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# Title: Leibniz-Handschriften zur Mathematik LH 35, 3 A 22
# Folio 11v, page id 00068235:0022
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sit irrationalis quantitas.
X aequ l+ m+ n. ubi l et m et n ponuntur esse irationales
cubice. firt: X aequ l +3 lm X + blmn aequ. 0 quod ubi si linspationalis et ima us est absoluto
sin minus ut generaliter procedamus, sit. Erit l aequ X- mXX + 3m: X - _m
eatl
+ l XE. 16 ma
2
t lam in A IM + Nm dat p l- n in U R IB + Il dut n m3 + n
Genimuliter
n
at 1em per mmt mnim dat l seu l + m per l - in + Mm dat 1- m
l+ m +n per A+ n l per I a = Im dat l - 3 lmra. Ergo firt : (3) L - 27 l2m n
I+m dat l- m 1 m3. 1 - m erc generaliter. l- m mulliplicando per l
lamtn dat quadr p -4l
22
quad ut B - 27 lmin gy P - 6412, cgcultaliter SI2 =y Imu Ejus
1
+ (12
mulliplicanido per 12  I2 m + I nim erc. in 1 SI2 
inu
ZImt S Z erc
+m + n + P mult per V ∞Im +A dabit M 13 -3Imn verum ita runsus qualuor adsunt termini, unus rationali
et lies nationales tam luum + emp + mp lnpemup. plures quam ante cui malo ut occunramus
dividamus in duas quasi Um + ntp, et fiet: l +3Im l. sda +R3 +3AH sed ita rursus quinus
Termini Cubl = 27 cub Ima est autem cubus Imn aequ Vmn +bC 1 12 l
lamen per l y plm fiet t
F +
X aequ lam fint: X aequ. l+ _ 2lm X + m seu X = 1= m aequ. 3 ImX. fiet (3) N = 12 m 3 lequ 27 13 m 3 
sit rego tm+n sit X aequ 13+ 3Im BX + 6Imn seu X2 (2 m aequ. X3- (2- m sit aequ. s.
fieo S. 3lm X aequ 6tmn seu et cubando N6 Bm n aequ. s - e 13) 3ImX
n
fier. S’ aequ 6 Im n + 27 X () im + + rts lua X f fiet  s- 3lmX aequ 6 lmn
et cubando 0 12m n aequ 53 - B) lmX + 3s  UmX s- 3lma) Ima
rusus lem (3) 3ImX, aqu 53- 6 l’ mn + 356 lmn s. bimn.
Ergo 3
deuique S’aequ 6 12 m dn + _ LBlmX + 3. 3lm. Imn X)
Xaego l + m +n. fiet l’aego B -M - 3mum = 3 in X X seu - l’+m En _ +X sfago X+m in 3m
n seu erit saequ 3 m +n mn + X. X- m-n
seu laequ. 3 X. I. ma +Xl. sec fago 3Xmn 1 dxxb Amn 3XV go siet X X sit manaequ. t
siet Saego 3hma + 3hXX = 3Xhh. Ergo shaepo isid aeqo _ m n + . ex 3 mnx+ tomn  + X 3
=Lx
seu ( a 27h3 rgo MhE + IalKX - EX = 3 mncmnAXX - LX. XX.hx hamm dudium
aequationum ope translata primum X colcitur h. Sed haec prolixiora quam ut absolvi
mereantur
lun olmap