Ever since commerce began, merchants, and governments for them, have, from time to time, established multiples and divisions of given standards; yet, for some reason, they have seldom chosen the number ten as a base. From the long-continued and intimate connection of decimal numeration and notation with the quantities commerce requires, may not the fact, that it has not been so used more frequently, be considered as sufficient evidence that this use is not proper to it? That it is not may be shown thus:--A thing may be divided directly into equal parts only by first dividing it into two, then dividing each of the parts into two, etc., producing 2, 4, 8, 16, etc., equal parts, but ten never. This results from the fact, that doubling or folding is the only direct mode of dividing real quantities into equal parts, and that balancing is the nearest indirect mode,--two facts that go far to prove binary division to be proper to weights, measures, and coins. Moreover, use evidently requires things to be divided by two more frequently than by any other number,--a fact apparently due to a natural agreement between men and things. Thus it appears the binary division of things is not only most readily obtained, but also most frequently required. Indeed, it is to some extent necessary; and though it may be set aside in part, with proportionate inconvenience, it can never be set aside entirely, as has been proved by experience. That men have set it aside in part, to their own loss, is sufficiently evidenced. Witness the heterogeneous mass of irregularities already pointed out. Of these our own coins present a familiar example. For the reasons above stated, coins, to be practical, should represent the powers of two; yet, on examination, it will be found, that, of our twelve grades of coins, only one-half are obtained by binary division, and these not in a regular series. Do not these six grades, irregular as they are, give to our coins their principal convenience? Then why do we claim that our coins are decimal? Are not their gradations produced by the following multiplications: 1 x 5 x 2 x 2-1/2 x 2 x 2 x 2-1/2 x 2 x 2 x 2, and 1 x 3 x 100? Are any of these decimal? We might have decimal coins by dropping all but cents, dimes, dollars, and eagles; but the question is not, What we might have, but, What have we? Certainly we have not decimal coins. A purely decimal system of coins would be an intolerable nuisance, because it would require a greatly increased number of small coins. This may be illustrated by means of the ancient Greek notation, using the simple signs only, with the exception of the second sign, to make it purely decimal. To express $9.99 by such a notation, only three signs can be used; consequently nine repetitions of each are required, making a total of twenty-seven signs. To pay it in decimal coins, the same number of pieces are required. Including the second Greek sign, twenty-three signs