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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 165r, page id 00068196:0327
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Qvaeritur summatio hajus seriei  +  +  + 15 + 25 etc. Hanc in rem
(mi
utor methodo mea consueta, ef et scribo LX + X +  X + = X etc = v. et fiet
(2do) fact  + = X +  X + 1 X  ete = dv. dx et (3tio) X; X + L X +  X + = X  erc = Xdv: d xsito -
edx
jam dy = 11 X + X +43, et ( (Sto) dy = 1 + X + X + X etc (6,  1:  L X = (7mo) WV, fiei
seu quia rescentibus Z. decre scunt B,
humf vertex R quadratum CAB
sa A CX)
asis impotae CA, CK, in recta cA, sit AX, X;  et CA. Iptus quadrati
sit unitas, erit et CX erit 1 -X et  et XW erit 1:,  - X = du:dx
et zena AW erit ut Logarithmus ipsius Cx.
describatur curva AY cujus ordinatae
XY sint proportionales ipsio Logariohmis ipsarum Cx, intique XY, erit is
seu erit B) XY, y = sdx:, 1-X = (9n0) (per 4.) xdv: dx = log. - X. Ergo
seu f d . X ( mo) v = ((AX: , 1 -X) dx: X (m) 1g(- X) dx: X = (13tio) v
quaeritur modus curvam AV describendi.
1t
et ectam XV; qua
CX vocetur Z fiet (q Z = 1 - X e s ) dq = - dx. Ergo p 13 fit 5 log2. 7: 1. Z - v
(17mo), s do = log. Z. dq: 31- Z seu dv  = d, log Z: , 1Z = (8B0) 4. logz, 1+ Z + 2 + 43 + 3t etc
= dz, logzue + logz. Z + logZ. Z + log. Z. 2 + log. Z. 2 etc. Hoc ergo ut summetur, seu ut redeatur
ad V, debet summari log. Z. ZC. dqu, et reperietur esse f (gno) sibgs Z. Z. dZ  op. 2. et ) et
Eomos
(per 10). 
(per 14)
e
fiet B = log. Z _ i q +  Z + 1 23 + 1 d ete = log. Z. log 1-7 = log. 1- Xhogu etr .
-1,
12= 42+ 12+ = 7Y et
(E1 mo),
(p 13 et E)
= fL9L Stogt1. 22. Z = (log. X dr. (1. ∞)
ergo V seu (Iog(. -X) Ex: X = log. Z. log. 1- Z = flogl. Z stogx dx: (1 -x). sed
ita nil discimus, nisi quod jam tum constabat. Nam si differentietur
1660
logz. 2og. 1- Z firt logst- x) dx: X + lg Xlxi (X logX. dx: (1 -X)
Quia ergo hac non successit aliam tentemus viam quaeritur nempe
esse (231. G e log. b = 1 ergo fit log. 1- X = V- b  ex 1s fit jam siet quia bx
cY
egs seu  AX = dy. 1-X seu dx = dy. b
et per 16 et 25 firt 26 = (5. b dy: 1 - b: jam (1:, 1 - b. = 1 + b + b7 + B3 + P etC
do B. b. dy = (2800) y. b. dy + Y. B du + 9. b3 ty + y dus etc. Zodbritur fg. B ty sy. V27 1
1-15
d, 1. b = (2gm.) b dy + b + y. b. edy. Ergo Bomo) e sy. (. dy = y. Be7 -  bey
seu (31 mo) fy. B3 dy = e y. b - + be3 Ergo per  ex ex 28 per 31. fiet
9. b. dy = (3210) + + y. b +2y. b +0 59 bL. 9b2 ea sed hoc aequatio revera
1- b
- = b3  B7 _ -  V7 er
nil differt ab aeq. 20
Si proponatur   = +  + = quaerendum et fiat X + + X3 + : X etc = v
se (2d0 sdv: dx = = X + L X3 + + X5 etc = (3ho) y fiet dus) d yedx = 1 + X + x + X etc
6)
= (5t0) Ce 1:, 1 - XX. Ergo fit v SC Ax: , - XX) dx: X -
Jam
di Sdx: (1 - xX) Sax iX = Sax: (1 - XX) dx: x + dx Sdx. Y:  , -xX. Sed quantum
apparit, nit lucrabimur, cum res in priora resolvatur,
(33tio)
uid si sic d, six flog.  Sdx logsr. X) _: X = dex log. 1- X: X ssa lostr 
=dx Sdx logsi - X) : XX jam Sdx logex) quaeramus quod ita fiet.
dy log. F=X X = B34to) log. 1 - x dx + C X dx: =- X et 5h)t X: 1X = C 1: 1 -x - 1
erpo fic ) log. = X X = f log. 1 = x dx + sdx: 1X - X seu. flog. 1 - x dx = log. L X X + X - log - X
Ergo ex 33 fit sex log. -X: X = (dx log(. x) : x)  logtu ostog (1- x) X dX. XX - six: X + silog. - X. X
An sic: de X log. - X : X = dxlog. - X : X  _x log. 1 - x: XX + d log. Fx. sed ita rursus evanesctt issitum
ubi evanescit quaesitum
si jam c- 120, seu e = 1. fiet
_d, log. 1 - X XY = dx LA N X: log 1-X C: 1~, x + plog. 1-XC X
dy log. 1 - X XV = dx, XV: P.X + r log. FX X sed ibi infeliciter evenit, ut non liceat facere X = X = 1:7
ut desideratur, quia tunc r=o, addoque terminus quo agitur evanescit. Itaque opponit se hic natura
quaesito, ut oporteat hate aliquid altius oub esse