# 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. 203–206 (from the shelfmark LH 35, 14, 1 Bl. 203-206.235-236): 8 pages, canvases 405–412, page ids 00068201:0405 to 00068201:0412
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068201&page=405
# Lines: 783 recognised on 8 of 8 pages, mean confidence 0.76
# 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 203r — 00068201:0405
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 204v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000405.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 91 lines, status machine
generalis
Aequationum resolutio, in eo consistit, ut inveniantur earum radices. Quae ut generales esse possint,
debent esse irrationales. Radicem irrationalem  aequationis quadraticae facil et du
statim patuere. Primus ad altiora gradum struxit scipio Ferreus, vel quisquis is est, qui regulas
Cardani invenit, quibus aequationes Cubicae resolvuntur. Sit aequatio cubica
2 - 22 = r10. seu Z H 9Z Fr. Sonatur ZB m + n. erit Z3 p 3mo 7 + n
quam formulam comparando priori, erit  mapi 1. et i + n BY. nH. d. et m + q eY
27m
sive m - rm + = B = - 9, et a B Z + sF. 9. jam ap r- n ego n B = - sF, 2
et
in lineis ex hiberi non possunt, tuncem q p , tentandum an eadem servata suppositione, Zmn
alia excitari possit aequatio similis datae, sed in qua non nisi mn,  extra terminum e ultimum
et in termino, non alia potentia, quam haectuae m. n3,  (quae simue esse debent) inveniantur
Quem in finem aequatio  Z. n Fo multiplicetur per  formulam sive quantitatem
sequentem x +co m Z + quo facto produdum, utique etiam aequatio erit, etsi multipligats
+ con 
A.
+ B
+ d’m
+ Sn
+  mn
non ponatur esse aequatio. Notandum autem in formula quam
elegis quod ea semper n et n, eodem tractentur hoc modo
+(2 et, quod ita et omnes varietates in a formula possibiles
exhibeantur; et canti. adhibila sit ne nimis alte exurgere
Z2 3 maz + n Quod qua ratione
effecturi simus, patebit in progressu operationes, facta ergo multiplicatione, habebimus:
Z
=B M -Omen 1 sed corrigenda jam haec aequatio
= B m =Omne
Bmn=Cum
emn - nn
2omn
= B  x H
n
2x
est, modo X explicando
per man, modo contra
pro men, aut ejus potistatibus ponendo
X aut ejus potestates.
sed video omnia destini
adeoque nihil hini posse duci
ergo sic fiat
m+n + emn. fiet: +m + 3mngm + 3ema in ( 3Enn in H Z
En hp C temn
E 3emne ehitam
3cdmn
bem   cma 
-35 mn
ciniia
adeoque poneno ZF,m cmn erit Z p dema Z + 3ma Z
conferenda datae:
E e p Ze
= 3emn.
4
rema m
toc henc
+ n3
+ 23
 cEn
=3enin2
renm C)
HHII
erit. Cma s E. et _ .3 Cnn e - P. et mm , f - 9. uli si 9p F eit na
quantitas negativa, et aequatio ractabilis Alia ergo via ad regulam Cardam redibimus
Nec credo aliud predibit si ponamus imn + b. loco emn. sed et polaramus simpliciter facere
Zn
c. Bonendo C pro cnn. vel omm + b.- sed haec methodus praeclare servit
pro altioribus, ut pro surdesolido facere poterituus: n  b1E  Cm n (+ L)
videamus et an non ad aliquid sallem sorvore possit multiplicatio, ut si in Z= in p70. ducamus
com Z- em
ema
 Bn
-dmn
ex duabus posthrioribus fiet: Z com
ririgen
malum quod ante, quod scilicet, explircatione desmendi causa facta.
destruentur omnia, A Res ergo non procedit cur ducuntur in se
invicem valores diversi, sed cum idem valor cubatur, tunc enim non evenici
unquam nen in Z. quod omnin tuibat
possit productum, ultra id scilicet quod est in 
Omn
-con
2comu
X + com X + B- com
150
+Co m X
1se + 1. g
m n+ adra
3mn X
sit ergo

## Folio 203v — 00068201:0406
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 204r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000406.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 93 lines, status machine
Sit ergo Z F +m + bmn + cmn
erit Z m + 5mn + 1omn +em n + 5mn + n
bina bmncn 1obminemn tobmne t ebmne n +n
5e m bmn 10 m  bmn
 cmin
1min 2bmacm
3mn emn
Subit ergo animum alia suppositio:
10a2b3 mn 1om
10 mbnn
Sed video c mn esse inutilem, quia nunquam ab altera  cum
fnctu separatur.
Ziam bmu n cmm
EM Faetum in
10ab nn 1onin
habm
n + n + b pZ nnB co mb B ~
Zr man + b2c (edom +bc)
erit , nn c. et bn3
et bn do
m + snn + om +0mn +5mnt +n e m omn-smn cd
b + 5 Btc 1 10b2 410b23 +5b e + c 1 B sbc v - 5b D
5 m 2 1
+mnc
Erit in universum ZY)
5 m  14
n 4bC
10 m  b 10 m = b3
non X - 5cov X
3mn bc
3mn e
explicanda autem necessario
2mn 3bc
n2 3b
Smn Co - 5nn Cd
5 bc v - - b V
quantitas CoD, Ilem ex duobus istis paribus  m. n. et b. c.
eligsendum est unum, quod reservetur Termino ultimo, eligemus exempli causa menfco
et ope reliquorum terminorum, quaeremus V. bc. mn.
( - cmn V.
D + 5mu3V.
5mn3r n
G 2 = smasvrr.
remhi v
 su v
5mn c03
-omn 3V, C0 - emazr (co 
V omniver. 8
-5 mn (Co
1cr 2
C0v3
aerm F bnn Jo tium
3Co v7 H
5xsu 
_ 5V 1c
- 1002 Z3
enn 3bmncnn
n 3bmn emn t
. =rmu v Z2
tort Z
ba cnir nB
quod fmihi, fateor nonnihit displicet. Nec v tamen victo qua ratione
et fiet: 21
54Z en c023 (5V 10L
rvo
Ubi bc. manet quantitas mus arbitraria, quoniam in ultirum terminum repiitur
imn cs
id fieri possit melius. Quoniam vero bc manet arbitraria, poterit poni pimn.
1 - 5mn cC0
trniepe- X
tip u
+ 5V) CB2
45Dpo. (.
+nireus coed
Aua
tar . enoe
Et colligendo in unum 
(Et Srenso co
qua polintea erta
cuilibe
signorum formae accommodari potest.
5VZ4 +smn
2 =sm
10btn. 20 t is
=15 ma VZ
mono) qio Cdea Z
 morne
m suq + sum
.5musr V,
mado cole
masi
2 B
eutque Z

## Folio 204r — 00068201:0407
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 203v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000407.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 102 lines, status machine
s jum fecissemus non Z co+ V sed _ ct Z, fiet.
1 50Z + emn Z -ienu vZ - emin qu  + m Sit jam aequatio data
erit ve P mne q
44 15, pq Y Y seu -3pq er
++ v+
Donamus jam in aequatione data etiam superesse
+ 15mnv
_ 7.
22+Z+S +t
-69 +p S sive p ( 125 + 7
525) 625125
arbitrarias quandam, facitas explicando, ut si initio ae inaegrita
fusset: Calculi autem contrahendi causa sic agere utile erit
Ultile autem videtur explicare Z et facere ZH PX +  f. fiet:
X + 5fX  + 10fX  105 2 + 5 fexf
- 5v - 50 4f - 5vbf - 51 4f3
emn. - smn2f
+15 mnn
Facie sit jam aequalu data,
et habebimus:
-5mns f
remnvis
+5mn
-  f
sunf..
+ienavs..
+5ninf..
X asqu X rrgX 48 X + pgt X
5 = b5
=1emnp3
2I
+stb. tasbus .. 3rbgs. 2969. + pbg
410tb. +65bg.. 3r69.
rast.. 45b9.
+ 10+b3 v. 5t b4
conferendo Terminos terminis, erit ex secundis; vi 5f c0 - 59 - rtb
ex tertis:  VH 10f c - rmn c, - r9 - 4sb9- 1otl quos duos valores confecendo
(20) fcoe
erit 1mnt (10 F C - rga - 45b9 - 10+ 1, -(20s co + 4sgto + 20+l. fco
c
Rursus ex terminis quartis Vr 493 + 3rbg2 +65b9 + 45 b4 10t b3- 10fco etemnsco
Quem valorem si rursus aeques valori primo ipsius V, habebis novum valorem ipsius
15mr = 30f2∞
mn, nemne mn H 4g7 + 3r b9 +65b29 +45b + 10t b, = 10 fc0, +30f  - 6f0s9 - 30fl tb
= 30059 - 15c0 +b
aequando duos valores ipsius mn. aliam rursus rabebimus aequationem, sed calculus videtur nimis polixus reddi
hit  et Z m  bc  + d FC H X. BZ+ h. fet
et ZB m er Zt Aemun Z3 ismavZ smo h
Ergu 25 n
irnmv
+Gvahh ahati
 poti ti
-Guatisti lL
timnyliti
+smn3 h,
=sR+tZ
5MZX  X.
1.  
-5u6ti 3
=5V4t Zh, 2tZ
5t X - 1kZ X
a 15  d 5de l - eae 
-Avate  Z 
+svatistit. h
-54sHd
Et olligendo in summam, fieb, goniva
X ex X- Zovh X + 3ot X 2ovhDX m bd
6t2.
1imah: 
- 15mnv..
+ 10h3.. +5V
sit jam aequatio data t.
explicata explicatione minori
Xr  10b 2X3 10b3 X
Apb 6p82..
398.
in
=Gcanzhah
+ismuvti
=rmnoh Z
Lsmnyti 2
+rmastiat
+ 15mox ah Z
Bmnv Z
+ 5tich. 23
Hanc aequationem comparemus
cum  data sed explicata
ita ut sit ZH = +d. tantum
Ponendelx loco ut alia plaga
ex qualibet c quatiope surdesolida auterendi
termingisi pares, tantum ibi posueramus X
pro Z. et contra: utile autem invenii
calculi contrahendi causa, ut fiat. d 70.
scheda deteram, ubi modum ostenderam
5mu 73
m u
+ 5th2 X
-58 2tk.
5Vb+p 5h. 5ma,, 1062 4p b. +9, + 20h5b r ohip.- rohst
3045b, 30t p - 3olish, = 15, 5b10b2-155b abp.- 15, 169, 155b20h56 rot
=155b 20hp + 155b 20h tr -15, 5b6t
1 10 b3 +6pb + 39b + 7

## Folio 204v — 00068201:0408
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 203r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000408.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 104 lines, status machine
Sedquoniam sic V. et h. distincte tractavimus, iio videamus et pumum v ipsi
Co junximus, postea h. superuddidimus, ideo sarius erit aequali ime tractare R. et Y ut videamus quis futuus
sit eventus. Sit ergo Xs in + x + b +c + d R
X n+ sum co emno
B 1 5bc v - 5 be p
 (+rdeti -sdER
L. r h C -eti 0
Y5Ca N -5id W X
51CX2
.
et habebine X5 GWX  + E maX
ec
smn Co
2vcoZ
3tizue X
-,  X
- 5 0 0 X
200.
=, 5 mus X
=10.. - 15 nn B..
+ 10103..
X3 - 5∞X4 = smn X3 +remu D X2
+ 10cd . - 1013
Si faciamus mapibe de fiet:
X5 Y
et
-5 vetr W
70
+1ymuv 
15mnvh
=1smio
= 51 coX
ncor.
+5ma co3
30C0Z
hZX
h. p co + cd
+5c 20dk
1emn 
+15muh) X
+ moosco e.
mnk3
temnirvk stico
Y5 mA3WWOX
+ 5mn3ti X
sni net co
= 5to 2t0 X
+ F colo.
5mnh
b5 d’C
5det3 - 5d et
SBh3-ENh 
- 5 mn h3= 1mnvh
+ 5 m n h
= Emn 3V 10 Z
=emi hZX
etc
50 X
tiemnt
= 15 nn 252X
+51 2 cdA.
=-10l.
+ismuva
rrennto 
= 15 mad loz
x +dm + e
XL + bm X
+cX
4, 1o+ X
+ismnirh
+ 10E2X
Smn .. . - 15mn V..
om. +db m + de m
eX + ob m + ec
c
emn X
X + abm X X + bum
e
cn
lmn
ex -bcc.
60.
x + ec
bc M X + bma X
X +X+ en +f1
n xre rsn
X + Gglim
X- imn -m
X3 + tmX -EmnX.Etmam
vn
+cm X
+ em X
=2emn.
E -ema X
 in
Emn
bisi pouas non explice, id enim sumia destruit, sed dastrua, superflus
faciendo CB1. Be. et fpp et 9ti temn e un. restnentur
pemque et peri X + d N 15 nt  
n unde reapse tantum fieret,
7 = 3mu X A
1
x
 X dem

## Folio 205r — 00068201:0409
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 206v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000409.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 107 lines, status machine
X = im 10.
sit b X + cm X + e m
siteer
bX3 + cm X
sit h X + lm
X-rmn -m pC
X3 + lm X - 2muh. X
+ β..=hmi.
2lma B
- 1m3
= 2mul m
Componantur hae duae aequationes, et ut superfluae m+n
vel mnt  destruantur, fiat:
c - b  + l r 0.
-f = β 1c
adeoque fH d. β H - d
Adeoque ex compositis aequationibus proditit:
_X Ac emB X - e m 
= l m3 sBo. vel
(+ 2lmud
almad
C - c ∞hr0
f- d10
=emn - 9 = -lmn Y0
bp et1
eti eh
X x - mn X
Sie X ( -3 ma X
Dr) unn
- e m3
et, +1
E) n n
Quare si id agatur ut summae exporestatibus  destruantur, non alia aequatio prodibit
quam quam quae fuisset si simpliciter cubum quaesivistemus ab X, es Nam si Xp in +n
erit X hmn X +
X e m + n + b.
Adeoque:
prosis
150m b
-27 mub
e eut X H +m
3mn X, suc X p in
(5 mn m+r, +b3 +31Y
n + b + m +3MX
hut Hi gnib nex
X m
- 5  (Bb X= Z (+emn3 b2. X
∞5nbEN
Jmub X
 +embin
(+enb3m
+15 mub X
- smn Bl - 5 mub 12
mb 73
b 13)
Calculus hic fit justo prolixior, et
ut arbitror sine fructu
3mX m . Imub
b 13mbin  inba
+ 3bX +3ma X
mub Error 3b
emnn ivmul 1bex 7 embm
Emb n 63  anx
Sub Rn 21z mub - 3n X
 ma X3
Emn
 
s interea memorabile est esse: on m + mbrm, + nbe H
ar  gmM X = 9mub, si XB mAn +b.
X  2 tdum i, +3ab X H m 2nb n + 3Bm X p vi + 3mb in + 3 In X
componantur inunum, fietque (3pi Astun n +  x 2 soenbX
+Sab n
G. 3mab mb im
re ien
ditm
In M X
n (x3
Xip m smn d niri a B
B HaBX  ca b X
Etror B6X 
XH hmn a + vit
N 1 3bX - 3b
FXbX
a +p
-3 mnb.
Anc patet theoremata morabiliu
c. g hmn a +3m B + 3nbY 
et r mna
+27 mub H 3mtr X
5mbB3 H 5ma X3
Snby
nE:
Gd-2m bn
um X
+ X
-10 mn b3
+ 150 in nb M
terende umn
romnbr X
Inter regulas artis hun
ut non ipsi eligamus sed
umnaturae ipsi relinquamm
exitum; unde caele
paribus tres calculi sunt
gungendi
Znl X. si Xp in + n + b
X H n

## Folio 205v — 00068201:0410
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 206r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000410.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 103 lines, status machine
imut 3nb.
arm+n. ap
2ma. a p,   3mm D
a m remin a = snin a
XS m cnn a -smina. +5aBX - 5a b X  n 5 nt β - 1 n B β3, + 5 m BX - 5 m p. X, m 5mbY 2 5mb.Y, + 5 nV X - snY X
ivc
3mS (imnaes) (yLeon m En cub n
3XF
I. 3n5 5nb (33..
3b1 (emb (73)
Emum
3nb n
5mb m 1 C) - 1mn b - 15 n. b m = 15 m b n
Embm
=15 mnb mn.
a G (srun n +smu n tabn + 5 mbB + 5m bmi +5 mbV. ) B sn m
Seo haec persequi mutile videtur.
(initro hujus pagine)
Cterum quomiam video ex destructionibus aequationum compositarum, in quibus n +n.i quantum li cet valian,
tur, nos ad veras formulas pervinire; ideo videndum est a. non aliquo modo,  per de dlestructiones
pervenre liceat ad formulam hujus modi, X C X X X b x X m. Ita scil cet,
ut M +nF solam sit in termino ultimo nullaque alia potestas ab. m. n
dione sit
altimo et penultimo, indistincte, modo in termino pultimo non excedant b+ et.
non reperiantur  nisi ma, et bc.
sit fX2 + 9d X + p a
+de
In caeteris brminis
et quomam prolixior foret calculus in sto gradu, experiamur
 +nc
3X + y d
2nC
X- m o-2de
Yd X2 Bm, x. 7.
Eas duas aequationes
mmn. Y mud+
 de
nn X
X2 m
XI i
XH  m 2mn
2nb
tde:
=dm
n27
Ebc.
mntile sd inuele hoc quiduet ad imposidie
g - f + 170. h - f + BB
e
 - 4 - a, = pe n0
- 9 - eA
=pmn -2bc v mu
Unde 91ln S(D) vn f-g. De fn
sges
Yp-hvis 29) (D (+3) - 2f + 2h r. f +h r
Ve - 2H + +f - 2.99 bc n - 2f + 2g, mn  4f + 4h. nn..
e 
Y n61
F.g h. Nam f. 4. 1. J. A
sequatio ergo produdae ert: 2fX3 X 29b X. videtu necesserii aiquis admipus erve biu at, rcuntur
r 2p - (2 (+9) +n
N - B - f + h - 2f + 29
Ergo p H g - f. seoita fiet B - f unde quae
 Aegonco3
t.
aequationes componi non poterunt, quin destruatur 3, quodesse tordeset
b+CH p
6’+ ex 9.
S m= faciemus b3 + 5 + 3bc _b - 3bd7 9. eritque,
in se bc, hoc modo Bc haberi posse pare lametsi
3 itoque vidiamus quid producat destructio sola aggregatorum potesta- G continent be bc.  tin ingrediatur, habebitur
tum, a m. n. erit g - f + Vpo P -g . B10 + -g- h - 2β70
-prmn  -2bc t - v - 37mu - 2 ybC. , - = q e - r b 2
25
2.03
cubica quidem sed nonpura pro varu, tamen casibus.
9- l70
qu ha3
F. 1
A 
i. 2
re 
264 6
Aem
9 - 17 0. S. 10  t-9 - 2311 0
Z ude BPHE3=f
Componamus inter se
urav
et videamus quid
per destructiones
obtineatur
er te (O)
assptnh q.
PHG. en noh
9n eps. rn. t PB- +
omn - 2bc+ - v0
+r - f p rn f verfun + 2bcf
erit 9pf. hef PHf. 4B0. l10. spo
P P- F +b- 7 +19
unde tp R- 2f + 9
solae ergo in explicatae restant
. b +c n s0 8+ 3b  1 Unde patet si  non contineat
id b

## Folio 206r — 00068201:0411
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 205v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000411.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 93 lines, status machine
im+ n +bm + b n
nn bmpo) rotiride +) 3de X
an 
q X n ib m, 3b mn 1
3miri. X A 3mub n3 + 3 Men b
X XH
XPom + Cme.
sive LF  telem  
ii +b m
X  b n 3bmo m tscmen n H 3bcm X +olcbm 
Sed ex hoc jam calcula illud video oriri
malum quod hoc modo, superior gradus
is
talm 
emut m et
3bm 3bcb mn m obccm obnum
bemnbm bmbmum bc ob ium
Sed ex hoc calculo illud video oriri malum, quod hoc modo in superiore grada  oriuntur termn
+
qui neque depri poss retineri possunt, nec in inferisrem rejici, ut in X, habebitur mnit
mi cu
ubif. ubi lerminus m retirendus est termini autem m, m. rejiciendi sunt, sed non possunt rejici in ulti
proxime inferiorem seu in X, quio in eo repericntur pit no a erc scilicet non nisi termini pares.  si fecisse mu
XB b m CMiZ + d. habrissems quidem et rmins pars mpaos, sed qui non ecessissent m8)
in eo non exceditur m.
Nec in X3, quia
Idem est si faciamus  XB  ero Num in cubo exempli qatia 5.
A m retinendam et m, quae nec retineri deber, nec rejici potest, quia in terminius inferioribus non reperitur in 3.
sit X bm + em. X p, Fa
b m, + 2bmn
e 
tiberim
b i 3bma in semum (+ 3bc mi X 3bc b m nbeb nin m + 3bcc m +3 VCanm
X_ b3 3bcb + 36d e1
.3 13  
sive XH 63 + 3bc X
X3 Br + q X
9- 3cd
c+ 3d
4
27C + 27, 3cdC 42783
et erit novissima aequatio: 4- 9cd9 2,d = 27ed), +278, 19) ed + , 3ed c apeid E  reme oi
ivc 63 - 5b 5 b + 5b d X - 5 b d X
lo e
Ecd 43
-5 b42 b
- 5ed c
+ 5tc.. 1
+54 b...
ebd bi cd
5 c
X27 53b, 353b0 H + 39 de d + 3f9
ononiam satis  agnosio aequationes omnes  hoc mlore XB mn cplicari non posse, idem est antem
si sufponas X B C in +Y. ( tantum enim rudices angentur et multiplicantur) ideo proximum est, ut (pro Zomito gradu)
elevemus grudum fsus X in suppositione, faciendo XH.  +b  X + d n  quam multiglicabimus
Abyne agmi X bpyi X
gem
hom
cp
X + om X + lm= L
+ Pin
X
bin 2bm + 2bc m + 2bcmm m
+2e mn
ero resolvendis qualnor
terminis aequationis
surdesolodae datae,
tot. enim supererum
secundo ulique subtato
in qua aequatione ad pam.
superfluenda tollenda opus
est quatuorderim aequationibus
Subsidi ariis. Habentur autem et
quatuordecim indeteminatae
exm Quibus tamen adjici Poterunt adhuc duae, non
plures, si aequation inventae addatur ex alia,
ex quadrato ipsius D. in formulam assanti,
titium X + cd m. Et ita supererunt qua–
m nabitrari  g. 
+ h.
dum
btni
alm X
+r m
Xr n
b +C +
X + b m X+ idm X
per aliam formulam cubici grudus, e
00.
AY 1
et mt
etm
agin

## Folio 206v — 00068201:0412
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 205r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000412.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 90 lines, status machine
Et hac methoous etiam alibi videtur usum, ad inquiiendum an aequationes quaedam per quasdam alias  dividi possint; sit:
vig. X + pXS + qX + rX + SX + +X + V. po. sit aequatio assumtitia
Nempe quaertur an aequatio haec _ sit divisililis per aliam X + bX +c.. po
Ducatur haec informulam quantum licet varialam, quarti gradus, ut producatur sexti gradus
aequatio quae eris D. Haec aequatio non statim comparetur datae, ut faciunt
Cartesius, Huddenius, aliique  sed suppositione eadem retenta X + bX + cppo. erit et
quadratum ejus p. quod misus in formulum assumtiliam ducatur, quae non eris nisi quadratua
unde producetur aequatio O quae priori addi potest, quia eadem iutrobique apposatio
est, et summa etiam erit aequatio, in qua eruut duae supernumerariae. Nec scio
eu non conducat ipsas prio e smmos lorminos per aliquid multiplicari; item tolaihamque
aequationem multplicari ante adbitionem. Quo facto adhuc plures utique supermmmerariae
Eiberentur. Si quaeratur an aequatio prposita datu dividi possit per
simplicem aequationem X + b 10. Tunc ejusmodi aequationes particulares
x + bX +c multiplicetur per
per X4 + d X + ex + fX + 9
fiet: X +d XF CX+ fX gX
b. bd. be. bf. bgX
c. cd. ce. f. 9
Quit porro quadentum assintitiae
uquatuor habebuntur, quae componi possunt; ubi quaeritur non tentum an singulae
Xt +2bX3 +2CX
b2. +2bcX, C
X 14
X6 +2bX 2CX4 2bcX E
h. 2bh. 2ct. abeh. Etit
l. 2v8. ab vul
multiplicari possint usilitirs sed et an arte compositionem, sed et a non componi possunt u
duae ac dinae et ine multiplicari et rursus aliae duae ac duae, et summa diam multiplicari; et; an non se variae earundem fieri possunt
Combinationes; quo facto haberentur plurimae arbitrariae; sed hoc credo nimium. Inferea notandum etiam posse non subi eo
duas cigmodi aequationes (vel unam cum facta ex duabus) ita componi negative, ita, ut utrobique evanescet terminus sube summus
sive aetimus, sive alius quatiscunque; multipliando vel dividendo redditus pris utrobique idem. Nobum ista profecto multis speculationeus
Campum aperire videntur.
XE + pX + qX FrY0
1b . +bc. +bs H0.
X + b1 0  XE+ CxF d Y, X + CX EdX
X + 2bX + b 170 X + e ⊓ x + 2b + B2X  po. summa harum duarum aequationum X + E X 4 d X + bd Bo
+e.. 72b + et
quam comparando datae, eut b  2p- - e
fier Cp 2p - 3b - e
derg - p pb +  ( ct) (, 6) - lebe
3+ 1b..
c. + et
Unde ex Term. 3tiofiui: 29b = pb2 + 1b3 4eb - ab + et erer sui b  e b + qb o
si b Undesser vioetur posse tolli terminum ultimum;
faciendo ers der, et fiec b2 + ser R + 9, 50. et b 4 13, b2 (+ 22 (, 2 rap 12r+ 4 ter _ap + ∞p +a- + 9
et b sir +  +.Tir. rpsor- 4 ser +np r0 pr +- 29
uaemmu
Hujy aequationis cum ujus terninus sit pure artitrarius nempe sicundus videsi saltem haberi posse solitio stura
Toleatuo cum ejus terrmnus secundus, fer fuceido bet 3= E - P, fiet: y3 + + e apt Y + etc
X ApX 2 + GX + Y Bo XEy = B
Xn 9 a 3y +3P y - p
2 X d Xt
cujus tolli potent terminus ultimus ope
ipsius e, faciendo,
 e - p - 1 1 Y39
ep +
sive eH 2P + 2 + 2 139. Quod semper possibile, quia in equatione ibica
plena ipsa q. semper intelligi potest reddita esse affirmativa
sed vereor e error subsit, et e destruatuur.
CH 2P - 3b -c.
d H. N 2X3 + 2p X F 2qX + 2YB0
+ eb
et de 29, - 2pb + (3)b (+ cb, ( b) - be
et  ultima aquatio erit, 24b - 2pb + 2b2 (b e, + eb) r. ubi sublata e reditur ad
asquationem prorsus datae coincidentem nec actum est quicquam, sit enim b - pb + qb - r .
miac YC B.  m+ n _ c. erit  up V6 b. et fet. e + 10b n c.
et fiet: m = cm + p E - 10b. et mp t O C - sBb. et ng c - sc - 19 b - c
1= = SC - 1Ob- sd spiendum si talo quid proponereturque non fore opus nolilia singularumin, dato enum
sumum et rectongulo semper habebuntur eat omnium siocestitum paria, licet non incalculum adhibitis
Ein un ubet
cs
ih
X EpX = 2GX ErX. AX et
Lp, 2p - 3b -h
anipe
CH29 - 2pb +3B (+B, 3C ( bd - 2th sed satus angura ob similem calculo
h. destrnctum iri. Non est ergo quod ab hac compositione arbitrarias Seremu:
Repetaruus: D. destructo 9. fier. X + d 24 +C Xh + f X +b
modraticun
X_ fFo. Ponamu
b. et d esss irrationales, et bB .d Eodem modo sit e rrationalis, dque Nempe esto.
bri dsh dp  gst
en lTha erique lsH fgrh et cep el sh. et debet
esse f9 + cl po sive lH =f9 et hoc modu nulta erit irrationalis quadratica in producta
Et fiet. X F. - Errorem video, res ita non procedit. exus est pluibus irrationalit . Recte b. et D. Eodem modo C et e. invicem relationem habere
intelliaentum erit autem  alia irrationalis, quae ducta in priotem dat rationalem. Opfi. Sed restat f quod turbat. deber et esse rationalis. debet
e+ c. esse rationais. Debet bs + ca esse rationalis. sim si summa duprum radicum quadralica rum binomiorum quadraticorum rationalis, tunc si etium
Aliquot jam
plaqu 
factum rationae