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# Title: Leibniz-Handschriften zur Mathematik LH 35, 8, 30
# Folio 145r, page id 00068124:0293
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Demonstravi summa eatio seriep rationalium utcunque infinit  disferentras esse
rationales; tum ex eo quod supra dixi quod de differentiis infinite parvis tum
ero etiam methodo generali hoc modo. sit figum ABC summatu, ABF
si ea est virrationalis, non potent generaliter evenire, ut du anm
ordimatarum quarumcunque ut BF MG differentia sit iatronalis
quod ita ostendo. poterit aliqua ex ordinatis quadratricis esse rationatis
enam potest haec curva alicubi secari aut Ma tiangulo recta aut
vel etiam assumi potest rationalis ordinata ut si a recta PG
parboca
b
secetur in G. sit AP PM rationalis ratioque ina laterum trianguli
uut numeri ad numerum, utique erit Msus rationalis. differentia autem
inter Merationeatem, et BF irratio nalem utique cirationalis cntra Htuspothesius
Eodem enim Nor ila facile applicatur hoc ad series numerorum poterit tamen
sit aequalio
y + by + XY +X +CK +d o et rursus
satque  X ec rationales
(Y) 3. Quaentur an Z. possibile sit esse racionalem
(y) a b (1 + (4(Y) (x  C(X) L d B.
quaeritur an
ubicunque quacurque posita relatione inter y. et (Y)
hoc patee esse impossibile
potest enim tatis utique poni ut sit differentia irrationalis
componantur aequationes + et = by + Xy + X ECX (+d
_ y - (7) 1Z.
(s) = b (y) a() (Y) =) + C() -
et  X - (X) Y co fier hine ut obiters dicamu derivari potent
+y + bZ +
+ (y)
aerq
y-
Y - (V)
2
regula slusii fangentium
1- c
iralionec les pleroque
rationales
patet hoc fore imppossibile si vel una
ex ipsis a. b. c. d. e. sit rationalis
Hinc Etia a = b, _ = b + C rationalus seu a + c - 2b
t differentia inter
a + b = 2C
ilem a +b +C, - 30
a et C sit irationalis rationalis. debet nulla esse
demonstrandum superest quod inqostibile sit iuutum esse quantitate ritionatem ad quadratius
litera
seu esse numerae qui vel rationem quae semper salis facias
X1- Si Y V
107
hunc
314159 ad ponatur esse fractio C reducatur ad decimater 
sit numerus quicunque mulliplicons 1ot infinitas poterera modo ipsae sit finitus qui si dividatur per finitum
innula
100003
frei fr 107C p 314159 H
 pro 10 subamus v,
pro v ponetur
seuper numeris priores examnandum stabit ila
producat
et
v7 C b c + ev + fv est
ponendo  infinatam
s t38utu