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# Title: Leibniz-Handschriften zur Mathematik LH 35, 8, 9
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1. seplemtr. 1690. Hhc alia
scheda ejusdum diei in
dy = ydx.
Difaquato calculus in differentialiber replicatus per semi substitutiones
cr
= Syax  Ponatur pro es sub vinculo ejus valor sydx
u
Et firt ex aeg. 2. fiet y=Sydxd ubi rarsus pro y sub vinculo ponatur ejus valor ex aeg. 2
fiat
= c (syax dx dx etc. Mn Setessy ax posito ax esse constantem, et e numerum infinit
e
Nempesi HC Hyperbola et AB, y, ft et BC, V
et AH, 1. fit V = 1: y s adeoque dx:du = v
Et tunc X est infinita repraesentans summam omnium BC ad
indo ab A, hoc est spatium e infinitum hyperbolicum FABCF
Et notandum quotiescunque falimus Say= y  ipsam y incipere
faciamus GH = Z seu y = 1 +Z. ita det  rursus incipiat a nihilo
a sipl ecatu
sum ax est fomana- a sidaci ea
tiam infinitem tunc sed summa omnium l: 1+Z sec omnium BC ert sump. a omnium BC inde a GH
sem area spatium HGBCB.  Et fiet, dq= dy adeoque dq = dx + Zdx
et
quia
et Z = X + sZax. Et substimendo Z= X + = XX + sZdx dx ali
4
ordimna Et semisubstituendo in oq. 3. ex 3. fit Z = X + ZXX + sszax dx
X incipit a nihilo
ileque et rursus
eisubstituendo in aeg. 4 ex 0g. 3. sit Z = X + = XX + 1 X+SRax dxax et
1. 23
x
7
datur ergo C= Z
+
ete +
X
34.
constatuintem esse X+
lrqu per 6fd, 6. fit
C
123
ete s)Zax dx dx ete ax
 posi 
Estcantem CB. seu posito esse
Sit fetc. F..
u
1+Z
7
x +)X+  X e
L dx +C
1X
seu
tx
da
1+2
en
(12)
HX-
(XaX = LX
SXdxdxx  LX
(8)
12
posito ex constantem
 ia at  du  log 1-
Ergo fia  X + f. 2+X
+
AX  1o 1C + ex eo i stA
(16)
(dx X
SXdcar = 1 X
Stu
posito esse X a2 seu posito de esse num  cujus loq sit
4
sit sete ssZu dx dx dx et dx = C = Nad 
(e
seu posito X esse log. de 1+Z. seu posito 14Z esse num. cujus las
(13
seu iq fode X
6
6)
ita porro in infinilum continuando, fit
= X + L X + L x F.
ala e
1.2
e
et ut alisas seirbamus
14
iceret
sert
posita a onstante eoc et C numero efinito
in datore ipsius (, aeq. 17. contiunalavere in infinitum supposita dissimulare Z. qua si
sites
ex. sam
evanescentem foret
La  L
7)
etc S) dx dx dx esc dx - HX log 1 - x - 1. adeoque
et valor habesetur insi ititi trunscendentis quantitatis in aeq. 4 superioris calculi, adeoque
absolusi hberetur tabs vator. Videamus in
serie infinta,
sit
CO L = KCC aY + bX+CH + d F + ex et fit  s2= ax + L Wx F. bX erc
sil
so
e
10
q= ax + LBX + L cx erc et
12
c
S.
an
l
lar
1 2.3.44
( 5. 
b etc.
comparataalis seu homogenea
tine quia qanlitas X estordi summa omnim bX 2CX + dx et
ipsa X. scribamus loco ejus mX fiet Z = aX + mX = nx, quanta ad aestimationem homogenertalis
ubi paet selisss fore = infiite parvo seu evanescenti ergo et 0 =3
infinitum Hu
quia Runditio
popio quan
titas  est
ordinara et
ax infinite par
va. Hinc et
 est infini
te para
Er go seu eva
esse, et ehumenim
X
itaq
sedesse
per dinitule a poten in
drvisa intelli
nati ob legem
homosentolu
manit y
quanblas 
constantem it