NOTATION. 
howrever, are aya fuch as to lead us te expect, or even to with 
alee mary Ser for os which long efta- 
dere iar to all our ideas of 
which iti 
Barlow’s Theory of Numbers. 
Prop. I. 
Every aaa N, may be esac to the form N = ar 
+ br" 74 &e. pr? + grt ase oman 
any number uate: and a, 5, c, &c. integers lefs than r. 
For let hy be divided by the greatelt power of r contained 
. it, as and let the quotient be a, and remainder N’; 
o that 
N =ar* +N’. 
Divide, again, N‘ by the next lower power of r, as r”~’, 
and, let the quotient = i which will be an integer, or zero, 
as N’ > or <r" d the remainder N", when 
ae +brt-i 4+ NM, 
Dividing, again, N" by r a and fuppofing the quotient 
é, and remainder N'', we hav 
N=ar™ + br?-* +er*-* + NM, 
And by thus continually dividing the remainder by the 
next lower power of r, we fhall be evidently brought finall 
to the form 
N=ar" + br"-' + er? -prtartaw, 
in which expreffion, as a, 4, c, &c. a rth quotients arifin 
from the divifion of a number by th e higheft power of r 
contained in that number, it necelfarily a ows that each of 
— henry aaa a, b, cy &c. is lefs thar 
—If r= 10, then a, b ly rage are the digits by 
which any number is expreffed in our common method of 
notation ; thus 
en 104 + 6.103 + 0.107 + 3.10 +4 
18461 = 1.104 + 8.103 +4.10°+4+6.10+1 
which form is always underftood in — the value of 
any number sa aoa that is, we give to every digit a local, 
as well a its original or natural value ; thus, in the number 
76034, the fecond digit from the right is 3, but we confider 
it as of are 30, on account of its local fituation, being 
in the fecond place from the right ; in the fame manner the 
6. pape 6000, and the 7, 70,000 ; fo that the value of 
each digit is — according to its local sae and 
its original value, the former indicating t 
and the aa the cucnker of thofe powers that are fe intendes 
to be ses ae 
Cor. 
den caanntone oe to the value of the ra 
Ifr= itis = the Binary foale, 
T 
ros 4. ernary, 
r= 4 = - Quaternary, 
= 5 - - Quinary, 
r= 6 - = Senary, 
r= Io - + Denary or common {cale, 
r= 12 - +=  Duodenary 
as many c 
number expre of the i le Thus for the 
Binary {cale, a cadens are ©, I. 
e 4 2 0, I, 2. 
Quaternary - - . I, 2s 
Senary Ty 2, 39 49 5 
ie or common fale . I, 2) 39 49 55 6, 7; 8y g. 
we mult 
he fymb 
whence the digits of the duodenary {cale w: 
Oy I, 2) 39 dy 5 6, tie 
bol o and 11 b 
ll be 
Pror. II. 
Given the we 
"4 br® 4+ crt? ....pr?+or+u, 
in which [ - rare given numbers, to find the unknown 
co-efficients the a, 5, c, &c. ; and the exponent 2, Or, which 
is the fame, to transform a number from the denary to any 
oiler fcale of notation 
Tt is evident that this may be done by Prop. I. 2 — 
by dividing N ae oe by the higheft power of r which 
s centained in it, but it . more Saal peforiaed By di. 
viding N fucceflively s r3 if 
Se sees prrtar+w 
be divided by 7, the quotient will be 
ar™-* + br®—? +er"™ 3... 
and vas remainder 
ae qunecne being again divided by r, gives for a 
--pr+g¢ 
ete 
ar™—? 4 br®-3 4+ er®-4+ 2. 0.. 
and a remainder g. And this quotient, divided by r, gives 
a quotient 
art-3 + br™-4* 4 er"—s 
and a remainder #. 
Whence it is evident that i {fucceffive remainders will 
be the co-efficients w, g, p, &c., or the ace 8 that exprefs 
any — in the fcale of which r is the radi 
= I. Se 17486 = + 6.6"-* 
Or, which i is the fame, convert 
oe c, &e. 
17486 5 trom the ¢ common to the fenary fcale. 
Here, by the foregoing propofition, 
6) 17486 
6) 2914 —2 = w 
6) 485 —4= 94 
6) % —s=p 
6) oe —2=c 
6) 2-186 
6 ee 
SS ty aed in the denary ae. is expreffed by 
12942 in the fenary 
Example 
