NOTATION. 
be sage out of the foregoing feries, to weigh 716 
pound 
Firft, 716, in the ternary f{cale, is expreffed by 
222112 
Add I 
222120 
Add 10 = 3 
fore) = 3° 
Add 
1000000 = 3° 
Therefore, 222112 = 3° — (3*+ 341); that i ‘iss 3 5 muft 
be placed in one Seale, and the ae weights 3°-+ 3 +1 
in the other fcale, with the body to be wei hed 
hat weights out of see above feries muft 
be felected to afcertain a weight of 1 
Firft, 1319 = 1210212 in the pied fcale 
1210212 
I 
1210220 
1¢ = 3 
I21IO0O 
100000, = 3° 
2011000 
1o90coo = 3° 
37 + 3° + 33 
IOOorIo0oo = 
oa d hence we conclude, that the es 37 + 3t + 3° 
uft be put in one fcale, and the weights 3° s+ 341 
in ‘the other feale, with ‘the body whofe ac is to be af- 
certaine 
Thefe curious numerical problems are mentioned by 
rat page 253 of his * Analyfis Infinitorum,’’ and 
yftem of weights is rigoroufly demon rated ; 
fey een in the two feu a is much fimpler, 
and they have moreover the advantage of indicating the 
ae of folution, which is not attainable by Euler’s 
met 
Be for we conclude this article, it will not be improper, 
a few general obfervations on omparative ad- 
ubje& of our inve.tigation. 
this head, ge is evidently the firft confideration to be 
attended to, for int ie atone confilts the fuperiority of one 
fyftem over another; but this ought to be eftimated on 
two asoes eee Vid. ‘fimpli icity in arithmetical operations, 
and in arithmetical expreffions. 
ee pectin vent the 
cry emba rraffin z 
oul proceed very flowly, on account of the number o 
hee that muft be made to enter therein, 
The next fcale that has been recommended is the 
mber or places of figures for expreff- 
ng a num not be very great; befide; that thofe 
quantities, equivalent to our decimals, would be more fre- 
uently finite 
re) 
rs of 10, is indefinite, and ae only are 
finite, that contain the powers of thofe factors; and it is 
xaGily the fa 
fyftem; therefore, in — decimal fcale only fractions of the 
_ form are finite, and in the fenary {cale the finite 
q” 5” 
fractions are of the form and as there are neceflarily 
> 
ae 3” 
more numbers of the form 2* 3” within any finite limit, 
than there are of the form 2” 5”, it follows, that in a fyf- 
arithmetic, we uld oi more finite ex- 
preffions for fraéhons than h the ree a 
confequently, on this head, the cee mutt be giv 
the fenary fyftem; and, indeed, the o only poffible objeto 
that can be made to it is, that the operations would eed 
a little flower than in the decimal fcale, becaufe in ee 
numbers a greater number of figures muft be employed to 
exprefs them. This leads us to the confideration of the 
expreffion ; ; and the only additional burden to the memory, 
is two charatters for reprefenting Io and 11; for the mul- 
tiplication table in our common arithmetic is genera ally 
carried as far as 12 times 12, although its natural limit is 
only g times g, which is a clear proof, that the mind is 
a able of working with the duodenary fyftem, without 
any inconvenience or embarraflment ; hence we ma 
conclude, that the choice of the denary arithmetic did not 
roceed from reflection and deliberation, but was the refult of 
fomec ego ti in an unfeen and unknown manner, on the 
inventor of this fyftem; and it may, therefore, be gonidaea 
as a fortunate circumftance, t 
an improvement introduced a long time afterwards, as is 
evident from the arithmetic of the C ecks ; who, n otwith- 
36 different characters, and we whic 
a long time able to exprefs a number greate 
this was, however, afterwards cane: ides by the 
improvements of Archimedes, Apollonius, Pappus, &c. We 
have given, under the article CHARACTERS, a general ir 
