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# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folio 135r, page id 00068195:0269
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068195&page=269
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Lavenantius quotutat qualuor
triz sunt faciliora. qualnor assurgunt
invenire qualaor numeros progressionis
Geometricae, sic ut summa quadralore
summa cuborum sit duta:
LeXG X y L
XZ= yy
X +s + Z b x+ yr BC
+Fya
sit xy+ XZ+ yZ = m. fiet: at 2m.
(9)
b+2 m = a. Ergo b + 2ya = aa
X3 + y + Z3 + 3Xy ∞3XZ + 3 5Z = a3
X)
x3
93
(11)
seu  C. + 3 Xy~X = 3XyZ = a
Z
seu C + 3am
093
 (n)
393 = a
seu c + 3aa y - 3y3= a
ex ex g et 10 inter se sunctis habetur  aequatio
pro y. vel si univis ex 8hraberi
potest aequatio pro quod optime fiet collendo
prigum a, fiet enius:
y LaC  b a: 2 - b: 2a, ex aey.9
un in aeq. 13. fit
a = C + 3 3a : 2 - 3 a b: 2
 - 3a: 8 + 3. 3 a b: 412a - 3. 3 ab: 2. 4 a 4. b: 8a
1b
seu 8 a =8c+ 12 a - 12 a b - 3 a + 92b- 9 ab +, b3
2
seu a 9a - ga b. a 9ta + 3b0
et inventa a facile habetur y per aeq 9
seu a-3b a +88c a - 9bb a + 3130
1N 729
o is
2d
648
problema duvena
5. 4x x
sent ra + a + a
Ergo V a N
 spril. b6ss
,X  aN
ax- 3aH,  F + 3axX
beE 108
tam idem  aN. ergo aequando
a N= 3a X, X  + 3axX -
1 Na
b.
E
seut cubus ab aV = X  X a 
seu
E 1 10 nane - a 
Quod si summam cuborum a
quatuor terminis. X. a x 4 X. a x
putes equalem dato a
tunc eadem orietur aequatio tantum
pro  substituendo a. et pro
X ponendo X, et pro aN ponendo
a B nempe.
cabus ab a B X a X aB
s
r3 B cubus ab a
eB aB
X3II a cub. ab et.
o sive
à B- X3 
Unde:
aB3 - 7a 8 X2 4 2 + 3a 8X 43 -x  px a B
sive  saB- aB a D neoque conferendo aeq. Z et E
sue areo avran f xaa )- a   a
 d. (BB- 2 av   4 2- a uZ- 0 a
EIta X + L X
a X à
a X + a X  a N. problema davenant
=b v + X+ isP = C. fiat v+xty+ Z =a et v exG=n
(3
fiat b + 2m =aa C+ 3am - 3  3n= a
L aa - b
3(107
fint c - 3ba 3n 4 3 da a. sent 2c - bn - 6ba + a -0
n exy + vygt vyz+ xyZ fint Cexs per 1. E4. () X gng.
x +Ky + xy +G = n
b (∞)
b
ryt 2 a aβ t. Pn. lin m
si suat quatuor R. y Z. v. X. y. Z ex v
sit V + X+ y +Z2
N Ay  = XX ~Xy + gya
uty
0
3
. -729 +, 648 = 729 + 81 = 0 cubi X + a x + a x + a)   a B ad lne 12 Apuil. 1695
probatio per calculum posito X-y- Z =1
G1b
inge
solvialibi adde ubi de homoeositotus ex. XaX
ad gradum nonum
tres
m = ya, (per 4. 5. 1.)
ton
M
X. Y
P y
py
vvare a
n  et virav
 Ve a give: C)
at
X
ixa
sive d’ a
adeoque: yak- atNax
Xay 1
sive yar-a xaya
hioc V aN
aX. XX a
Ergus aex
 calriti
+ 3a V X
XoaDaS
unde conserendo aequatione
C et D tollitur y
(3) Xa NM 3a.
Na X  Z3 at
Quaeritur X. et b
istota oldenbum
ex vetdstm= a
ex
 2y 39
XX +Ny + hy + ex + 9Z + vZ + xy
artut niutque
vra.
Ox + XX + xG Fgy + yZ = m = _ aa- b