NOTATION. : 
2) 1810 3) 1810 
2) 905 —o 3) 603 —1 
2) ne 3) eer 
2) 226 —o 3) ‘Gy ase 
2) 113 0 3) a. ek 
2) 56 — 1 3) 7 2 
2) 28 —0 3) a end 
2) ve eas eo 8 
2) % axe 
2) a ea 
2) to 
Oo 1 
be ahaa 1810 = I11Q0e@I100I0 : the binary {cale; and 
IO = 2111001 in the ternary ie 
mple 3.—-Transform the numbers 844371, and 
21 58 . . from the denary to the dacdensey {cale. 
12)844371 12)215855 
ie) eee — 3 72) ade —Il=y 
12) 5863 — 8 12) “1498 —i1l=y 
12) “488 —7 12) 124 — lo = 
12) 7% — 8 12) 10 — 4=4 
32) 2 —4 °° — 10 = 9? 
— 
Hence 844371 is epics by 348783 in the duodenary 
le and 215855 by ¢4¢ 
And thus a a is readily transformed from the denary, 
ag des other fyftem of which the radix is given, and hence 
nd 1000 is exprelid in in the following manner, according 
o ne value of the radix 
If r = 2, 1000 = IIII1O1000 
= 3, 1600: = 100 
= 4, 1000 = 33220 
r= 5, 1000 = 13000 
r= 6, 1000 = 4344 
r= 7, 1000 = 2626 
r= 8, 1000 = 1750 
r= 9, 10co0 => 1332 
r = 10, 1000 = 1000 
r= If, 1000 = B29 
r= 12, looo = 674 
” Hence i it is evident, as it is, indeed, from the nature of the 
fubject under inveftigation, that the ers the dpa is, the 
lefs will be the number of digits neceflary for expreffin 
any given number, but the operations of mulasheaune 
and on a jutt eftimate of the Leer the radix 12 will be 
found preferable to any of the other fyftems: but on this 
fubjeG we fhall add a few remarks at the conclufion of this 
article. 
Prop. III. 
To transform a number from any other fcale of notation 
to the denary, or common fecale. 
This propofition is the conve of the ems one, and 
it is save effected by the reverfe opera *i 
n— 
Forletar”? + br” + wy re- 
prefent a number in any known {eale ‘ notation, whofe 
radix is 7; then fine are alfo known, we have 
only to goles the “foeeettive values of the Giterent terms, 
and ss hee will be the number transfermed as required, 
le 1.—Transform 7184 from the duodenary to the 
common nice of notation. 
= 7.137 1. 12> +8.12 + 4, 
ee we have 
7.123 = 12006 
r.12? = 144 
8.12 = 96 
, = 4 
Duodenary 7184 = 12340 
nay he 2.—Transform 1534 from the fenary to the 
denary {c 
1534 = o = 5 Ma + 3- aaers 
I 
5. & -_ as 
3-6 = 18 
4 = 4 
Senary 1534 = 418 in the common feale, 
Prorv. IV. 
In every f{cale of notation where radix is r, the fum of a 
the digits exprefling any number when divided b 
will leave the fame remainder as the whole number divided 
N = art + br®* + er"? prtgrtw, 
then will N — (r — 1), leave the fame remainder as 
(at+bte...ptqa+w)+(r—1). 
For make r ~1 = r', orr = r! + 1, then” ~ (r — 1) 
leave a remainder y, becaufe every 
= (rf + 1)" 7! will 
term m of the expanded eee (r' + 1)", is divifible by 
r except the laft, which is 1, bape sonfeguently ( vo ryt 
—r',orr® 1, and this 
1), will lea 
property is eely’ independent ae the value of 2; and hence 
it follows, that every power of r, divided by x — = pia Jeave 
a remainder 1, or each of the powers r", r?— —*, &e. 
x i 
