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.

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