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# Title: Leibniz-Handschriften zur Mathematik LH 35, 3 B 6
# Folio 2r, page id 00068256:0003
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20 Decemb. 1699
pag. 1.
pro Fractionibus dyadice exprimendis
105 (001 seu 100 = e1 + 7, et 1 = 0 ot o101 ete
14++
u
Nam Ndm = 4B (0∞ 00f s + vel 2010101
eF + 1 orsiaoI
l000000
aetrectea
100000000
vel 6 01 010101+
 ei eb
ex his patet periodum quidem semper descnere in 1. Qualis hoc loco est 01
AXA X,
atet Tibobobo
166 (00101B7 vel 0. 010101 vel 0. 0101010101 etc
Cat
nu
seu
101 + 
seu
10101 + L
100000d
101010101 et
1 0000000000 en
patet ex his periodum, ut, hic o OI, semper desinere in 1. et resitrii numeralorem
10000
()b) esto antib, ita ut si b sit 1 tunc (b sit 0; et si b sit 0, func (6) sit 1
si divisionemfinias semper esse 1
b Theoremat. b + 1= b (6)
. Nam A 10
6b1
1(1) = 4
b + (b) = 1 Nam B si b = 1 erit (6) =0 et fit 1+ 0 = 1
b -bb = B3 etc sed b = 1 peo 4
si si diffi
seu
(6) = 1 - b
b (6) =0
P 4.
9(es -0s1
sin b =0 ert (6) =1 et fit 0+ 1 =
et b =  = b) patet ex praecedenti
Nam si b = 1 erit b(b) = lin 0 -0
si b= 0 eut b(b) = 0 in l20
enempe 0 si b =c et 1si b = (C1
B.) BC20 = b uc
Ci b2c erit b(e) = 0 et (b)C 20
(C) et (6) C= () (B) = (6) = c et
Hic quaelibet t quantitas
si bec ent b∞) Cs t vel 0 si a - 6)
(1b)C)
si molet cec, ert (a) =b
si b = C = 1 erit b-c70
si  N b posito = p sit C = (K) et b posito (p) sit C = K fint BC) +d) =
b(C) + (b) C = 0 vel 1 nunquam
+ peA _ + = (b) C + P b (p) + C(K) = 1, et (p) = b - bc(K)
semper b + c = c+ b = b(C) + (b)c
bc erit (1) = (bc) = cb) (C) sit t() B)C
dtrosquod sucito ipistma
si b =c fit (b c) = (6) CC
sin (b) =C fit (b d = bc
Nam _ sit, b =c, fit bc = b et (bc) = (b) et (b) (C) = (b) (6) = (b1
Hbc-o
X- 2Fh +nh = X + h = 2Xh+ h
=Xh4h = 0 X = 0 +h
X -X + 2h - 2XF
sin (b) =c fit bc = bb)   - 0 et (bc) = 1 at 1b) (C) = c() - 0
 pordiseum
()
er
de de
by
ba
ce ep
u
du dn dta
E s S B 
ete
il
bn
n m
1
an am al
bm
V
V
d vid 8. pag.
nas
si  b- a sintque ambo positiva
erit b20. Nam s
si b (c) erit bet c +1) = 6+
) +
si b= (1) erit b()
P
X, X- 1 =0
a, 4- 1 = 0 = aa - a 22. AXX -2xh-h
xu
0, 0-1 =0
1, 1-0 =0
h.
ex
X- 3Xh+3Xh - h) 
-x + 2hX = h 20
2lx ata
seu X = 0+
b1
smu
XX-2Xath
X-h.l
X3 2K CXth
XL+ 2hh h
EX +2xh - h
 3xh tgh Eh
xect. +t2b
=1+3h
nite a  bsc =0 sequenti coliadoud . postsequenti 0
al + 1 = 6 150 Ergo n al =1 adeoque a = l = 1
am +bl + i = i + b + 1 = B 150 fit m+ b = 1 seu m = b)
p + b(C) + c(b) + d ∞ 1 = iso mlpo si b = c sit b (C) +b) =0 et p + d + 1 = 1so et prd- 1 et p -Cd)
n +b m + c + 1 = n + c + 1 fit e at = 10 fit, n = (c)
Sin (b) =C fit b (C) + (1b)= b + (6) = 1 et p + 1+ d + 1 = 1s0 vel ipso
, erit p = (d) sin  = (b) = P erit p = d
unde b b(C) + p(d) = 1 ret p =  , - b (C), : (d) = ad3 b(4)d) - () (() )) (d) (b(1)) = p
10 et p - d,. itaque si b =c
1. + b() Cb(0)) 4 Ln) 1 d(b) + e ubi si p = (d) item s seu si P +d = 1 item si p+ d = 0 fit additur 1 in
Ergo p+ d = d l
sequentem columnam. sin p+ d = 10, addetur 10 in sequentem columnam. Aa poterimus hinc aliquid
generale ducere in sequentem columnam addendum 2 id st X. si p+ d = 0 addetur et d + + fiet X = p + d + 1
si p+d_ = 1  fiet X = p+ d sin p+ d = i0 fiet A = p+d Ergo ducantur in se invicem haec tua
quibus ductis
rationatiter, nam X et xx hiac non sequantur
xah L0 vel X-h=0
d per compendium sit p+d = h esto  vel X - h - 1 =0 vel
+XXh- H. sed ita qu x potest esse 10 seu 2, toem est de h
ideo sic
sed quia 10 aedi ad columnam sequentem est potius 1 addi columnam post sequentem; = p+ d = 1
it
inficem fit aqualio pro X, nempe X= 22x + Exh- 
procedendum ut columnae sequenti addatur 1 si p+d = o et erit X = p Ad + 1
ad fiet X= p + d sed si p+d = 10 erit X = 0 Ergo X = ++d p p- d et Nam tunc p = d = 1 et
ita ducenda in vicem X - 1, X - b - d, X +d - p et fiat XX - 2p X - Dd - pp), X- 1. = X3 - 2 pXX - X (dd PP) + dd-pp
et quia P, d, X, sunt ovil 1 fit +X -d 
ita si pa vel
aequatrs. p
Cx + 2pX.
+d- p vit Y pod vel polius x= +p +d. nam
 quae aequalio salis facit, nam si d p seu si dep dividatur s
2P∞ - X(d- p( (+2pX
  sigi si de et pe0 utcunque, sed parum apta calculis
-d
s Ansic: si
es
3 1,d
X -d + p, fit denu X = pd - P
bra
eus erro iam hpli bitu  erte
crys cugo jam applisibitum cotibe
ionum ex sedib
t
anta
generaliter
p+ d
et
(0)
 conden
Quite linea ben
ticalis,
mini separat
X- k - 1
X = h- 1