The Atlantic Monthly | Page 3

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 are required; including the compound signs also, only fifteen. By Roman notation, without subtraction, fifteen; with subtraction, nine. By alphabetic notation, three signs without repetition. By the Arabic, one sign thrice repeated. By Federal coins, nine pieces, one of them being a repetition. By dual coins, six pieces without a repetition, a fraction remaining.
In the gradation of real weights, measures, and coins, it is important to adopt those grades which are most convenient, which require the least expense of capital, time, and labor, and which are least likely to be mistaken for each other. What, then, is the most convenient gradation? The base two gives a series of seven weights that may be used: 1, 2, 4, 8, 16, 32, 64 lbs. By these any weight from one to one hundred and twenty-seven pounds may be weighed. This is, perhaps, the smallest number of weights or of coins with which those several quantities of pounds or of dollars may be weighed or paid. With the same number of weights, representing the arithmetical series from one to seven, only from one to twenty-eight pounds may be weighed; and though a more extended series may be used, this will only add to their inconvenience; moreover, from similarity of size, such weights will be readily mistaken. The base ten gives only two weights that may be used. The base three gives a series of weights, 1, 3, 9, 27, etc., which has a great promise of convenience; but as only four may be used, the fifth being too heavy to handle, and as their use requires subtraction as well as addition, they have neither the convenience nor the capability of binary weights; moreover, the necessity for subtraction renders this series peculiarly unfit for coins.
The legitimate inference from the foregoing seems to be, that a perfectly practical system of weights, measures, and coins, one not practical only, but also agreeable and convenient, because requiring the smallest possible number of pieces, and these not readily mistaken for each other, and because agreeing with the natural division of things, and therefore commercially proper, and avoiding much fractional calculation, is that, and that only, the successive grades of which represent the successive powers of two.
That much fractional calculation may thus be avoided is evident from the fact that the system will be homogeneous. Thus, as binary gradation supplies one coin for every binary division of the dollar, down to the sixty-fourth part, and farther, if necessary, any of those divisions may be paid without a remainder. On the contrary, Federal gradation, though in part binary, gives one coin for each of the first two divisions only. Of the remaining four divisions, one requires two coins, and another three, and not one of them can be paid in full. Thus it appears there are four divisions of the dollar that cannot be paid in Federal coins, divisions that are constantly in use, and unavoidable, because resulting from the natural division of things, and from the popular division of the