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# Title: Leibniz-Handschriften zur Mathematik LH 35, 3 A 3
# Folio 12v, page id 00068216:0024
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Murficas esta habent usus pro  problematis uniformiter
compositis inveniendis ex. qu. quae
sit problen, X+ y + 93 (1) C B
Jl. X + y4 Z M
fiet Cn m p - 29. Ergo q n m et
fier r El + 3N- 2n ni et fit:
et fiet X  - n X + : n X -
et
212+3n2 3nm2
Xryt ZF n.
l7 p = 3p9 + 3Y
0
 m
N L3p 3uni
Si aequatio fuisset X +y3F 13 et X+ y pi m fei 4B( p
(11
 m 1 p - 24. et l3 p p - 3paet et quia 47 p - m
5
fiei 21 i p  3p p + 3pm, seu A p3- 3m P+ 2l0
Res ergo redit ad has tres aequationes:   _ pX + 4 0
et X - pX +q0
s
15
et p - 3m p + 2V 0 et qH p= m
quid si et ita. sit aequationes X+y B B, et xcy m
ascribamus X+ y 1Z2p 12+3 X+ 4Z m +Z denique Xry+ Fn
et faciendo n.B compendii causa
Hini etiam innumera problemata Methodi diophanteae
nullo negotio solvimus: ex. g. X + + ZHB. X+ y  + Z F
invenire bres numeros ita ut summa cuborum cubus, et summa quadratorum
quadratos assumatur certia aequatio x+y +Z i n manentibus
et n aibitrannis, fret. habebitur aequ. g. Quam si aequemus
arbitrariae divisibili X3+ B x + bd X + cd po
faciendo dp. n b p 212 qn + 2nm B _C n - m - 2c
habebimus bp  - 213 +313 -3nm -6Cn 3Cn - 3Cm = 6c
2(3- 3n’ + 3nmi
ubi faciendo bgo. seu 13 3n - 3nm - 6cn. set  n
fiet d - d fiet dgi n et fier dp n n 2c
fier dp - n. et n e 2C + m. et posito mp C+n.
fier nH 2C + e+ NBren et n, 2+ C. et dp
no rete
2e
et Fo qui posito
fient X
7c
c
=c’ esse quadratum seu - C B S. _. 154C
erit de,
+ 2L
+ f
e
e
sed in eo jam male quod sic +f et
concurrunt unde aliquis terminus est
regativn unde non lice facereto
quia omnes tres radices aequationis u sunt rationales
videndum potius
us
Hinc nihil efficitur comparatione
quomodo aequ. 9. dividi possit, in duas
ex. g. pontudo X n fieret
Sed jam video
non magni hnc subsidia resci ad diophantea
satius erit ex aequ 40 I1q inveniore aequ 1. 2. facere
aequ. 10. 11. faciendo Lp 1 Z et MB
pro l re. m ponendo L. M. et faciendo L (3Z3
. et M  m- Z
(19)
(20)
(21)
fiet ph- 3M p +1 Pp0. YB p - M. X pX + qu
i seu p - 3m P + aB 10 et q p = m +Z et X px + P nirp( 2)
+337
-2
et et si aequ. 22. 24. rationales solutum est publema
quaerantur conditiones generales aequationis cubicae et
quadralicae ut sits ralionalis. illud dubium an si
irralionalis etiam x irrationalis. Imo necessamu
X+bX+C
de + 1+
+CX