312 
NUMBER. 
of 12 months, containing each about 30 days, the round 
number, 360, was chof'en as the rnoft convenient for fub- 
dividing the ecliptic into degrees. But the radius of the 
circle was naturally employed as a standard for the inea- 
furing of lines and arcs in general; and this being nearly 
the fixth part of the circumference, and comprifing, 
therefore, about 60 degrees, the arcs themfelves, or their 
multiples, when exprefied by degrees, came to be reckon¬ 
ed by fixties. The notation was eftefted by annexing a 
fingle dafh for lixty, two dartves for its duplicate, and 
three dafhes for its triplicate, as explained under the arti¬ 
cle Arithmetic, vol.ii. p. 164, 5. 
Ptolemy and fucceeding aftronomers, haying adopted 
the fexagenary fcalein their meafures and calculations of 
the lines about a circle, likewife employed its defcending 
terms to exprefs the fractional parts. This fexagefimal 
fubdivifion, called fometimes logijiic arithmetic, is (till re¬ 
tained in exprejTmgarcs by degrees, minutes, and feconds; 
though it has at length been fuperfeded by the decimal 
fyftem, in the exhibition of chords, or of the fines and 
tangents which came to difplace thefe in the modern tri¬ 
gonometry. Nor was the latter improvement embraced, 
or even contemplated, at once. The Arabians, though 
well acquainted with the advantages of the denary fcale, 
appeared fatisfied with the fexagenary numeration de¬ 
rived from their Greek inftruftors. The great rellorer 
of mathematical fcience in Europe, George Purbach of 
Vienna, a man of original and extenfive genius, who died 
at an early age in 1462, in the tables for lines which he 
computed to every ten minutes of the quadrant, diftin- 
guilhed the radius into 600,000 parts, thus blending the 
"fexagefimal with the decimal notation. His difciple and 
fuccelfor, John Muller, commonly ftyled Regiomontanus, 
from Koninglburg, the place of his birth, after fome he- 
fitation, laid afide this fubdivifion of the radius in 1464, 
and enlarged it to a million of parts, having re-calculated 
the fines, and likewife joined to them tables of tangents. 
But his work lay many years after his deceafe in manu- 
lcript, and was not printed until 1533. A very long pe¬ 
riod Hill elapfed before mathematicians were trained to 
the ufe of decimals. Simon Stevinus, a celebrated Fle- 
milh geometer and engineer, was the firft who compofed, 
in 1582, a diftinft treatifeon the theory of thofe fractions. 
Nothing more clearly difclofes the properties of num¬ 
bers than the transferring of them to different feales of 
arrangement. Reckoning them, for inftance, byfucceffive 
braces, lealhes, warps, &c. they are difpoled on th shinary, 
ternary, quaternary, and fucceeding, feales. This procels, 
requiring only the continued divifion by two, three, four, 
&c. is readily performed. We fhall now, 
therefore, in order to bring all thefe feales, 
which we have been fuccelfively treating of, 
into one view, give an example of transfer¬ 
ring the fame fum into each ; and, in order to 
procure a greater variety in the refults, we 
iliall take a number that is rather large. Let 
it be fought to exhibit the number 2138507 on 
fuccefiive feales. Its decompofition will be 
Binary. 
2)2138507 
1069253 
534626 
267313 
133656 
66828 
33414 
16707 
8353 
0 thus effefted: 
0 Ternary. 
1 3)2138507 
Quaternary. Quinary. 
4)2138507 5)2138507 
4176 
I 
712835 
2 
53462613 
4277OI 
2088 
O 
237611 
2 
133656.2 
85540 
1044 
O 
79203 
2 
334 3 4 o 
17108 
522 
O 
26401 
O 
8353 2 
3421 
261 
O 
8800 
I 
208811 
684 
130 
I 
2933 
I 
522J0 
136 
65 
O 
977 
2 
1302 
27 
32 
I 
3 2 S 
2 
32 2 
5 
36 
O 
108 
I 
8° 
I 
8 
O 
36 
O 
2 O 
0 
4 
O 
32 
O 
0 2 
2 
O 
4 
O 
I 
O 
I 
I 
0 
I 
O 
I 
Senary. 
6)2138507 
Septenary. 
7)2138507 
Oflari). 
8)2138507 
356417,5 
305501 
O 
'267313 3 
59402,5 
43643.0 
33414 » 
9900 ( 2- 
6234 5 
4176 6 
*650:0 
8 90'4 
522 
O 
275. 0 
127 
1 
65*2. 
45,5 
18 
I ' 
8 
I 
7 3 
2 
4 
I 
O 
2 
X 
O 
4 
c 
I 
Nonary. 
Undenary. 
Duodenary. 
9)2138507 
11)2138507 
12)2138507 
23761i 
8 
194409 
8 
178208 
Tsr 
26401 
2 
17673 
5 
148 50 
8 
2933 
4 
1606 
7 
1237 
6 
325 
8 
146 
O 
103 
I 
36 
I 
*3 
3 
8 
7 ’ 
4 
O 
I 
2 
O 
8 
O 
4 
O 
2 
Hence the 
fame number 2138507 
will be thus repre- 
fented on the different feales: 
Binary 
- 
I00000I0IOOOOIIOOOIOil 
Ternary 
- 
I IOOOJ 22 I 10222 
Quaternary 
- 
200220X2023 
Quinary 
- 
- 
I02I4I3012 
Senary 
- 
- 
- 
113500255 
Septenary 
- 
- 
241I45OO 
Octary 
- 
- 
- 
10120613 
Nonary 
- 
- 
- 
4018428 
Denary 
- 
- 
2138507 
Undenary 
- 
1230768 
Duodenary 
- 
8716855- 
Here it is 
evident, 
as it is indeed 
from the nature of 
the fubjedl under inveftigation, that the greater the ra¬ 
dix is, the lefs will be the number of digits necefiary for 
exprelfing any given number; but the operations of mul¬ 
tiplication, divifion, &c. will be the more complex ; and 
therefore, in judging of the advantages and difadvantages 
of different fyftems, we ought to keep both thefe circum- 
ftances in view, as alfo a third, which is the number of 
prime divifions of the radix ; and, on a juft eftimate of the 
whole, the radix 12 will be found preferable to any of the 
other fyftems. 
To transform a number from any other fcale of notation 
to the denary or common fcale. —This propofition is the 
converfe of the foregoing one ; and it is readily effefted 
by the reverfe operation ; i. e. not by divifion, but by 
multiplication and addition. We have feen (p. 303.) 
that, mi, in our common notation, is the fame as 1 X 10 4 
= ioooo-j-i X io 3 =iooo -f 1 X io 2 =: joo-J-io-j-i. By treat¬ 
ing the root of any required fcale, therefore, in the fame 
manner, the value of any denary number in that fcale is 
readily found. 
Example 1. Transform 7184 from the duodenary to the 
common fcale of notation. 
7184=7 X i2 3 -fiXn 2 +8X 12+4. 
Therefore we have, 7Xi2 3 =i2096 
I X I2 2 = 144 
8Xl2 — 96 
4 = 4 
Duodenary 7184 = 12340m the common feale. 
Example 2. Transform 1534 from the fenary to the de¬ 
nary fcale. 
1534= 1X 6 3 + 5 X 6 2 -f 3 X 6+4. 
Hence, iX6 3 = 216 
5X6®= 180 
3 X« = 18 
4 = 4 
Senary 1534 = 418 in the common feale. 
