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