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# Title: Leibniz-Handschriften zur Mathematik LH 35, 7, 19
# Folio 5r, page id 00068212:0009
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divisio sit ZF Z: a = gv: Y fiet Z = v - gl1v,: Hist
demonstratio 2+ 4Z: a = v+ deD, y+ dX et sublats fractionibus
(Z+ γdZ F Zdy + dzdi = av + ado, et utrinque auferendo aequalia e
et an, restabit dividendoque residuum per dx, fiet 
adv xdz Zdx,  = dZ adv. XdZ,: dx = Z+ dZ, seu
a (dx - X (4)Z,: 1d) X = Z+ de a ita sola superest quae evanescere
um
(d) Y
avit
b9. dx
et ois se peoporcaitrg 1d 2. D  ax Qa s: XL IE WB rcl = .  us. n l av.  pia simer alius
se
s
es
(elsZ. (X = AZ. deo, lirebi poloriam pngere inftasu evanes celitium du dlesi et at et havere Az. do exar d fictis
unde prodibit uc  av XX, proponebatur
AX
e-t
t
ntates
sed . cc
selentiae autRadices Bodinitaise
sit  =2. X td y: C 1D X =
_ n.y1
siB  n fet X
vatio
a s+X AeX dx + L, C- X dxdx + C. - 1. P-2 pe-
n - C
st fiet
Quod sic paulo fusius quam priora demonstrabo
a
132,3
ita porro per veniatur usque ad e - e seu 0.) aequ. = V +   dus
192
donec
+ non =L p dy dy + 2, n-1, n.2  dydydy, (et ita porro donec
1, 2, 3S
perveniatur ad usque ad n = n seu 0.) auferatur ab una parte
120
X, et ab alterat  y cum sint aequalia, residuum
duas
dividatur per dx, et pro dy: dx ponantur ratione inter, quantitte
n - C, l
a
deinde, c
ratio d)g. _X ei ratio inter, quantitates, quarum una cx semper permanet
duas quae continue imminutaar
duas
eadem, durante imminutione e differentiarum seu accessu ipsius
2X adpermanens punctum 1X et fietque
x + ege l X
Aete. L p
B
nat (d
T) (I)X. BE
+ 1 n∞I y 2. 2 (dy dy + 2, n- 1, n. 2  23 y dydytele + et c.
1d)X
Quod verum est in omni casu, etia etiam etiam cum differentia
CdX
1, 2, 3
sunt assignabiles. Cum ergo (per postulatum) in hac regula
generali comprehendatur et casus  dff
quo differentia fit aequalis suunt nihilo aequales, seu quo puncta
2X, 2Y coincidunt respective ipsis 1X,  1Y, ideo eo
ly =0. fiet eX
a p. 
ut
L -1 n n - 1 (d1y
ponebatu. Est autem lidiX
ans
demons Hac demonstratio locum habet sive diguitates sint potentiae,
sunt numeri et sive sint habe aut
quarum exponentes supt smacti,
nam cum potentu
sive suit radicefero
exponentes rationales, sive sudos. Quiaas
 ex aequatione toller exponentes furdos
utrinque exaltando,
ut aded non C et n non significent sivenii potentias exponen
une
tium rationalium, qui continua subtractione numerorum, 1, 23 etc
tandem ex hagriuntur ne opus sit serce progrediente in infi
tun
nempe
Licetit autem salte per fictionem modo supra explicato, n etiam redum
umC pr
quidrationem itae po
quia haec ficta seu
veritatem
indubitabilem
1, 2, 3
x
ex sasi inspertim ot
ae qualtux pasur ipsprum )
faciendo ut seme
a ete
entiu asu
demonstratus
gorithnus differentiarum
primi gradus, nunc  ostendendum est, eandem
caeteris evanesentibuseu Caus: (dX
reducius potes