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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folio 198r, page id 00068199:0395
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C
ractavi alicubi de differentiis inter eodem modo denominatas, tam Arthmeticas
quam fluxionalibus seu Geomebricis. Ooservavi autem ifiliter fortasse hine aliquid duci posse
ingendneris, n pita eorum s aggregata aut alia ab iis formata. Exega. et
1
s
tos Et sane intendum fit ut quadratura aliqua redecatur ad aliam smiliter formatam. Exempls
gretia log summ loc sex log +b) et (dx CogX pendent ex se invicem Nam ( lo loq Xlloq X+s
s
diferentati gedabit dx logX+6) + de logX. Ergo log. X. loq (X+b) =  (ax logk + b) + (at logX. quaeramus
x+b
A+b
Ex logse +b)) = y et ((aX log X) = v, fiet d dy: do = (xt b log (Xt b): XlogX
X1b
sin dy ad dv ut lognt logX logk+ B ad logX logx+B M+b 
p
aliam inter haec relationem, sit
1
seu talia sunt reciproce ut sua XlogX
ay: do = :  dy + dx = dl Ergo du = dl- dy = mdy: a
tognt Xeb, log +b)
a ceris, e itemque ipsas ay et deu esse hetervienems quantitates sed
s  d y +. v = 1
non esse
logX
at.s
cicet pit
 et v- (X logX + b) b93
logX. log(+b
dit hogotentus quantitates tuli modo hnam y. (ax ex lo0x
Notandum
pti X +b, log (xb
nat tamen otest  neque enim ut unius nominator ita et numerator ex altero formatur
lonoee
sed sintetemem differentiae tamen seu dus et do sunt inter se ut homogenea quantitates ejusdem
tamb
e
foruet, sunt enim ut 1: XlbgX ad 1: (X+b, log(X+ b),
x ita secari in partes, ut ex logX log(+ b) forent quantitates
constantes, quod utique in potestate est, foreut sa y ut ad v, reci proce ut
summle X logx ad d  summ. X+b, log( +b). sed tales summalum exscressiones quibus
si poneretur 
ds
enes homogeneorum violantur  juvare non solent
 e
(X+b)  =y. sit X loq Y log(X +b) =  log y
 = y. Ergo V logZ = logy. Ergo  vaz. Z + ZB logque dv = dy: y. sit
b + X+
am v aa: Z+b sit jam
Aliter
qu l0gX. (X
( Glogxa
19
et D = caBe (ee aa 10G +b)
fiet vv 9udzue. Z +1log2 dv = dus = d
C
s
Vd + 08
1
Vv-
ado du
xidque + ~ logdze dv
logx d =y fit dy = dxlogx AdeslogXdx . Ergo A hujus summatatur. M, datur summa)
D = y
sot si foret gd. L
  XyR(Lu t) dx = y Xy- P5 Coqxib  et Xdu +ydX-
G
G
b
N+b S
logx ax mg axu)
y - logX dX C Xt
A+b
logx ex
g
(1
a dx. lo1 + dx
Ergo du: dx = log. X i Xtb
datur e summa C
afxib, 9) xib dy  ydx logx ax