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# Title: Leibniz-Handschriften zur Mathematik LH 35, 14, 1
# Folio 67r, page id 00068201:0133
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aequatio ergo erit talis (16)
co X - r O Cd
3 a
In qua aequatione videamus an sem quadratum a semisse ultimi
possit esse majus cubo a triente penultimi,, fiet fcL vO+ a- vO
sive (8) V- 9.
o 7a8
+ L debet esse p 0.
Cd 1 veO vO
17a eu + Y + 
729 a 
27a
Nam valor ipsius cd erit talis:
+ 4 + 10
zaxit v = 310 -0
7ax2 14
Unde patet eandem reperiti determinationem, hac via posteriore quae reperta erat priore hempe aeque
Unde patet nihil aliud denique requiri, quam, ut  sit quantitas negativa, ita enim
O erit quantitas affirmativa:
affiruntrucn
S verum enimvero scimus quantitatem C esse
est negatious; pater ergo ex his duci posse nihil
E Resumta ergo Aep. 15. fiet:
 2m n’+ m 
ay 1)
mn -2mp m +2mn nt
em sm
nt -3ne
(32)
ax me-umere-nen n unde ma - m-n-
Videamus an haec aequatio serungi possit commode in duas:
ut ponendo Y + M n C62O
Sive (3)
et fier 54)
mn-0
ag
as, 3mn c
nd
au 17- omtntane + 3rs em
umn
.. n4
quae aequatio dividi potest perm 7
 = 3mn + 3m + 3n. Quod si jam in locum n tn substituatur
eorum valor ex ex aeq 32. orietur absurdum
Aliter ergo instituenda, erit divulsio, nimrum:
frat: mn C. 32 +, Andenam -hmtre -hmin n sive na  1 aua t, emern
non ergo sic destruere licet:
C’ara
TacoDdI)
Residua aequalio erit (37) mt mnt- Bn nn - 3m  mo +Emn m
a qu 0a
et divisis omnibus per mn, fiet: min + mn a= aa B = m- n nere - 3mn m 4 2 m +
inetisque omnibus per n Quae aequatio divisa per nt n dabit
(9)
2 F
mn  -O
En
ca   
n. umn 
=m ne- dmm + 3m + 2n et pro 3m + 3n, substituendo eomlore
da
eorum valorem exaeq. 36, fiet  mn -0
-men - hmn - un 350....., et divisis omnibus per m
=mn - 3 - 319.... atque inventa erit ratio nn sine cosp mna ponendo ov a este
a52 II da
eoque valore inserto in aequatione 36, tei ibi: ( nn 1 +o
a
o A
nn 11
etiamnum ergo in nostra potestate est, alteratiam, in vel in, ata sumere pro aabitiio, ut tamen
nullum postea oriatur absurdum, adeq utile est semper in ergmodi explicationibus novas adhibere
arbitrarias; ita enim semper contratitur calculus, net et certi tamen sumus nullum deterius
rando committi errorem postea vero cum demonstrationem in fine enim apparet eomum quomodo
arbitrariis sumi liceat atque idem servit ad artem inveniendi tegendam: nam ubi ad demonstrationem
veniendum erit. Positiones illas sive explicationes arbitrariarum, in fine inventas ab vartro
ad hibemus:
71 + netu 00 B3  mn D  - m- 15
mna
p (AB)
E) ponamus ergo? 2O
ergo ex a, cit metn e maa, 8i t.-28
us   -m- uB
nen
et erit aeq. 47. quae valorem dabit ipsius -n2
sive ipsius , nempe . )) 63-a 
extribebitur vulor ipsis B. inod  sit mijus quam 46b + 4
ing vr
ma ingredi valorem ipsius  60. Est autem 65+
03 -4. 202
na 1 a oa 6 ax
- a H0 quae aequatione
-203a
 ex Notandum autem
27= 3 - 35B per aeq. 41. et ex aeq. 17. fiet:
44) D3 +3.B +300
3a ..  2022 N prueudoque  V + - a. fier: a B3 A  XN (-2a) 41a = 2a 49
3aB 1
Siffecnitero hutin d’epet noijum quin g. Sed sta in mineis conquoib anda sunt
67 49.
V (+ 3aysjo
Ea