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# Title: Leibniz-Handschriften zur Mathematik LH 35, 14, 1
# Folio 211r, page id 00068201:0421
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 Acquatio Cubica
proveniet aequatio alia
X ahX + ael H0. Sa si per quantitalem X + m multiplicetur.
X + mX -ah X- alim X + ael m p o. Nam si al ulp
Nam si a quod nihilo aequale est, per aliquid mustiplicetur, etiam productum nihilo aequale erit
Hanc aequationem ut resolvamus, comparare e re erit cum alia aequatione quadrato quatratica
Sed ad quadraticam reducibili, producta scilicet ex purgatione aequationis hujus:
X ( Hac +Vac + avat. sive X - Hac  Sac + a tas, sive (X -2 ac X + acppae + atas
sive X = 2Sac X + ac 1 a faf. et ruryus quadrando: X - 4 ac X ( aC X
onferantur jam duae aequationes qquadraticae, uma data
resolvenda X +m X etc. altem factitia resoluta X- 4ac X etc.
ut scilicet appareat, quomod. explicatis  per collationem
arbitrariis m. c. e. f. effici possit ut ostendatur eas coincidere,
+4 nc.. = Wac X
+gaera
ac proinde et ipsam X4+ mX prorsus ut ipsam X +Vac X etC. resolvi posse. Conserendo
Jam singulos terminos unius, cum singulis terminis respondentibus alterius erit aequatio collatitia
p ex cllis terminis secundis mata: m  ac Unde ex terminis tertiis pro ac substituendo
ejus valorem inventum fiet: 6m - 2ae p - ah. sive 6m + .6ah 32aC
sive ac 3m +Bah Quos valores quantitatum ac et ae inventos substituendo
in eorum locum in aequatione ex terminis quartis collatis nascitura, fiet:
Fah im + ael e C e em-(4) ac Hac + ae sac
et multiplicata aequatione per 16. fiet:
(=16 ahm) + 16a l ⊓ C m) + (m + ( h im.
m3 + 1 ah M - 8 ael 10.
un gati
ut quae est aequatio (ubica per
resolubilis, posito datam resolubilem non esse. Nam si peuultimus terminus cubicae
datae, qui est an X est negatias quantitatem cugnitam ati, habet negativam, posito
scilicet h esse quantitalem affirmativam,  seu numerum verum, tunc 12ah quantitas
cognita pemultimi termini imenton aequationis n ete invenda, erit affirmatio ac poinde
aeqratio ipsa resotubilis erit.
Superest ut ex terminis ultiis duarum aequationum Quadrato quadraticarum coincidentium
collatis quaeratur valor ipsius f. explicando scilicet ipsas ac et ae per valores
inventos, et ipsam m pro cognitaer sumendo, quoniam valor eju per aequationem cubicam
resolubilem  nullam aliam incognitam contimentem haberi potest.
af ac - 2 ac ae
5 Im r8al
Itaque fiet
+ aee
om +4al n x 6gati
et multiplicata aequatione per 256. fet seu 1621 6.
= aln
25b, a fH (m - 6m  -16 al ) + (9) mt + (+B) ah m +64 a tiv = 256 al
Adeo qu
af
m +gah m + 16ae h2,  = 64 ae lm
Qui valor ope aequationis cubicae m + 1ah m-gal po. Supra inventae ad huc simplicius
efferni potest: restituta enim aequatione:
et pro m ponendo ejus valorem: 8a l - 12ahm.
64a’ af  m m + Bah m + 16a h - 64 a lm)
Ba l- natim
fret equatio: 64 a af e, (8 a lm2ntim (+ 8ah m) + 16a k- m
sive:
16 a a f  = ah m + 4 a h - 14 a lm. Unde af
Unde a fpi ah m = 14 a  lm + 4 aeti
Quod si porro libeat ex valore iptius f. omnino eliminare vlmip ipsam, hoc ita poterimus: +9a fr94a
M 1 (4 + 14 9 lon + f 14 à lm + 4 a ti - 16 a f  8à l sta hm. unde fiet:
=14a lm  4 aeti m - 16a f m ( 8a lh - (13 a m. et M
et m’i
+ In + Ba f m (p + 14 a Im  34 t + 16a5. adeioque (+ alh)  Vastim ri 349 aem a
adeaque m p
at 56af l - 16 tel-. cujus valoris ope utique ral ipsis  ipsa m tolli potest ex valore
+afn 491
ipsius  af. sed quia ita ipsa f. age assurget ad Cubum. poterimus Ergo imvento
fupra ipsius at hoalbre esse contenti ex oppopta m inventa
X X -alX +ael
X X -alntaelm
puciam
X - ax fal
B rse
Bfiei L 27
FP. ianipp
scribendum potius:
cq uacpsatas
P, s1d
161 - 64 p p - 649
n+l
ina
N. Sac p- m
quod postea in calculo
reliquo non mutar–
dum, nec ponendum
sac H m
Adeoque m - tarm + 8a l 0.
si aequationem quadento quadrabiea
LEm X3CM. reduiere velis
ad cubicam fiei (Huddem. Tys
VfLy- ay-Im
caporuo
compendicam
scributum:
-3  hl
tm.-am h
+ hl. =tm
atm.  nd
tm. en
=um-+alm + ag
ti den agliaaluM
us
+ liagm Bhbalaun
= 3hibalo.. 64 213..
-3hat M - 3, tala m
aela.
+ag
ion +2k +X
cacquudiatu ab
13+ ha m- pM
3 af, V30
Cm6 -6It M 1, Calh  +- 6(3 fa
Oropez
F h+ .n - 21gah
e 2h.3afm
+6401e..
2, 8 Alh, 3afM
Ibatim e ael e eri-ond. pat
Enonca ergo reliqua omnia