The Atlantic Monthly | Page 4

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 from this cause, like the destructive effect of the constant
dropping of water, though too slow in its progress to be distinctly
traced, is not the less certain. The economic value of binary gradation is,
in the aggregate, immense; yet its moral value is not to be overlooked,
when a full estimate of its worth is required.
Admitting binary gradation to be proper to weights, measures, and
coins, it follows that a corresponding base of numeration and notation
must be provided, as that best suited to commerce. For this purpose, the
number two immediately presents itself; but binary numeration and
notation being too prolix for arithmetical practice, it becomes necessary
to select for a base a power of two that will afford a more
comprehensive notation: a power of two, because no other number will
agree with binary gradation. It is scarcely proper to say the third power
has been selected, for there was no alternative,--the second power being
too small, and the fourth too large. Happily, the third is admirably
suited to the purpose, combining, as it does, the comprehensiveness of
eight with the simplicity of two.
It may be asked, how a number, hitherto almost entirely overlooked as
a base of numeration, is suddenly found to be so well suited to the
purpose. The fact is, the present base being accepted as proper for
numeration, however erroneously, it is assumed to be proper for
gradation also; and a very flattering assumption it is, promising a
perfectly homogeneous system of weights, measures, coins, and
numbers, than which nothing can be more desirable; but, siren-like, it
draws the mind away from a proper investigation of the subject, and the
basic qualities of numbers, being unquestioned, remain unknown.
When the natural order is adopted, and the base of gradation is
ascertained by its adaptation to things, and the base of numeration by
its agreement with that of gradation, then, the basic qualities of

numbers being questioned, two is found to be proper to the first use,
and eight to the second.
The idea of changing the base of numeration will appear to most
persons as absurd, and its realization as impossible; yet the probability
is, it will be done. The question is one of time rather than of fact, and
there is plenty of time. The diffusion of education will ultimately cause
it to be demanded. A change of notation is not an impossible thing. The
Greeks changed theirs, first for the alphabetic, and afterwards, with the
rest of the civilized world, for the Arabic,--both greater changes than
that now proposed. A change of numeration is truly a more serious
matter, yet the difficulty may not be as great as our apprehensions paint
it. Its inauguration must not be compared with that of French gradation,
which, though theoretically perfect, is practically absurd.
Decimal numeration grew out of the fact that each person has ten
fingers and thumbs, without reference to science, art, or commerce.
Ultimately scientific men discovered that it was not the best for certain
purposes, consequently that a change might be desirable; but as they
were not disposed to accommodate themselves to popular practices,
which they erroneously viewed, not as necessary consequences, but
simply as bad habits, they suggested a base with reference not so much
to commerce as to science. The suggestion was never acted on,
however; indeed, it would have been in vain, as Delambre remarks, for
the French commission to have made the attempt, not only for the
reason he presents, but also because it does not agree with natural
division, and is therefore not suited to commerce; neither is it suited to
the average capacity of mankind for numbers; for, though some may be
able to use duodecimal numeration and notation with ease, the great
majority find themselves equal to decimal only, and some come short
even of that, except in its simplest use. Theoretically, twelve should be
preferred to ten, because it agrees with circle measure at least, and ten
agrees with nothing; besides, it affords a more comprehensive notation,
and is divisible by 6, 4, 3, and 2 without a fraction, qualities that are
theoretically valuable.
At first sight, the universal use of decimal numeration seems to be an

argument in its favor. It appears as though Nature had pointed directly
to it, on account of some peculiar fitness. It