# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/00068207:0160
# Permalink to this reading: https://leibnizlegible.com/page/00068207:0160?run=74
# Title: Leibniz-Handschriften zur Mathematik LH 35, 15, 5
# Folio 80v, page id 00068207:0160
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 79r); this page is its left half.
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068207&page=160
# Source image: https://digitale-sammlungen.gwlb.de/content/00068207/jpgs/default/00000160.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 121 lines, status machine
# Lines: 121 recognised, mean confidence 0.86
# 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.

investigavi etiam an non Tuae roefficientium formationes
per additionem combinatoriorum possenit exprimi generaliquodam
 aquatioge. Incipiamus p Columna D, inde pergimus ad 5, et 
quod tufficiet ad habendam expressionem generalem: 
sit e ) = 22 ab + 23 abc + 24 abc + Z5abed + etc.
fiit 22 = + = 23 = - 2 24- + 3 - 3.3, 1. 32 = + 3 - 1 = +2
425 = - 2 , 4. 4. 1 4- 2 - - 4. + 4 =0 et 26 - 5 - 3. 5.1 5-2 , 5. 5. 4. 5. 2. 5. 3. 5. 4
= 5- 10 + 1 = 4 et generaliter erit
EBZXH = x
2iZ
1. 2. 3.4.5
XX 1. X- 2, X. X. 1,. . X- 4 - X x la. X = 6 , etc.
c. 5
s
te
2XF = +1, X- L. 4 - 1. X 2, X x. pu. X - 4- X. X. 1. . . X -6 + etC
17
5
ubi in casu X imparis + est +, in casu apparis + est-
Termini autem spon Membra autem quae desunt, eanm calculis
gen in unoquoque casu per explicationem calculi generalis sponte
evanescunt, quoties x scilicet XX non est major, quam
maximus subtrahendus in membri numeratore. Et in hac observatione
obtenato situs fuit modus dandi expressionem cujusque in columna
eadem coefficientis expressionem generalem. Simili methodo
reperietur pro 8, aequationis aequatio generailis hujusmodi
3XF = 4 1, X. x-1. X. 2 - X. X-1.. x - 6 + X. X - 1. . . X -10- X. X -1. . . X- 14 t en
2
Et pro  prodibit aequatiu talis
1.2.-, 15
∞ 1.2... 11
AXF = +1, X. x - 1 . . . X- 6 X. X- 10. - 14 X. X -1. . . X - 22
1. 2. -. 7
1.2.-. 15.
1.210,23
ofine ergo tandem pro quovis eoefficiente cujusque
columnae post primam, habebimus expessionem.
generalem: numerus columnae sit y, trumerus  gradus
X. X- I∞, - X - 30, etc.
combinationum quam coefficiens afficit sit X, adeoque
coefficiens designabitur per yx,  vel yxrI fiet
X. X - f10  Z - (23 = - 2) X. X -1.. X - (2 - 2), X. X - 1.. . X.
GXH = +B, 1. 2. 3.-. (2 = - 1)  1. 2.3. -. (27 - 1)
1.2.3.1. (274 - 1)
generalis additio Dyadica spondet; operae prtium erit
diversas proprietates considerare et inter se conferre, Licet
enim certi modi inquirendi prolixiorem dent constructionem;
hoc ipsum tamen utilissimum est diversarum inquisitionum proventus
conferre inter se, ea enim novis noc spernendis meditationibus
viam aperire solet. Memineris  inter alios modos Columna
investigandi columnas, ad columnarum additionis dyadicae generalis
coefficientes, unum fuisse a posteriori, ex casu speciali aliquo simplicissimo
Nem qui est a si a, b, c, d, etc. ponantur esse unitates; ubi
acttadent 
sunt detrahendi in numeratore, membra deficientia
ubi ut dixi in casu X imparis = est +, et in casu X paris es= est = et in casu X
Caeterum ob usum latissime patendum, quem nobis
pro col.Ima 1. H = 1 2. 11 4 1. 12 = 0 3.11+ 3. 12+ B. 13= 1 4. u + 6.12 + 4. 13+1. 14 =0
Prodeunt aequationes hujus mod
2da 1. 3 = 1 322 +1. 23 -p 6. 12+4. 22+ 1. 24 - B 1031+1023 + 5. 24+1. 25 =0
ubi 1.0 10. 10
ubi 1. 1. 8. 0. 1. 1. 0.0
Itia 1.34 -1 434+1. 35= 1 1539+35+ 1. 36 =1 2531235 + 36+13) 21 o 3435, , 61837+138=0 iti 11. 1.1. 00-
tripe haberi possunt 11, 12, 1
tc. Item (22, 23, 24 et, item 24, 25, 26, et. Natsi compendio sum
i t
on facile aliquid cemp hinc ducatur compendiosa constructio
ludet calculi
a vatoribus facile quidem valoribus arithmeticis
a
_ 110,110, 10; eti alternis
sed non facile analyticis, seu ita ut in illis appareat lexprogressionis
caterum cu
loefficiens columaae ex praecedentium additione et
generalissimam us caluma
subtractione intervenieatibus numeris comsinatoriis,
conclatur: eum quoque hic exponere operae pretium erit.
nuta
probe notasti in valoribus ipsorum , (3), 6, item
+, OB, et similibus, plerosque terminos evanescere
ex Vatoribus ipsorum (0), (3) 18) etc. si ducantur invicem
si (()) ducas in (8) idem est ac si Be (C) ducus in S, cujus ratio est quad ( = 8+ Op. .. et (O) = diffo inter C et P
producta prodire simpliciora quam quis expectet. Ex. gr.
ratio est, quod excessus ipsius (B) super  formatur
mulsiplicatio dividi potest per 10 seu = pC multper... sed
Omnis autem differentia inter C et P
 = P, ea multiplicata per Cp dat +Op + ep = 0 itaque, Ea dem ratio et in caeteris signaturis valet; Eod
mut mulsiplicata per Op, producit 0. in dyadicis, Esto haec differentia
Cd) est differentia inter O et p
(∞) maltplicardo per S + pC.. sufficit 
1. 2... 31
2. -2
X. X - 1. - - X -( + 2- 2
1. 2.3.. (
t ete
majoris quam es
evanescunt
(9) -C)+p Oxdi siter C et p
CS) LS +pCD p O96- diff. inter B et pC
(136)diff. inter C86 et, 6
2 essent non unitates sed
1,1,0, 0, 1,10,0, et c. vel lit,
vel 1, 111, 1, 0,
el
si a essent
0, 1, 191, 0,0,
0,0, 1,1, 4, 1, 0,0,00, etc. simile
inde labulae pro ou aequatio
nn
ver
entis coeffilientium
idemque foret; si
ve notdeg
regulariter a
si possent
seut quaevis aliae suo
ider notae ex 1, et C
 dispositae.
=1
Nam XO L p = qu C +0 + p) = O +p. 2p
serarune