of weights or of coins with which those several quantities of
pounds or of dollars may be weighed or paid. With the same number of
weights, representing the arithmetical series from one to seven, only
from one to twenty-eight pounds may be weighed; and though a more
extended series may be used, this will only add to their inconvenience;
moreover, from similarity of size, such weights will be readily
mistaken. The base ten gives only two weights that may be used. The base
three gives a series of weights, 1, 3, 9, 27, etc., which has a great
promise of convenience; but as only four may be used, the fifth being
too heavy to handle, and as their use requires subtraction as well as
addition, they have neither the convenience nor the capability of binary
weights; moreover, the necessity for subtraction renders this series
peculiarly unfit for coins.

The legitimate inference from the foregoing seems to be, that a
perfectly practical system of weights, measures, and coins, one not
practical only, but also agreeable and convenient, because requiring the
smallest possible number of pieces, and these not readily mistaken for
each other, and because agreeing with the natural division of things,
and therefore commercially proper, and avoiding much fractional
calculation, is that, and that only, the successive grades of which
represent the successive powers of two.

That much fractional calculation may thus be avoided is evident from the
fact that the system will be homogeneous. Thus, as binary gradation
supplies one coin for every binary division of the dollar, down to the
sixty-fourth part, and farther, if necessary, any of those divisions may
be paid without a remainder. On the contrary, Federal gradation, though
in part binary, gives one coin for each of the first two divisions only.
Of the remaining four divisions, one requires two coins, and another
three, and not one of them can be paid in full. Thus it appears there
are four divisions of the dollar that cannot be paid in Federal coins,
divisions that are constantly in use, and unavoidable, because resulting
from the natural division of things, and from the popular division of
the pound, gallon, yard, inch, etc., that has grown out of it. Those
fractious that cannot be paid, the proper result of a heterogeneous
system, are a constant source of jealousy, and often produce disputes,
and sometimes bitter wrangling, between buyer and seller. The injury to
public morals arising