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Page 36
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} (two _l_-curves in the fifth and third places of _l_-curves)
plus 2^{18} + 2^{14} + 2^{6} (three loops) plus 2^{19} (the _r_-curve
at the extreme left); while the absence of 2^{3}, 2^{2}, and 2^{1} is
shown by the vertical stroke at the right. This equivalent expression
may be verified, if desired, either by adding the designated powers of
two from 524,288 down to 64, or by successive multiplications by two,
adding one when necessary. The form of characters here exhibited was
thought to be the best of nearly three hundred that were devised and
considered and in about sixty cases tested for economic value by
actual additions.
In order to add them, the object for which these forty numbers are
here presented in two notations, it is not necessary to know just
_why_ the figures on the right are equal to those on the left, or to
know anything more than the order in which the different forms are to
be taken, and the fact that any one has twice the value of one in the
column next succeeding it on the right. The addition may be made from
the printed page, first covering over the answer with a paper held
fast by a weight, to have a place for the figures of the new answer as
successively obtained. The fingers will be found a great assistance,
especially if one of each hand be used, to point off similar marks in
twos, or threes, or fours--as many together as can be certainly
comprehended in a glance of the eye. Counting by fours, if it can be
done safely, is preferable because most rapid. The eye can catch the
marks for even powers more easily in going up and those for odd powers
(the _l_ and _r_ curves) in going down the columns. Beginning at the
lower right hand corner, we count the right hand column of small
circles, or degree marks, upward; they are twenty-three in number.
Half of twenty-three is eleven and one over; one of these marks has
therefore to be entered as part of the answer, and eleven carried to
the next column, the first one of _l_-curves. But since the curves are
most advantageously added downward, it is best, when the first column
is finished, simply to remember the remainder from it, and not to set
down anything until the bottom is reached in the addition of the
second column, when the remainders, if any, from both columns can be
set down together. In this case, starting with the eleven carried and
counting the number of the _l_-curves, we find ourselves at the bottom
with twenty-four--twelve to carry, and nothing to set down except the
degree mark from the first column. With the twelve we go up the
adjoining loop column, and the sum must be even, as this place is
vacant in the answer; the _r_-curve column next, downward, and then
another row of degree marks. The succession must be obvious by this
time. When the last column, the one in loops to the extreme left, is
added, the sum has to be reduced to unity by successive halvings. Here
we seem to have eleven; hence we enter one loop, and carry five to the
next place, which, it must be remembered, is of _r_-curves. Halving
five we express the remainder by entering one of these curves, and
carry the quotient, two, to the degree mark place. Halving again gives
one in the next place, that of _l_-curves; and the work is complete.
It is recommended that this work be gone over several times for
practice, until the appearance and order of the characters and the
details of the method become familiar; that, when the work can be done
mechanically and without hesitation, the time occupied in a complete
addition of the example, and the mistakes made in it, be carefully
noted; that this be done several times, with an interval of some days
between the trials, and the result of each trial kept separate; that
the time and mistakes by the ordinary figures in the same example, in
several trials, be observed for comparison. Please pay particular
attention to the difference in the kind of work required by the two
methods in its bearing on two questions--which of them would be easier
to work by for hours together, supposing both equally well learned?
and in which of them could a reasonable degree of skill be more
readily acquired by a beginner? The answer to these questions, if the
comparison be a fair one, is as little to be doubted as is their high
importance.
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