1 4 8 -
-----
1 6|2
7 4 -
---
9 0
. .
_Answer_, 0.02439.

C.S. PEIRCE.

* * * * *

[SCIENCE.]

EXPERIMENTS IN BINARY ARITHMETIC.

Those who can perform in that most necessary of all mathematical
operations, simple addition, any great number of successive examples
or any single extensive example without consciousness of a severe
mental strain, followed by corresponding mental fatigue, are
exceptions to a general rule. These troubles are due to the quantity
and complexity of the matter with which the mind has to be occupied at
the same time that the figures are recognized. The sums of pairs of
numbers from zero up to nine form fifty-five distinct propositions
that must be borne in memory, and the "carrying" is a further
complication. The strain and consequent weariness are not only felt,
but seen, in the mistakes in addition that they cause. They are, in
great part, the tax exacted of us by our decimal system of arithmetic.
Were only quantities of the same value, in any one column, to be
added, our memory would be burdened with nothing more than the
succession of numbers in simple counting, or that of multiples of two,
three, or four, if the counting is by groups.

It is easy to prove that the most economical way of reducing addition
to counting similar quantities is by the binary arithmetic of
Leibnitz, which appears in an altered dress, with most of the zero
signs suppressed, in the example below. Opposite each number in the
usual figures is here set the same according to a scheme in which the
signs of powers of two repeat themselves in periods of four; a very
small circle, like a degree mark, being used to express any fourth
power in the series; a long loop, like a narrow 0, any square not a
fourth power; a curve upward and to the right, like a phonographic
_l_, any double fourth power; and a curve to the right and downward,
like a phonographic _r_, any half of a fourth power; with a vertical
bar to denote the absence of three successive powers not fourth
powers. Thus the equivalent for one million, shown in the example
slightly below the middle, is 2^{16} (represented by a degree-mark in
the fifth row of these marks, counting from the right) plus 2^{17} +
2^{9