# Leibniz Legible — https://leibnizlegible.com
# Piece: https://leibnizlegible.com/api/records/27497/text
# Catalogue record: 27497 — für Overbeck: Mathemat. Aufzeichnungen
# Incipit: [Tabelle] Summa columnae ab [Tabelle]
# Date: 1712-05-16
# Sender: Leibniz — Addressee: Overbeck
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=27497&katnr=56087
# The Leibniz-Archiv's transcriptions of the letters of 1712 (pre-edition): https://www.gwlb.de/leibniz/digitale-ressourcen/repositorium-des-leibniz-archivs/laa-transkriptionen-1712
# Work: https://leibnizlegible.com/work/DE-611-HS-861255
# Title: LBr. 705 · Overbeck
# Folios: Bl. 131–132 (from the shelfmark LBr 705 Bl. 131-132): 4 pages, canvases 261–264, page ids DE-611-HS-861255:0261 to DE-611-HS-861255:0264
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=DE-611-HS-861255&page=261
# Lines: 278 recognised on 4 of 4 pages, mean confidence 0.80
# Catalogue: Leibniz-Katalog / Arbeitskatalog der Leibniz-Edition (https://leibniz-katalog.bbaw.de/), ed. Arbeitsstelle Potsdam I der Berlin-Brandenburgischen Akademie der Wissenschaften, CC BY 4.0.
# 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, page by page: a "## Folio <label> — <page id>" line ("## Canvas <n>" where no folio label is recorded), "# " lines naming its source image and model, then its recognised lines in reading order, one per line.

## Folio 131r — DE-611-HS-861255:0261
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 132v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-861255/jpgs/default/00000261.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 137 lines, status machine
satuma ipsarum columnae ab
a b   14
a
at
ab ab
rag
0
ba
a+b- ab
a+ b + ab
+2ab,- 4a
ab -ab +a
atb -2ab
2ab
a + b =
atbrca
ab 24
 -2a
+ a+ b - 2ab
+ a +b = ab
a4- b +gab
b = 2ab
a+b - ab
adtim
et praecedens eas Nilla vel Mitta finalis in ) sed quia a caret altera columna, hinc fiit bl m =0 et
Ergo a - b = dyadice + 4. 
N
- ab- b
fiet X = 1
1-2b
=1
+
m
Ego d -b. dat a, b-
itaque si b sit 0 manet a, si b sit 
et si b sit l, et a sit ex a, l, fit
sed decs pliquid nempe ex a, o non fit
e a
bpt
a + b -ab, si b at 0
si b sit l, et a
inqirendum est in generalem expressionem subtractionis, quomodo scilicet
exprimatur a - b = l sit a = b+l erg musus a = b +, fit b
= Elbl, seu l= a- b
videamus an lieeat sublata fractione exprimere l, per
integrum, is sit xa + yb +Zab, quae formula
qui multiplicatus per 1- 2b debet producere a - b
y =Ca 1.
quod in integris effici nequit, nam fit Z-
l= vtxa tyb +Zab = a si b est 0, et = 1 - a, si b est 1.
Sit b =0 fiit v + xa = a. Ergo V = 0 et X = 1
2
sit b = 1 fcet A + y + Za = 1 = a. Ergo y = 1 et et 1 - Z = - l Ergo Z= 2
et l = a + b - 2ab, ila si b = 0 fit  l= a, si b = 1 fit l = a + 1 - 2a = 1= a
Ergo a - b = +ab set ,. nam 2ab - 2b + a2 a +b - 2ab =c
2b
e
t
Ak5 b
ne a
b - a  si tab sa d -c = adc
=eli t 
-gca
atoora
-2ab
Fab at b
+dc. etd
acd
Agh  +gth =at
ut
ad ud
abd
20
Lraberab
ac rad 4
ab
-
b
aed
b=1-
detd
E
+rabed F 0,
tina
g- tr = Fgh ) 9+h
229l
Et hanc jam video in Totamea regula subtatur
manet a,
= bl = ab F b =2ab
=ac, batue Lsd
-24, 1a b, qu(C. d.
C- 604
 1-C + 1-C + - 4
potest; nec atiter et cotssu prcedenum, donee modent aliqueo, oste tsuccebere spb trnctionem
e+d2c
ifficultatem. Revera, hameus aliter res inpititi non
e
c
a
b
Xya
l
a +b - 2al
Cdl .d
a
a
t
a
atb
a+b
X sit 0 et y
uinus
ibus
ergo
Xty -279- u
O subtractum reliaquit, 1 mulu
7-2 2Z =0 seu -2 -Z-0
7 = - 2. itaque
et m = bl
xa tyb +7 ab
yb = 2x ab
22ab
ta =b Z.
quaesitum est a +b- rab quod sucdedit
b- ab
seu
qu
si C=c
quomodo exprinermus. c - 1. lple enim membru
non permittitur in generabibrs,
lcE. idque lamdeu cotannand
in infeniur done
foret quantitas nihilo minor. fiet 
se laC.2
aa
c- 1
cal

## Folio 131v — DE-611-HS-861255:0262
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 132r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-861255/jpgs/default/00000262.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 53 lines, status machine
1 01 et
0 1 etc
0000 1111 et
avaitat
co
.
2
cl. 9
d0Cd
Quad ratorum differentiae sunt impares col. 1.  11 11 ete
Horum p lex talis
col. 2
col. 3
col. 4
cl. 1. 810 10 17 col2 00000d
10 1000d
8Bocd
a b a b a b, id est o101
1
00
defgr loboefghC
o mafers tuuxi
00 c dl
O +C DH  Fit C- 1 = 0 Ergo c = 1
d+ C - 2cd = d + 1 - 2d = 1 - d = 1 fit d-9
symma ratorahumseu
trangulares colisu  Natualium commma 1. a b ab ab verc. uti b
col. 2 cC d
col. 3 cee Affff
se potius: Naturat. Cot a b a b a b a b ab
a a b b aabb
a a a a bb
b
4 aa aa aaaa
m n
6bbb. 6bbl
etg.
C
e
e au ei lueo b- d
col. 1 Lmr
qusta 2 grs turoxy
s
101
Itn - 2bn = 1 - b seu B (. - 2  1u 2b
+ M- 2m = b est autem (=0 fit m - b
1
ed non petust de 1n 2b= , Ergo fit
m + n.
p = 1.
=D
aa
mol

## Folio 132r — DE-611-HS-861255:0263
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 131v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-861255/jpgs/default/00000263.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 52 lines, status machine
uia
Numeris quadtatilis dyadice expressis investi-
tandae sautae sunt pariodi Columnarum
primo, ut demta prima Nulla, Ceriodus prorsum et retrorsum sit eadem Ex. gr. 060OTCOtOTDID
inductio ostendit hanc esse cujusque periodi legem duplicem
00001101
secundo, ut Periodi prima et secuida peris conveniant et discentiant alternis. Exgr 3+ 0, 1.00
Hae binae dseruationes suspiciunt ad inventendam totum periodum. Nam
 tot habis praebent aequationes quot sunt notae periodi, deantis tribus
duabus; sed sempa dia aequationes suppeditantur aliunde, semper enim dantur binae prinae
enim dantur prum im notae periodi, quae sunt 00, o Hatemus ergo
prn
tot aequationes, easque simplices, quot notas. Ergo semper habentur
quaevis noto perisdi patet autem ex prima observatione, dari
tot aequationes quol sunt notae in dimidia periodo una
quod ita patebit clarius: sit periodus abcdefghitmnpor
e  b =r C9 d = p C=n f- m gel h= k i=1
sit notae Z, erunt hinc aequationes 7- 2:2, nam ultima nil docet
sunt aequationes
Ex secunda observatione
1-m
a-i Lk c- l d =I C=n f = p g-q h = B1 - Y
b 121
quae sunt Z TE enR Z. 2 I nam una ex his jam debet con-
sunt aequationes
tineri in praecedentibus, unde fit Z-E,: 2. quo addito ad quantitatem
numerum aequationum pracedentem, qui etiam erat Z 2,:2,
fiet G-2, numen aequationum, idem qui notarum Z, derutis duabus
imo reperio nos semper habere netas cognitas to sequentes:
plures, invenio columnas semper incipere per Nullas sequentes:
prima et secunda per
ytia
sta
7ma
gla
6tu
qva
et generaliter
a
27+ (ma Ht 2X F2 per
8
0d00
seu per  O (2
00000000
O(
C(2
0
a
Nota autem columnae ymad sunt ,7= seu 2 vel 2C +1. EAo ex his
taveum 2F semper habenturile 2F. prima, attoque multo plures quam ne cesse est.
on omnibus cosumnis
post tertiam

## Folio 132v — DE-611-HS-861255:0264
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 131r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-861255/jpgs/default/00000264.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 36 lines, status machine
cotutmm summma
tota etiam quaevis a numero suo ordinali
vocetua n eait semper
Notaperiodi generaliter vocetur n locus
autem seu numenis ordinalis notae vocetur , i Numemi notaum Z erit
bsit. (n = n, si sit (() = ZL l- 1 et hoc quidem ex obs. 1.
Sed ex obs. 2. Conserri potest nota cujes locus L, cum nota cujus
locus est Z + l. et si. l est par, et sit (l)  Z _Z it, era
itaque si sit (l) = Z + C, et l sit per impar, erit (1) = n
et l sit Epar erit (n) = 1 = n
Et genemliter erib (H) Etgeeraliter si invicem ducantur (n9 - n = 0
et (n) 1tn =o fiet o(r)  (n) un L   0
nZn
videamus an notam ne aliquam possimus investigare
ope exempli superioris in literis expressi, videamus quomodo
ex nota proxime praecedente
ex seinvicem  ducantur notae
n  f = m d = 1 - m fit d = 1 -f et f - 1 - d
et 9= C et h = 1 - b et i =a. Habemus ergo novamamoniam
abCdefgh i-a h = , - b g = c f = 1 - d e= e
quod si tolamus notas medretatis prioris, servemus notas medietatis posterionis,
fiet p = 1 - m  L g = l Y = 1= K. itaque absolvendo periodum
_ KLmnpgr r= -K q- l p = 1 -m n = n
I = a r= 1 - h = b h = 1 - b K = h. hinc K= -- b. Unde patet
+ - a Y = I - K
Hh
aequationem b = 1-K, quae in datis occurit, nihil dare novi, sed ex
aliis jam consequi.
olu Terminus sequens non differt a praecedente
in eo, quod ultima quotcunounitates termi–
et
in 0
num claudentes convertuntur
praecedens eas 0 in l, quod si nullae sint
Unitates finales vertuntur in 1, et prae
a a aa aa a