# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/00068196:0767
# Permalink to this reading: https://leibnizlegible.com/page/00068196:0767?run=74
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 386r, page id 00068196:0767
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=767
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000767.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 190 lines, status machine
# Lines: 190 recognised, mean confidence 0.78
# Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German.
# Licence: CC BY 4.0, Leibniz Legible. Machine output: do not ingest it as verified text; keep this provenance with the text.
# Wording rule: say "the machine reads it as …", never "Leibniz wrote".
# After the first blank line: the recognised lines in reading order, one per line.

XX +X - X 1+ p-1 =
XX + X - XX + 2X - 1 - X + 1 = X.
2 X dx + d - 2(x - 1) dx + dx. = 2X dx + dx = 2X dx = 2 dx  = dx = dx
 O KNL +XHI - XXHX = X. Ergo _ i(x + 1) ax +. ax - 2X dx = dx = ex
d (X+ B) + XF1) = A ExEX = dx
Ergo 2X
Ergo d, 2(X tns + 3(x t n) + Xtn, :6
2. 3 (X+ n) E F 3. 2(x n) ax + dx 8, : 6
unde R(X+ r) + XFn -
t3 XX + 6X 13
a nntX - dx fi
aequ.  omn d, XX = aequiomn 2Xdx
E) XX + 23. 2an + 3. 2X F 3. en + 1 :6
2. 3 XX + 2. 3. 2X, + 2. 3, 4 3. 2X + 3. 2n + 1, :6
aequ
+1X+ 1
omn 2X dx
aequi 2 omnX
ceq
omn X
2.6+ x 16 Xx 1523 (XF 1) +2.3(+ 1) + 1( 23xA 3X +  aeg. 2X + 2 aequ 2, Xt 1
Nam
GXXI OEXX
Hnua
si dip () Alindv = (y) si (X + csisa) = (X sit a) = X + e FD (X + Clis sa)
si des e si st (∞) v
erit (X + nclisa) =  fa t csist ) (X ; a) = omn X + C (5i
li diff v e
omn. dx
2X + 6XX +bX + 245 - 1H( - 3XX -x, es - XXAEX +1
_XEr + XrD - AB (2+G. + XF1 = dx
2X3 + 3XX + X, :6 = omn XX
aequ.
 (Sad) + 12X +6+    = bRx (bX aequ 12X4  - 2x+ 2
quia
seu
s
 seu (V) = V H Numericam differentiam notabo
d0 = (V) = p. v
Et sydx sit om
do = (1)   =V (∞) v (V) - ∞
dif v nota arithe differentia Arithmetica, et do geometrica.
quoni sucum  sit ergo rem cunt complectermur si Eodenmodo  (uda =  sum geom. et suum ydx est sumattutimet
XFC- X
iste nexte
exxt
l X. XrE XZ OXKIEeX
siff V = ((∞) ()
fiit dooe= summdusm.
si  V = summ. G idem quod () vman emin  vinax aminim = omn. X
Theorema ergo tale = sit v Z summ 
S EF
V. = suum y idem quod  supra vulterius  insimum  oun y ab infima ad
summum assumta
simu
it
dy = (y) - y
Hinc ita demon
10 = () - et Ergo du - vicinde = (V) - vicin (B) - v + vicino jam v- viein v = 9. Ergo-
y = dy
pen vmovissa vprim  sum tunde
c
erit do = suum dus copi
vultra noviss.
=ju aenno davys
ad novissimum,) et conrectitur cum hoc
Demonstratio pendet ab explicatione summus Ce connexa sunt v summus, et 
Theorema.
sit V = summ
quo noviss
cummu a primo
dy = (7) - y
Theorema magnum
v- vicin v
Hinc ita ratio cinamur
A(per 3) () = (y) - y  per
 By) y et (4t)
et  dy
et connectitur eum dla) - x  viein v) + vicius v  
9 Edo-
it v = 2X1I
 et ge X sit B = 2X + 3XX+ X:  et y = XX, ubi
Exemplum
XX, do = 6XXdX +6xdx + ax ib et dy = 2xdx
reperietur x (positu dpomstante) dx manente constante (d vi do- vicindo = dy, niam fiet
e
sit = 2
2X3+BXX+X
sit n X a 3xK
= 2( - 1)3 = 3(X-
 6(X - 1) - 6 (X - 1) - 1
:6 = 2X.
veniendas. Nam quia
t Ergo connexa sunt sux = summ sydx et
vutis nunerice euntet
summas arithmeticas c
Adeoque haec tres aequationes inter se connexae sunt, quin et
videtur via aliqua apparere an
6XX +6X +
FF +  po connexa sunt v = summ. y
in =C: Hinc jam
2
 ulterins novissimo = V er
fit Aua vicindo= (ass victi) n y+ vicin yn  Y
etrabitur
est v
per qost de unvicin do
2
quia v- vpcin = y
V= summ g idem quod
m resse qudq
 sim
d AX
5
a i
et de = summ dy
itaque si
Et ila porro.
quart
uadraturis. Sit series
Ax  smum (AX r
indentas vel descendenties numerice summabiles intervententibus a ratur et summ suxt
summ y.
axsvdx = sum faxsyax
43
 demes
+ 2 3
i2 5 12 20
ete seu  = V
ea sen 
et ita porro. sed non deo
sum de= suum sum. dus
avenitur ad summ v sum
num sum t1 SX 4L
  u ntala alicest
F EE 2Hinc F  summ  = summ sum du = ffrum  auua 
2tFe
infinete semcal
Xda contranii
BXHI) CI a
daretur sumo
Hinc s or
reciprocat rferentias arthmeticas
datur et summ. du et summ daus, etc. Sc
gressionis harmoniae, darentur et summae potentiarum
noc summatio
non est per
si 2 = 5 fit 271, f. et
dque p + 17 9 sZ Cf.  (2, A  + (Y 1) (f- 1)
aot s 1+= + 3
oxd a
eries
.+ 4
(Erd flr (Z +d Z +1)  f. d R+) fYtdE
= 1ru + 3ad +4 etc
ent differenta
itt = (7+ 1) f = u ptca + (41) f2C Fa2C. Y t 1) 12ta (21 Fs
Z dq + (Z+1) (fd
EtdZ
E
du -
d
(2+1) C
+(7+ 1) (FZt a2 fZ) (= q7 1. f.
q+ dZ)
Hie per summas numericas videntur reperdi posse tales feries summari, et quidem in terminis tam finitis a
enque est 
uper
intinitisctitem
50=q 5, 5 Y -1)
(E  ed ( 1, ubi si A Ea
d
cer
iit 2
mimas  ronessins 
. numerita
edi
lo consti
summam ipsius
+Z(+B- 1)
20 si do non
i
EF ∞
disfeculiae
FF. EHine si detur summ.
F+
1 X veri (7+ d
F57B (= dZ
Z + dz +1
eito cum a spostus bil