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# Title: Leibniz-Handschriften zur Mathematik LH 35, 7, 11
# Folio 2v, page id 00068082:0004
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des raitons
l’Europe raisennable, ou l’indien informé de ce qu’il y a de plus consideroble dans nos creames et practiques
cum quadruto ametri seu cregmscripto incommen farabilia sint Octogonum  sedecangulum, Brangulum, ec quam et omnuo
soligonum quodlunque, quod bilecando fit, quaeritur an hinc demonstrari possit, apsum sirculum qui est ultimus horum polysguncram
etse inconmmensurabilem quadrato dametpi
etunc quantitas-
Si circulus est quadrato diametri commensurabilis 1-  + 1 - 2 + -  +  - = etc
multiplicata per aliquem numerum, integrum, dabit numerum integrum, seu e - e + e - e + e dc poterit
esse numeus integer posito e esse integuum; seu 2e + 3C + 2C etc. erit numoius integer e cum quilibet horum sis.
fractus, seu in integris exprimatur per seriem infinitam, necesse foret omnia destrui additione seu abire in o
8 1o. 9
u 10. 5 1. 7 1on6 10.3 1. 4 1o03 ont 10. 10r0 Io+I 10t2 1013 1otq ton hons hor ton 3
le nadi1
1
ago iost Ddt ocst nomt unno pAe aequ i0  hoco roreoo uoreoeo  rovroromo
B tonus aequ 10 200 ot
A+B aequ
militer 1ost 10r8. C . . a9. + + 100o + E0 + 8 + 203 etc
10
1odo 1000or
3 3r 41 15 19dc+ +  eu
pro iculorit si debeamus aodere L + + + +  + 1 (5, 25 2g
59q euit   + + dic + u + i + i ec + L sed inciiculo summaaddendu destructio illa divisore
G81
oetb
6obi existente decimali oaa que procedit
1
C
1000eoo - 10-3
1 +
1 + 3
10 10
- 7
1
1
1
a L1o25 16-9 10- 13
Hec summa tauto propius accedit Circulo, quanto altior
quantitus est numerus e
3
+3
103 +
10
125
i aegu 10  7 oe 1  3  pxrarise osisiarsi ideoir
1029 59 1 10C + 7e  1e hc
Et pro quantiale quae magnitudini cicculi in tantum accedat, in quantum
1 + d + 11 + 1
10
a
Sc
U
O
1te
+ 112 etC
10
numeius e assumitur magur, habebitur:
1+1=171 = 1+ 14 1 +1 aequ. 0
- 3 + 5.  - 7
1
+9 P. 11 +13 ∞ 15. 11 Xa19+ 21Vc ag.
-3+5= 7 + 9 = 1+ 13  15 4 11 19 + 21 Ec 
345 - 7 + 9 - 11 + 13 - 15 + 17 - 19 + 21 dc
3+ 5t - 7 + 9 - 15t + et - 15 + 15 - 19 121 hc
+2 + 2 +2 + 2 dc.) se psum
10
16 +32 + 48+64 + 80 48
10
98 + 366. + 866. + 1586 H,
040
2.43 La-3
sum de
sum o4 x + 3. R ix + 3
(x. (X, posito X esse numeros naturales
utile autem erit rem ita institui ut per 10, inveniantur omnes summae protermini
se
constat autem alunde quae sit
quorum denominatores sunt infra 10; per
10 omnes quorum denominatores sint infra 100
ita porro.
per 10 omnes quorum denominatores, sunt infra 1000. Ditems 2= item rono- qu s 1
Et ita terminus ultimus addendorum, O, est 
10-9
100-90
summae autem omnium X, X, X, erc. praecedentium semper in caleulo sunt anterendae.
(1 egx posito X termino ultimo seu
sX aego fX- t + si ego (xt + x. jam fx + sai aequ X
Superest ut quaeraumus summas omnium hA, X, X, et.
maximo
oem adhibemus.
(X sii aequ
et (X- (X-, aequ. X etC.
(N - 2sX + Si aequ SX - 25X + X.  Ergo per 1 et 2 fiet sx agu X + Y seu B apo AX+
et TX - (X - 1, ai
Jam SX = 1 aequ. SX -  51. Ergo per 1 et 10 erit  si aque
Manifestum est, essex (X = sX1 aq. X
-
491
1X + 5 X 1 X
r 198Gden 3 scarear X
SX.  agb. sX  4GXL 6 SX- 4 sx + Srop s 4s+ (6. 3, X , 3.  nda -n - a  XEx 1 ∞) y 2 2 S aqu. X3+ B- s ausi
item SX aeq Sx3SX + 3 Sx - si aequ sX a 35X 4 L. ) X L  X. Ergo  per 12 et 3 erit 
Et ita habetur progressio in infinitum qua
a
similiter sx-1 aequ
(x sive D aequ. X + 6C - + B+ . A
Lceno)
quod in nostro
se invicem derivantur.
Daequ. X + 6C - 4B+ 14. E aequ. X + 15 D - 15C + 5B - 1A. Et ita porro
et eodem modo olinuando: E sive Sxt aequ. X +15D - 45C + 5B - 1 A
adtoque per sere et
casu sufficit nec opres est rem reduci ad ipsas potentias
Aago X Baego   X + 1A
C ag. 1X + 3B - 1A
Baequ AIX At aX. Caequ.  21 t 3. Ox + 3i x
.2.3
AX = 153, 2 1X = 15. 6. 3+ X  15. 6.9. X
4 +X
- 15. 3. 4. 1X
44 15. 2+
33 17
2. 3
2. 3X + 69. 2,
2
eti
Daeg. 3. 2. 1X + 6. 2. 1 X + 631X = 6. 21 X
= 3. 4. 1 + - 3 . 1X
1. 2. 3. 4
F5. 2. 1X
de summis potentiarum
qu
-3
16-7
aqu
144
erc la e est X
9d
3. 7. 11 dc est 4X + 32 diff. 2. f. 53 2e
5. 9. (3dtc est 4X+ 5
16X + 24X + 9
18
diff. 19245
23. 7 64X + (49X + 108X + 27 2di 96X g2 +9B
33. 97 S4 x 2+ 24OX + 300X + 1 25) (en 3. 165 X+ 3 4 5. 5 X
+4.4.  X+64. 3 xt 4+ 3r) Bisserentia ent
s isis
2
29Z
491.
22G
125
323
343
12
34.
386
306+
169
7
169
od 
386
. 48 19
88 288
Era  rit
sum. 4. 473 +
18- 41
Haec (alia scheda accuratius
exhibui, et penitus absolvi