The Atlantic Monthly | Page 3

between men and things. Thus it appears the binary division of things is
not only most readily obtained, but also most frequently required.
Indeed, it is to some extent necessary; and though it may be set aside in
part, with proportionate 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 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