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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 3 (shelfmark LH 13, 3)
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69 martis 1679
idendum an liceat definire, Quadratices analyticarum
transscendentes ope Exponentium qui nonrationalium
Exempli causa quaeritur an inveniri possit series cujus feries
differentialis est aope alterius assumtae exponentis inationalis
1+y
sit  aquatio ad curvam: fR+  +2X + 3 X + 10 f + 1y+ 1
+21 X V + 22X y + 23 X  aequ. 0.
hujus aequatio differentialis est:
e+ 1 X R+ 2 X +3 X + e dx
V1Y + 12 y + 13 1
et
m- 1C. C
+21X y +22X 3 + 20X +
m
1 r.t ∞ p5qu
dy
a, ay
npt. 1)
X 21. X. V +22X V + 24)
)
Et fiet: d X aequ.
= D = c
+0 + V. E
Ex qua aequ. 2. vel 3. tollamus
literam X ope aequ. 1
et primmus ut rem in facillimo casu experiamur, ponemus in aequi
literam x ad nullam assurgere potentiam, erit 2. aequ. 3. aequ. 0.
et aaeque l. aequ. M. aequ. n. aequ. 2
valor ipsius a hujus modi:
X aequ.
d C
ty
aequ.
unde ex aequ. 1. fiet 
1 v +127 + 13y + f ago H et ex aqu. 3. fiet sequens
= 21 V + 22y + 231, + 1 aequ. t
h-
g- 1
- a. Y - 12 y - 13V
c
+ 1, + 2y + uV + 23y
albi pro X substituendo in aequ. 5. ejus valorem ex aequ. 4 habe bimus
id quod statum possius ducere poteramus ex aequ. 4 hoc modo:
X aqu 3 sergo ex aequ. +2  7+) (4 seu (4G- 21)  2+ at () G d2
seu: dx (9 (( )IV + 12V 4 133
y
aequ.
d
-
K
dt
in r fa rs st n a taturis
uyZ.y+ny+.
+G Verum 14y
non est quadrata quantitas, nec putandum est in aequ. 6. posse forte
dividi Numeratorem per latus nominatoris.  seu per seu
+ 4 dF(+) t d4 dividi posse per 2. sed hoc impossibile est nam
sequeretur vel 2 vel ati dividi posse per F. non 3 alioqui _ valor
ipsius X ab initio non fuisset fractio, quod tamen est necessarium nam
alias nunquam differentialis series fractio fiet. vel ati dividi posset
per  quod est impossibilae.
Itaque si valorem ipsius
 quaerimus debemus. recurrere ad pavationem l’vel 2
l
. ab, et ea al  e mers paisa v
cy