30 1 
N U M B E R. 
fent notation. Such, however, being the cafe with re¬ 
gard to the divifion 'of numbers, the choice of the radix 
10 was by fome perfons thought the moft natural that 
could have been feleCred, though it was not, perhaps, 
the heft adapted to arithmetical calculations, iz being 
by many thought much better fuited for this purpofe: 
the ad vantages of if, however, are not fuch as to lead us 
to expeCt, or even to with, that it fhould ever be fubfti- 
tuted for that which long-eftablifned cuftomhas rendered 
■fo familiar to all our ideas of numbers. But, if not the 
heft, the pretent fcale is as convenient as can be defired ; 
ir will, therefore, not beamifs to in vc-ftigate the principles 
on which it is founded, and the advantages and diihd- 
vantages of it, compared with other fyftems. 
The earlieft and fimnleft mode of reckoning was by 
pairs, avifing naturally from the circumftance of both 
hands being employed for the fake of expedition. It is 
now familiar among fportfmen, who ufe the names of 
brace and couple, words that figirify pairing and yoking. 
To count by threes was another ftep, though not prac- 
tifed to the fame extent: it has been preferved, however, 
by the fame clafs of men, under the term Icr/h, meaning 
the firings by which three dogs, and no more, can beheld 
at once in the hand. The numbering by fours has had a 
more extenfive application. It was evidently fuggefted 
by the cuftom, in rapid tale, of taking a pair in each 
hand. Our fifhermen, who generally reckon in this way, 
call every double pair of herrings, for inftance, a throw 
or call; and the term warp, which, from its German 
origin, has exaCtly the fame import, is employed to denote 
four in various articles of trade. Thus it is evident that 
we may have a binary, ternary, quaternary, &c. as well 
as a denary or decimal, notation. 
Binary Scale. —In this fcale, the characters are only 
two, o and i. Very fimple and eafy, therefore, all calcu¬ 
lations mutt be upon this fyftem ; but the.advantage in 
point of eafe is more than counterbalanced by the intri¬ 
cacy of exprefiion, on account of the quantity of figures 
neceflary for reprefenting a number of any confiderable 
extent; for 1000 in the binary fcale would require io 
places of figures; and, to exprefs 1,000,000, we muft have 
20 places, which would necefiarily be very embarrafling ; 
at the fame time that all calculations would proceed very 
tlowly, on account of the number of figures that muft be 
made to enter therein, as noticed under Arithmetic, 
vol. ii. p. 164.. 
The advantages of the binary fcale of notation were 
prodigioufly extolled by the celebrated Leibnitz, who con- 
defeended even to write a difeourfe on its ufe and mode 
of adaptation. This very learned, original, and inde¬ 
fatigable, philofopher, had the fatisfaCtion to difeover, 
(as he thought at leaft,) from the refearches of ingenious 
miftionaries, and particularly from the letters of father 
Bouvet, fome feeble traces of the binary notation in the 
early hiftory of China. Fuu-hi, the firft emperor, and 
founder of that vaft monarchy, is venerated in the e’aft as 
a promoter of geometry, and the inventor of a fcience the 
knowledge of which has been iince loft. The emblem of 
this occult fcience coniifts of eight feparate clufters of 
three parallel lines, or trigrams, drawn one above another, 
after the Chinefe manner of w'riting, and reprefented 
either entire, or broken in the middle. Thofe varied 
trigrams were called Koua, or “ fulpended fymbols,” 
from the circumftance of their being expofed in the pub¬ 
lic places. In the compofition of fuch varied clufters, it 
was not difficult for the fagacity of Leibnitz to perceive 
the application of the binary fcale carried only to three 
ranks, or as far as the number eight. The entire lines 
fignify one, two, or four, according to their order ; while 
the broken lines are void, and ferve merely to indicate 
the rank of the others. Before this invention, the only 
mode of reckoning ufed in China was by knots tied on a 
fingle cord. The next emperor, Chen-nomg, devoted his 
attention chiefly to the improvement of agriculture. But 
lie likewife extended the Kona, or d italic fyftem, of his 
predeceflor, 'employing hexagrams inftead of trigrams. 
Thefe hexagrams were formed with fix parallel lines., 
broken or entire, and placed one above the other. In 
fhort, they were the firft fix bars of the binary fcale, of 
which the laft term would reprefent thirty-two, and all 
the terms together would reach to fixty-four. Chen- 
noutrg contrived that the hexagrams fho-uid likewife ex¬ 
hibit fixty-four varieties of combination, by placing them 
in eight rays iffuing from the lame central fpace which 
was occupied by one of the eight trigrams. See Iviemoires 
fur 1 es Chinois, tome ii.—We muft fay, that we think it 
very lingular that a Chinefe enigma fhould find itsCEdipus 
only in Europe. But probably there is more ingenuity 
than truth- in the whole explanation : if a whole line re- 
preCents 1, why muft a broken one be fuppofed to de¬ 
note o, rather than twice 1, or the half of r? 
The binary fcale appears beft adapted to the'defeending 
progreffion ; for the fractions produced by a continued 
infection, are, from the equal competition of buyer and 
feller, naturally introduced into commercial tranfaCtions, 
and employed almoft exclulively among the Eaftern na¬ 
tions. This fubdivifion is likewife ufed with convenience 
in Europe, for afeertaining the fmaller weights. 
General Rule fur all the Scales. —Divide the number, as 
exprefled in our common decimal notation, continually 
by the root of the fcale you would bring it to, and referve 
the 
remainders. 
Thefe remainders, taken in a reverfe 
order, will form the number in the required fcale. 
2)1 
2)5 
zf 
2)1 
o> an 
d 
1000 
5 °° 
O 
250 
O 
125 
O 
62 
I 
3 1 
O 
15 
I 
7 
I 
3 
I 
I 
I 
O 
I 
10, therefore, on the 
binary fcale,will 
be reprefented 
by - 1010 
100 by - 110100 
1000 by 1 m 1 ioiooq 
Ternary Sc ale. —This fcale, which reckons 
by fucceffive threes, or triads, advances with 
fomewhat more fpeed, as having three cha¬ 
racters, o, 1, 2. Still, however, we fhould re¬ 
quire feven figures, 2111101, to exprefs the 
date of the prefent year 1819, as will appear 
from the Example in the margin. 
3)1819 
3)6 o 6 |i 
3)20210 
3)67:1 
3 ) 22 U 
3)7T 
3 ) 2 ;i 
02 
Quaternary Scale. —The quaternary fcale may be 
conlidered as a re-duplication of the binary, fo that the 
fame number will be comprifed in juft half as many 
figures. Thus, if the exprefiion for 1000, which in 
the binary fcale is 1111101000, be articulate at every 
alternate place from the right, thus, + ) IOOO 
ii , ii . io , io , oo ; and the value of each period -— 
2 C0'0 
62L 
1 
152 
33 
0.3 
two of the 
be fubftituted, as 1 for 01, 2 for 10, and 3 
for 11 ; the whole will thereby be transferred 
into the quaternary arrangement, and will 
ftandthus: 33220; to prove which, it is di¬ 
vided in the common W'ay in the margin. 
We are told that the Guaranis and Lulos, 
very loweft races of favages which inhabit the boundlefs 
forefts of South America, count only by fours; at leaft, 
that they exprefs five by four-and-one, fix by four-and- 
two, and fo forth. We may gather from Ariftotle, that 
a certain tribe of Thracians were accuftomed to ufe the 
quaternary fcale of numeration ; for he fays that they pro¬ 
ceeded no farther than four, which they would doubtlefs 
continue to repeat. If fuch was the hiftorical fad, thofe 
fimple people muft have never advanced beyond the early 
practice of reckoning fucceffively by caj'ts, or warps. It 
1 feems 
