} (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 no