measured by real measures only; constant, because the determination of quantity requires a standard of comparison that is invariable; conveniently proportioned, because both time and labor are precious. These rules being acted on, the result will be a system of real, constant, and convenient weights, measures, and coins. Consequently, the numeration and notation best suited to commerce will be those which agree best with such a system.

From the earliest periods, special attention has been paid to units of quantity, and, in the ignorance of more constant quantities, the governors of men have offered their own persons as measures; hence the fathom, yard, pace, cubit, foot, span, hand, digit, pound, and pint. It is quite probable that the Egyptians first gave to such measures the permanent form of government standards, and that copies of them were carried by commerce, and otherwise, to surrounding nations. In time, these became vitiated, and should have been verified by their originals; but for distant nations this was not convenient; moreover, the governors of those nations had a variety of reasons for preferring to verify them by their own persons. Thus they became doubly vitiated; yet, as they were not duly enforced, the people pleased themselves, so that almost every market-town and fair had its own weights and measures; and as, in the regulation of coins, governments, like the people, pleased themselves, so that almost every nation had a peculiar currency, the general result was, that with the laws and the practices of the governors and the governed, neither of whom pursued a legitimate course, confusion reigned supreme. Indeed, a system of weights, measures, and coins, with a constant and real standard, and corresponding multiples and divisions, though indulged in as a day-dream by a few, has never yet been presented to the world in a definite form; and as, in the absence of such a system, a corresponding system of numeration and notation can be of no real use, the probability is, that neither the one nor the other has ever been fully idealized. On the contrary, the present base is taken to be a fixed fact, of the order of the laws of the Medes and Persians; so much so, that, when the great question is asked, one of the leading questions of the age,—How is this mass of confusion to be brought into harmony?—the reply is,—It is only necessary to adopt one constant and real standard, with decimal multiples and divisions, and a corresponding nomenclature, and the work is done: a reply that is still persisted in, though the proposition has been fairly tried, and clearly proved to be impracticable.

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 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