onate inconvenience, it can never be set aside entirely, as has
been proved by experience. That men have set it aside in part, to their
own loss, is sufficiently evidenced. Witness the heterogeneous mass of
irregularities already pointed out. Of these our own coins present a
familiar example. For the reasons above stated, coins, to be practical,
should represent the powers of two; yet, on examination, it will be
found, that, of our twelve grades of coins, only one-half are obtained
by binary division, and these not in a regular series. Do not these six
grades, irregular as they are, give to our coins their principal
convenience? Then why do we claim that our coins are decimal? Are not
their gradations produced by the following multiplications: 1 x 5 x 2 x
2-1/2 x 2 x 2 x 2-1/2 x 2 x 2 x 2, and 1 x 3 x 100? Are any of these
decimal? We might have decimal coins by dropping all but cents, dimes,
dollars, and eagles; but the question is not, What we might have, but,
What have we? Certainly we have not decimal coins. A purely decimal
system of coins would be an intolerable nuisance, because it would
require a greatly increased number of small coins. This may be
illustrated by means of the ancient Greek notation, using the simple
signs only, with the exception of the second sign, to make it purely
decimal. To express $9.99 by such a notation, only three signs can be
used; consequently nine repetitions of each are required, making a total
of twenty-seven signs. To pay it in decimal coins, the same number of
pieces are required. Including the second Greek sign, twenty-three signs
are required; including the compound signs also, only fifteen. By Roman
notation, without subtraction, fifteen; with subtraction, nine. By
alphabetic notation, three signs without repetition. By the Arabic, one
sign thrice repeated. By Federal coins, nine pieces, one of them being a
repetition. By dual coins, six pieces without a repetition, a fraction
remaining.

In the gradation of real weights, measures, and coins, it is important
to adopt those grades which are most convenient, which require the least
expense of capital, time, and labor, and which are least likely to be
mistaken for each other. What, then, is the most convenient gradation?
The base two gives a series of seven weights that may be used: 1, 2, 4,
8, 16, 32, 64 lbs. By these any weight from one to one hundred and
twenty-seven pounds may be weighed. This is, perhaps, the smallest
number