The Providence news. (Providence [R.I.]) 1891-1906, 1896-04-30

While this mode Of expressing numbers
has some advantage, yet II would be at
tendedswith the inconvenience OF requir-
ing too many figures, as twenty would be
necessary to express a number equal TO
about million. IL the binary arithmetic
OF Liebnitz Is entitled TO no further notice
than to be cIassed among the curious
arithmetical speculatiops, the case. how
ever, is not the same with duodenary
arithmetic, or that kind which, as al
ready observed, would have been brought
into use had men been born v ith twelve
fl ngers. This arithmetic would indeed
have been as expeditions as the airthme-
tic now employed, and even somewhat
more SO; the number OF the characters
which would have received an increase
only OF two to express ten and eleven
wosld- have been ss little -bt4FtieBsoHfe to
the memory as the present characters
and would have been attended with ad-
vantages which ought 10 make us regret
that this system was not originally
adopted.
1t is not probable, however, that this
duodenary system would have been pre-
ferred had philosophers presided at the
invention, for II would have been readily
seen that IL, all of the mum ers from TO
20, is that which possesses the advantage
of being small and OF having the greatest
number of divisors. for there are no less
than four divisors by which 1t can be di-
vided without fraction-viz., 2, 8, 4 and
6. The number is, indeed, has four di-
visors also, but being larger than II, the
latter deserves to he preferred for measur-
ing the periods of pumeration. The first
OF these periods. from / TO II. would
have had the advantage OF being divis.
ible by 2, 8, 4 and G, and the second, from
I to 144. by 2. 8, 4, q g., ". 16, Is, "a,
36, as, 72; whereas, in our own system,
the rffst period, from to 10, has only
two divisors, 2 and 5, and the second,
from TO 100, hos only 2, 4, 5. 10, 20. 25.
90. 1t is evident, therefore, that frac-
tions would less frequently have occurred
TO the designation OF numbers in that
way namely, by twelve.
DUoDEcIMAL PROGRESSION.
But what would have been the most
convenient in this mode of Enumeration
is that in the divisions and subdivisions
of measures IL would heye introduced a
duodecimal progression. Thus as the
foot has by chance been divided into
twelve inches, the inch formerly divided
into twelve Ines and the line into twelve
points, the standard pound avoirdupois
would have teco the same as the apothe-
carries pound OF twelve ounces. The ounce
would have been divided into twelve
drachms and the drachm into twelve
grains or parts of any other depomina-
tion. The day would have been divided
into twelve equal portions called hours,
the hour into twelve otner parts, each
equal to ten minutes, and SO on success
sively. The case would have been the
same in regard to measures of capacity.
1t is well known by daily experience
that when 1t Is necessary to divide any
measure into three, four, or six parts an
integer number In the measure OF lower
denomination cannot be found, or at
least only by chance. One thirn reduces
to .33333--anu so on ad infinitum, and
no figures can ever express the exact
qusntity. They can only suggest IT,
which is all that any intermediate deci-
NAL can do. But with twelve as base,
every fraction having denominator
smaller than the base Will reduce TO
precise decimal equivalent, except IL. 10
and ILL, but it Is not serious matter
even il these do fail, because they are
used less, perhaps, than any others.
With ten as n base the more important
fract ions, such es l-3, 1-6, ETC., are 81
ways intermediate decimals. In reading
decimals having twelve as base, Il
would be one-twelfth instead oi one
tenth, etc.
IL Is evident that In duodenary erith-
metic the nine first numbers might be
expressed as usual by the nine known
characters, but as the period ought TO
terminste only at twelve IL would bc
necessary TO express ten and eleven by
some new and simple characters, cay L
70 Il