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 pound, gallon, yard, inch, etc., that has grown out of it. Those fractious that cannot be paid, the proper result of a heterogeneous system, are a constant source of jealousy, and often produce disputes, and sometimes bitter wrangling, between buyer and seller. The injury to public morals arising from this cause, like the destructive effect of the constant dropping of water, though too slow in its progress to be distinctly traced, is not the less certain. The economic value of binary gradation is, in the aggregate, immense; yet its moral value is not to be overlooked, when a full estimate of its worth is required.
Admitting binary gradation to be proper to weights, measures, and coins, it follows that a corresponding base of numeration and notation must be provided, as that best suited to commerce. For this purpose, the number two immediately presents itself; but binary numeration and notation being too prolix for arithmetical practice, it becomes necessary to select for a base a power of two that will afford a more comprehensive notation: a power of two, because no other number will agree with binary gradation. It is scarcely proper to say the third power has been selected, for there was no alternative,—the second power being too small, and the fourth too large. Happily, the third is admirably suited to the purpose, combining, as it does, the comprehensiveness of eight with the simplicity of two.
It may be asked, how a number, hitherto almost entirely overlooked as a base of numeration, is suddenly found to be so well suited to the purpose. The fact is, the present base being accepted as proper for numeration, however erroneously, it is assumed to be proper for gradation also; and a very flattering assumption it is, promising a perfectly homogeneous system of weights, measures, coins, and numbers, than which nothing can be more desirable; but, siren-like, it draws the mind away from a proper investigation of the subject, and the basic qualities of numbers, being unquestioned, remain unknown. When the natural order is adopted, and the base of gradation is ascertained by its adaptation to things, and the base of numeration by its agreement with that of gradation, then, the basic qualities of numbers being questioned, two is found to be proper to the first use, and eight to the second.
The idea of changing the base of numeration will appear to most persons as absurd, and its realization as impossible; yet the probability is, it will be done. The question is one of time rather than of fact, and there is plenty of time. The diffusion of education will ultimately cause it to be demanded. A change of notation is not an impossible thing. The Greeks changed theirs, first for the alphabetic, and afterwards, with the rest of the civilized world, for the Arabic,—both greater changes than that now proposed. A change of numeration is truly a more serious matter, yet the difficulty may not be as great as our apprehensions paint it. Its inauguration must not be compared with that of French gradation, which, though theoretically perfect, is practically absurd.
Decimal numeration grew out of the fact that each person has ten fingers and thumbs, without reference to science, art, or commerce. Ultimately scientific men discovered that it was not the best for certain purposes, consequently that a change might be desirable; but as they were not disposed to accommodate themselves to popular practices, which they erroneously viewed, not as necessary consequences, but simply as bad habits, they suggested a base with reference not so much to commerce as to science. The suggestion was never acted on, however; indeed, it would have been in vain, as Delambre remarks, for the French commission to have made the attempt, not only for the reason he presents, but also because it does not agree with natural division, and is therefore not suited to commerce; neither is it suited to the average capacity of mankind for numbers; for, though some may be able to use duodecimal numeration and notation with ease, the great majority find themselves equal to decimal only, and some come short even of that, except in its simplest use. Theoretically, twelve should be preferred to ten, because it agrees with circle measure at least, and ten agrees with nothing; besides, it affords a more comprehensive notation, and is divisible by 6, 4, 3, and 2 without a fraction, qualities that are theoretically valuable.
At first sight, the universal use of decimal numeration seems to be an argument in its favor. It appears as though Nature had pointed directly to it, on account of some peculiar fitness. It is assumed, indeed, that this is the case, and habit confirms the assumption; yet, when reflection has overcome habit, it will be seen that its adoption was due to accident alone,—that it took place before any attention was paid to a general system, in short, without reflection,—and that its supposed perfection is a mere delusion; for, as a member of such a system, it presents disagreements on every hand; as has been said, it has no agreement with anything, unless it be allowable to say that it agrees with the Arabic mode of notation. This kind of agreement it has, in common with every other base. It is this that gives it character. On this account alone it is believed by many to be the perfection of harmony. They get the base of numeration and the mode of notation so mingled together, that they cannot separate them sufficiently to obtain a distinct idea of either; and some are not conscious that they are distinct, but see in the Arabic mode