arithmetical computation, depends upon another consideration entirely; as will be seen in the sequel. The English system of pounds, shillings, and pence, is the best system with reference to convenience of use. The American system, dollars and cents, is the best with reference to facility of computation. It may be supposed by many persons that the ratios of the several denominations of the English currency to each other are accidental. Four farthings make one penny, twelve pence one shilling, and twenty shillings one pound. Whence come the four, the twelve, and the twenty. The prevailing impression probably is, that they resulted fortuitously from some unknown circumstances connected with the origin of money, which occurred in a rude and early age, and that these numbers are retained only because they are established, and it would now be inconvenient to change them. This, however, is not so, for on examination we shall find that the system bears the marks of high scientific design. If a company of mathematicians were to be set at work to devise the most perfect system, we mean with reference solely to convenience of use in ordinary transactions, without regard to the question of the facility of compulation in written accounts, they would adopt the English system, and no other. They would be driven, in fact, to the English system by inexorable mathematical laws. This will be made evident by an analytical examination of the system itself. In dividing commodities in the small transactions of trade, we have occasion most frequently to halve them; that is, to divide them into two equal parts. Next we have occasion to quarter them, or to divide into four parts. It is true that the number three comes next to two in regular succession, but still we have occasion for a quarter of an article or a quantity more frequently than for a third of it. Thus, at a shop a quarter of a yard, or a quarter of a pound, &c., are much oftener called for than a third of a yard, a third of a pound, &c. The fact that we have more frequent occasion to employ the fraction one-quarter than one-third is shown also, curiously enough, by the fact that we have a distinct word for dividing a thing into four parts; namely, to quarter it, while we have no word for dividing it into three parts, though the latter is, in respect to magnitude and number of the parts, a simpler division than the former. Next to dividing a commodity or a value into two parts and into four parts, we have most frequent occasion to divide it into three parts. And next we have most frequent occasion to divide it into five parts. The penny is two times two (2 x 2 = 4) farthings-so that it can be halved and quartered. The shilling is two times two times three, (2 × 2 × 3 can be divided by two, and by four, and also by three. The pound is two times two times five, (2 × 2 × 5 be divided by two, by four, and by five. 12,) so that it 20,) so that it can The numbers two, three, and five being thus brought in as primes, in regular gradation and combination, the result is a system which, in respect to divisibility without fractions, is the most perfect that the nature of numbers will allow. That is to say, the numbers denoting the component parts of the various denominations of the English currency can be divided without fractional results by a greater number of divisors than any other numbers whatever, of anything near the same magnitude. Thus, the number of farthings in a pound is 960. The number of cents in an eagle is 1,000. The divisors of these two numbers, under one hundred, are as follows: That is to say, there are twenty numbers under one hundred that will divide 960, the number of farthings in a pound, without a remainder, while there are only eight that will divide 1,000, the number of cents in an eagle, although the latter dividend is larger than the former. The result is much the same if we compare the other denominations of the two currencies. The number of farthings in a shilling is 48; that of cents in a dollar is 100. The comparative divisibility of these two numbers, taking divisors under fifteen, is as follows: Thus 100, the number of cents in a dollar, though more than twice as large as the number of farthings in a shilling, has only two-thirds as many divisors under fifteen. It may, perhaps, be thought, at first view, that these differences are only theoretically curious, and that they have no important practical bearing on the question of the comparative convenience of the two systems. We shall see, however, on more mature reflection, that they have a very practical bearing indeed on the question, so far as it relates to convenience of use in the ordinary transactions of trade, for it is in these that we have such frequent occasion for divisions. The advantage that was aimed at in the adoption of the American system was facility of computation in written records. The decimal ratio makes it very easy to add columns, and to multiply and divide large written numbers. This was the object for which it was designed. The convenience of a currency for ordinary shopping transactions depends on totally different properties from those which determine its facilities for rapid computation when the numbers are written; and it will be found, on a careful consideration of the subject, that what its excellence really does depend upon, in the former point of view, is this very principle of divisibility. In order to present this principle of divisibility in its most practical form, we should compare the English shilling, (which is, perhaps, the most common coin of the small transactions of every-day trade, and is thus, as it were, the unit of value, for what may be termed the pocket currency,) with its American representative, the quarter of a dollar. The shilling may be divided into halves, thirds, or quarters, the very divisions which are most frequently needed to be made. We may almost say they are all that are ever needed to be made. The quarter of a dollar is divisible only into fifths -a division which we may almost say is never required to be made. If a purchaser does not require the whole of a yard of cloth, it is almost always half a yard, or a quarter of a yard, or a third of a yard, that he asks for; not once in a hundred times is it anything else. He can have either of these without a fraction in the use of the English shilling; but in the use of the American quarter he can have only one-fifth of a yard, a portion which he never wants. In other words, the English coin gives him all the convenience that he requires, while the American, so far as the quarter of a dollar is concerned, gives him absolutely none. As, however, innumerable instances occur in the ordinary transactions of business where commodities and prices must be halved and quartered, we are compelled to halve and quarter our denominations of coin, and the result is an endless confusion of fractions. Purchases come to six-and-a-quarter cents, and twelve-and-a-half, and eighteen-and-three-quarters cents, where in England it is simply three-pence, six-pence, and nine-pence. The amount of it is, that the shopkeepers and their customers, in all the stores in Broadway and the Bowery, are kept in constant confusion with fractional amounts, in order that the clerks in the banks in Wall-street may have an easy time in adding up their columns. The same difference exists between the two systems in respect to integration of numbers as in the subdivision of them. If a single article in England is, in price, two-pence, two will be four-pence-a third of the shilling; three will be six-pence-half the shilling; four will be eight-pence-twothirds of the shilling; five will be ten-pence, and six will be a shilling. Again; if the price of a single article be three-pence, it is a third of the shilling, and then two articles will be six-pence-half the shilling again; three will be nine-pence-three-quarters of the shilling; and four will be twelve-pence-the whole shilling. And if the single price be four-pence, a double price is eight-pence-two-thirds of the shilling; and a treble price twelve-pence the whole shilling. Thus, everything goes smoothly, and comes out even. On the other hand, where the decimal ratio governs, all works wrong in such cases. If the postage of a single letter is two cents, a double rate is four cents, a treble rate is six cents, and a quadruple rate is eight cents, neither of which numbers is an aliquot part of a dime. The half-dime will not pay exactly for any one of the letters. In same manner, if the single rate is three cents, a double rate is six, still avoiding the half-dime; the treble is nine, and the quadruple is twelve. Not one in either series can be paid for with any one coin of the Federal currency, whereas, in the English system, every one of both series can be paid for with a single coin as soon as the amount becomes large enough to reach the lower limit of the silver coinage. There is another view of the subject which will put the difference between the two systems in a clear light, and that is a comparison of the proportional value of the coins in relation to each other. In the English system every small coin will be found to be some simple aliquot part of the larger ones; thus That is to say, the English subdivisions of the coinage represent the fractions,,,,, 26, the fractions, of all others, most frequently required in the transactions of every-day life. Each of the three denominations has a coin to represent one-half, and another for one-quarter of its value, and the most important one has also one for one-third. On the American system the result is very different:- Thus, it will be seen that the subdivisions run to halves and tenths, almost exclusively. The half is a useful fraction, but the tenth almost utterly useless. How seldom it is that a tenth part of a yard, or of a pound of any commodity, is asked for. We cannot even quarter anything in the Federal coinage below the dollar. The coin which, more than all others, is to be considered the unit of value for the every day transactions of life; namely, the quarter of a dollar, corresponding, in this respect, to the shilling of the English currency, and the franc in the French, is wholly unmanageable. You cannot get a half of it. You cannot get a quarter of it. You cannot get a third of it. You can have a fifth of it, if you should ever have occasion to use such a fraction as that, but that is all. But we find that we must have the half and the quarter of it, in some way or other. The inexorable exigencies of trade demand it. There are a great many commodities for which the price will be a quarter of a dollar a pound, or a yard, and there will be a great many occasions when people will require half or quarter of a pound, and half or quarter of a yard. So the tickets of admission to public exhibitions will be set at a quarter of a dollar, and children will be required to pay half-price. A thousand other emergencies constantly occur demanding a division of this coin into halves and quarters. The only way in which the people of this country have to meet the exigency, is to abandon their own system at this point, and use, instead, the old Spanish coin, which furnish the necessary subdivisions. A great many ingenious financial movements have been made to compel people to use the dime and half-dime as subdivisions of the quarter-dollar, instead of the Spanish coins; that is, to employ in trade the fractions and , instead of those of and; but such efforts, it is obvious, never can succeed. In fact, the partial success which attends the experiment shows visibly the resistance which the nature of numbers makes to it. The Federal coins are occasionally seen, it is true, but the half-dime is almost always accompanied by a cent to make it up to an even quarter of the quarter-dollar; and the dime, in the same manner, is supported by two cents, to bring it into a tolerable condition to represent one-half of the quarter-dollar, while in the meantime the old Spanish coin, under the various names of ninepence, shilling, levy, &c., in the different States, holds its ground, and will hold its ground, in spite of all efforts to drive it away, simply because it is more convenient to have a representative of half the quarter-dollar coin in one coin than in three. Thus we see that the reason why the people of the United States do not adopt the Federal coinage and currency in their ordinary dealings is not, as is sometimes supposed, the fixedness of old habit, and the consequent diffi culty of changing them. There is a substantial inconvenience that is inherent in the very constitution of the currency itself. That this is the true explanation, is evident from the fact that the Federal currency was at once and universally introduced throughout the country in keeping accounts; for that is a function which its nature admirably adapts it to fulfill. In the day-books and ledgers of merchants, brokers, banks, and treasuries throughout every State in the Union, the Federal system reigns supreme. In regard to this field no difficulty was experienced in the universal introduction of the system, for here was a purpose that it was fitted for. On the other hand, all efforts to introduce it as a circulating currency in the ordinary transactions of life have everywhere failed, and must continue to fail as long as tenths and fifths are less convenient fractions than halves, quarters, and thirds. In fact, the government itself seems at length to begin to yield to the inexorable necessity which demands other multiples and divisors than five and ten, in a currency for popular use. We have now a three-cent coin, the is suing of which is a flagrant departure from the decimal system, or rather the introduction of a wholly new element into it; namely, the prime 3. The number three is a very important element of the English system, as we have seen; and the introduction of this new coin is, therefore, an attempt to incorporate a feature of the English system upon ours. It is extremely doubtful, nevertheless, how far this limited and partial attempt at a remedy will succeed. It is yet too early to see the practical result of the experiment, but all the theoretical considerations which bear upon the subject indicate that it will fail-making the coinage more confused and complicated, without gaining the advantage intended. That is to say, the two systems, namely, the one in which 2 and 5 are the elements, and the other in which the elements are 2, 3, and 4, are so entirely different, that a part of the one cannot be grafted upon and made to harmonize with the other. The threecent coin, for example, is incommensurable with every silver coin in the whole Federal currency; that is, no number of these coins will make either a half-dime, a dime, a quarter of a dollar, a half dollar, a dollar, a quartereagle, a half-eagle, or an eagle. Observe, now, the striking contrast when we turn to the corresponding piece in the English system, the three-penny piece : 2 of them make the sixpenny piece. 13 of them make the half dime. 20 of them make the crown. 40 of them make the half sovereign. three-cent piece 16 of them make the half dollar. 33 Thus, in the one case, everything is commensurable and simple. In the other, the results are all perplexing and unmanageable fractions, showing us that the whole system must be constructed with the element three as an essential constituent of it throughout, and all attempts to introduce it incidentally into a system formed from the elements 2 and 5, will lead to endless intricacy and confusion. It is curious to observe how the elements 2, 3, and 4, which are the elements of the English system, reign everywhere in the construction of almost all the tables of weights and measures in use among civilized nations, and |