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# Title: Leibniz-Handschriften zur Mathematik LH 35, 3 B 5
# Folio 43r, page id 00068255:0085
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ter
Hic
 sint, a, b, c, etc. notae dyndicae, seqificantes salicet
quarum quaelibet, nihil aliud significare potest qu
 seunon potest, ex cedere unistatlem
investigeturque modus addendi generalis a + b = 2ab + a + b - 2 ab
Et faciendo inde duas columnas _ab)  = a + b. Nam ab
in secunda columna idem est quod 2ab. oteco quantitas
2a
autem hoc modo in quavis columna posi rursus est c nota dyadica
nec excedere potest 1. Nam  si a et b non excedunt I, etiam ab non
excedet 1. porro a + b - 2 ab non excedere 1 manifestum est,
modo a et b singulae non excedant 1. Nam si a et b sint unitates
fiet 2 -2 =0 si alteratia sit unitas, altera o fiet 1+ 0 - 2. 10 =
si utraque sit 0, fiet 0+0 - 20 =0.
Hinc jam via aperta est, eodem modo procedendi ad plurium adde
potarum duadicatum additionem generalem per columnas non nisi
ex dyadicis notis constantes. Nam quaeritur a + b +c. Habemus
a+b nempe ab (a +b. 2 ab. addi debeta hioi
Addal. 2 ergo C ad a + b - 2 ab seu ad X, fiet, XC X+ C - 2XC sed XC debet
a seu habemus ab +xC (X + C - 2xC. Sed Ergo ad huc ab et xC, a
(quae quantitates ambae sunt notae dgadicae in unum addi debent. Et
fiet ex ab +xC fint abcx ) ab + XC - 2 abcx. ad eoque t fiet.
a +b +C = abcX )ab + XC = 2abex ) X + C -2xc et ex plicando
ZX per valorem a +b - 2 ab, fiet Facibe - 2abC) ab + ac + bc  2abe - 2 abc - 2abc + gabes
ab, a +b. 2ab-0
abetabe
 a+ b - 2abu = 2ac - 2bc + 4abc. Ergo collectis omnibus erit
tandem aB ab - 2abe ) a - 2ab. +  abe = a Hic iterum
p
et hoc modo aequationem fieri identicam. Nam sublatis columnis,
fiet 2ab - g abe 4 a - 2ab + 4abe = a quod identicum esse patet,
prodest tamen identicismus, dum ita separantur membra, ut quaevis
columna non nisi colum notam dyadicam constit habeat
observandum et aredob a pro uno a - 2ab pro
duobus, a - 2ab + 4 abc pro tribus literis  esse notam dyadicam
Itaque quantitates his minoles et a - 2ab +4 abc - 8 abed pro quatuor
literis esse notam dgadicam seu unitate non majorem, si ipsae a, b, c, d
sint unitatibus non majores. Demonstratio veritatem hujus propo–
sitionis calculus ostendit. Sed inveni tamen et demonstrationem a priori
Nam quia (p + q) = p + n p 19 + n. n- 1 pn - 2 92 + m. net n ~ 2 p. 3 a et C.
2
Hinc si p sit - 1, et q sit +2, fiet (p +G) 3 = (.1 + 2)3 = 1 = (-15 + a (-1  2 + non. 1 (., p 2, 2
1.2
n. n- 1. n - 2 (,)- 2 2 etc seu 1 - (-1) = A 1) a  + n 2 + n. n - 1 2 Ce
X n. n - 1. n.2 IC. (Nam f, (1), (1) h. 2  snt 4 1, 41, 41, +
1. 2. 3
(Nam ((- 1 ),  (1) n. 2, (- 1) n. 3
. 2. 3
-1) n-4 etc. sunt
4 1, exc. nempe  si n sit  impar, fienit , - 1, +
etc. sin n sit par fiet = 1, + 1, -1, +1 etc. ) Ergo +1 + (- 1 = n. ( - nn a 1 2
=
+ n. B - 1. n - 2  - n. n- 1. n. 2. n- 3 2d en p ubi _ si n sit impar + est +, sin par est -
.2. 3
1. 
1. 2.3
1. 2. 3. 4
1. 2.3.4
sin n sit par set hiet t +1 = n - 2 + n. nal. M 2 2 - n. B - 1. n. 2. a -2 2 et = 0
.n
1. 2, 3.9
s posito ajbit,ere elle unitates sunt ut au ni det idem quid ab ,et n. no l.net idem quod abi et ita porro
ui