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# Title: Leibniz-Handschriften zur Mathematik LH 35, 5, 23
# Folio 4v, page id 00068041:0008
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Habemus erge jam tros aequationes ad Logarithmicam curvam
quae sunt:
X aegu Y Sidy: y et V. raequ. sh. r(sit a aequ
Posita faequ r ftah: h)
et X aequ. log. y. posito log. Y esse 0.
Sed possunt adhuc aliae inveniri aequationes per series in finitas
et quidem quia M P erat Z. est Peaequ rr: Y + Z. et PB aequ r (aZ. r
est autem in integris PZ aequ Y= Z + Z + Z etc. vel q
vle  esse  Ergo PB aequ Z - yt + 73. A
2t
5Y
etc
ar3
itaque Numerus cujus Logarithmus est 0 sit Y. et quivis alius
quidam numerus propositus sit Y+Z. ipsis Logarithmus Hyperbolicus
potentia
posita Hyperbolae rr) erit
2
= +
posita MP _ q et PB, X.
est investigare Logarithmum.
et + 23 -
Er
r
74
+ 2
et c. aequ. X
Et ita ex dato Numero facile
Videamus jam quomodo ope alterius seriei infinitae ex dato
Logarithmo possit inveniri numerus:  Quaeritur scilicet
linea in qua sit dx aequ dx r + Z aequ. dZ r. Scribatur
Zaequ. (a) + bX + cX + _x + f x ete et rZaquta + lX + CX et
et rdZe aego 0 + rbdx +2rXd t 3re xdx +0 fXdx etC
49
dx rFZ aequ() dx + bx dx + Cxdx + ex dx etc
+r
et comparando singulos terminos valorum O et D fet poterit poni a
1. 2Y
et ita porro in infinitum, et  habebimus:
Z aequ X + X + X
12Y 12382
+ X
+ Y
723473 123,4574
12. 3Y
+. etc.
posito X Logarithmo, et Y+ Z bg Numero.
numerus sive hibcoqu tunc erit Y=Z aq seu fiet Zgo
et erit 1 seu Z aqu e seu haequ Y - r + Y - Y + Y
seu h: Y aequ.   -
1.2 1. 2.3
1. 2 1. 2.3 1.2.3
Bere
Eadem methodo potuissemus invenire valorem ipsius X, ex Z. nam
 scribatur X aequ (a) + bZ + cZ + ez3 + f29.
dx  aequ 0 +ibdZ + 2C247 13C2 47 + 9533 42
dx rtZ aeq. (4.brdz+ 2eiZtZ +ser z  12sr2 42
11
dz Y
fiet b aeq.  1.
Faequ +
473
= 1b212+2cZ47131743
Y dque x X X
Caeq+
caequie
et ita porro
seu Xaeq (D) +  2 = Z
+ 53
+
12
et
C
eaequ 
Cnonstum
cessunt
numer
Et quidem in valore ipsius q rationali per X ponitur a esse 0. alioqui alia
a seu constens supernumerria assumta ponitur esse oAat in vulore ipsius X per Z
omitti quidem sed tamen et retineri potest.
Hinc si Logarithmus X sit tenitas, seu si CF aequ CM, erit FH
sit aqu
aegu
falioqui alius
prodint sertes,