NUMBER. 
311 
fures to our prefent fyftem of numeration, by making 
them decreafe in decimal progreflion, was made by Stevi- 
nus, a Dutch mathematician, in the 17th century. 
Undenary Scale. —If we advance beyond the denary 
fcale, it is evident that we (hall w'ant more characters, at 
the fame time that a lefs number of them will be required 
to exprefs a given fum, if large. Thus, in the undenary 
fcale we (hould want a character to exprefs 10; and in 
duodenary fcale, of which we are prefently to fpeak, we 
(hould have occafion for another to exprefs 11 II ) IO oo 
alfo. Mr. Barlow affumes the Greek (p for 10, —— 
and it for 11. The procefs of reduction or 9 °|P 
transformation is the fame in thefefcales as in 
the former: 1000 will be exprefled by 829. °l® 
Duodenary Scale. —This fyftem muft have been in¬ 
troduced at an advanced ftage of fociety. It plainly drew 
its origin from the obfervation of the celeftial phenome¬ 
na, there being twelve months or lunations commonly 
reckoned in a folar year. The Romans adopted that in¬ 
dex, to mark their fubdivifion of the unit of meafure or 
of weight. They diftinguiftied the foot into three hand- 
breadths, or palms, and each palm into four lengths of 
the thumb-joint, or digit; and, in like manner, they 
firft bifeCted the pound, next bifeCted this again, and 
then divided the quarter into three final portions. The 
twelfth part of a foot and that of a pound were alike 
termed uncia, which has branched into the modern words 
inch and ounce, applied more difcriminately with us to the 
fubdivifions of meafure and of weight. The mode of 
reckoning by twelves, or dozens, has been very generally- 
adopted in the wholefale trade. Nor is its application 
confined to the firft term of the progreflion, but extends 
to the fecond, or even the third. Twelve dozen, or 144, 
makes the long orgreat hundred of the north¬ 
ern nations, or the grofsof traders. Twelve 
times this again, or 1728, forms the double 
grofs. Of courfe this fum, 1728, muft be 
reprefented in the duodenary fcale by 1000, 
as in the margin; and 718 would require 
thefe marks, 47 r(p. 
The duodenary fyftem of numbers, while it poflefles 
all the advantages of the fenary in point of finite frac¬ 
tions, is fuperior even to the decimal fyftem for fimpli- 
city of expreflion; and the only additional burden to the 
memory, is two characters for reprefenting 10 and 11; 
for the multiplication-table in our common arithmetic is 
generally carried as far as 12 times 12, although its natu¬ 
ral limit is 9 times 9, which is a clear proof that the 
mind is capable of working with the duodenary fyftem 
without any inconvenience or embarraflinent: and hence 
we may conclude, that the choice of the denary arith¬ 
metic did not proceed from reflection and deliberation, 
but was the refult of fome caule operating, in an unfeen 
and unknown manner, on the inventor of this fyftem; 
and it may therefore be confidered as a fortunate cir- 
cumftance, that, for this unpremeditated index, that par¬ 
ticular one (hould have been feleCted, which holds at lead: 
the fecond place in the fcale of general utility. 
To perform duodecimal operations by means of the duode¬ 
nary fcale of notation .—Transform the number of feet, if 
above 12, into the duodenary fcale, and fet the inches 
and parts as decimals. Then multiply as in common 
arithmetic, except carrying for every 12 inftead of for 
every 10; and in the refult transform again the integral 
part of the produCt into the denary fcale. 
Ex. 1. Multiply 17 ft. 3 in. 4'. by 19 ft. jin. 11'. 
17 3 4 = i 5’34 
19 5 n = 17'jTr 
13908 
7248 
{poir 4 
1534 
240-6688 = 336 ft. 9 in. .6'8" 8'* 
12)1728 
I44j° 
120 
1 o 
o 1 
Ex. 2. Find the folidity of a cube, whofe fide is 13 ft. 
7 7 '- 
1177 = 13 7 7 
11-77 
77 T 5 I 
77T5I 
1177 
1177 
135-93:51 
11-77 
9049867 
9049867 
13593-61 
13593-61 
1571-281417 = 2533 ft. 2 in. 8'1" 4'" 1 Iv 7 r anf. 
This method was firft publifhed in Nicholfon’s Journal, 
vol. xxv. and it appears to poflefs confiderable advantage 
over the common rule, both on account of the facility of 
operation, and the accuracy of the refult ; as like.wife 
that it may thus be lubmitted to proof, the fame as com¬ 
mon multiplication, which it is not poflible to apply to 
the old method. The fame principles are equally appli¬ 
cable to the extraction of the fqtiare root, as is evident 
by the following example. 
Ex. 3. Having given the area of a fquare equal to 17 
ft. 4 in. 6'. required the length of its fide. 
15-46(4-2029 
14 
82 
146 
2 
144 
8402 
20000 
2 
14804 
8404 
733-800 
-— 
671-4404 
4733-3 
Therefore the fide 
is 4 ft. 2 in. o ; 
And thus may any other numerical operation be per¬ 
formed with as much facility as in common arithmetic. 
The main advantage of this fcale, confifts in its fitnefs 
to denote fractional parts. Its root has indeed no fewer 
than four faCtors 2, 3, 4, and 6 ; while ten is divifible only 
by 2 and 5. Several attempts, accordingly, have at differ¬ 
ent times, been made to carry the Duodenary Scale into 
aCtual practice. It is a Angular faCt, that the famous 
Charles XII. of Sweden, whofe conduCt was always 
marked by an irregular grandeur of fentiment, is reported 
to have occupied his leilure moments, during the depth 
of winter, in the trenches before Fredericklhall, on the 
Norwegian frontier, with deviling the means of introdu¬ 
cing the duodecimal fcale of arithmetic into his heredi¬ 
tary ftates. Had he lived to attempt the execution of 
that fcheme, he would probably have encountered no lefs 
difficulty, though attended by (ewer difafters, than lie met 
with in his chimerical projeCt of effecting the liberation of 
Europe. 
Sexagenary Scale. —Another progreflion, (till fwifter 
in its operation than that by (cores, and long familiar to 
aftronomers, was introduced into the Alexandrian fchool 
by the famous Ptolemy, who had the merit of digeft- 
ing the refults of celeftial obfervations into a body ot re¬ 
gular fcience. The lexagenary fcale, proceeds, as its 
name implies, by the fucceflive powers of fixty. This 
arrangement of numbers appears quite artificial, and was 
no doubt fuggefted by the divifion of the circle, founded 
on aftronomical phenomena. Since the year is compofed 
4 of 
