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ing 3s. as before. Many a good housewife, whose only practice in arithmetic is adding bills once a week, will adopt this plan for a time.

The opponents of the pound system, whether they hold by the farthing or the penny, are very careful not to come to close quarters with its advocates, in comparison of the modes of learning the systems, and especially in the case of the uneducated. As before noticed, they talk only of coinage, and not of calculation. They assault the arithmetic of their opponents, but no amount of defiance will bring out their own. The Russians will come out of Cronstadt and attack the allied fleets, before they will dare to put forward the way in which the poor man is to learn and practise one of their systems, in opposition to our single rule of 25 new farthings to the halfshilling, and everything else as now. All the modes of attack which they employ may be reduced to five, as follows:

First, they parade the mathematical mode of writing decimals, and charge the advocates of the pound with forcing this mode and all the higher notions of arithmetical process upon the uneducated world. Mr. Lowe, after talking more learnedly than any expert arithmetician would have done about reducing fractions of a pound to decimals, observed that this was a pleasant sum for an old apple-woman : and some of the conscript fathers cheered him.

Secondly, they persist in the tacit assumption that contracts and sales will still be made in old pence, and ask how they are to be exactly rendered in the new money.

Thirdly, they magnify the advantage of retaining exact expression to the utmost fraction of a farthing, and diminish the disadvantage of losing the shilling and the pound as coins of exchange and of estimation.

Fourthly, they exaggerate the difficulties of detail which will arise in the adjustments of postage duties, stamp duties, bridge and other tolls, &c.

Fifthly, they introduce sarcasm and something approaching to reflection upon motives. To this there is no great objection, as they thereby render the task of the other party somewhat more easy, by the power of reprisal which they give.

As to the first point, the charge of forcing decimal fractions on the poor by Act of Parliament. No system, pound or farthing, decimal or common, forces any fractions at all, in the sense in which the word is used by the accusers. There are two ways of treating the relation of part and whole: in one, a foot is compounded of 12 inches; in the other, an inch is taken off as the twelfth part of a foot. There seems not much to choose, and both methods are convenient to a practised arithmetician: but there are two kinds of persons to whom the matter is not indifferent. The first kind, including the uneducated and beginners in arithmetic, find multiplication more easily conceived than division. For them was contrived that excellent old mode of expression by which “seven-nineteenths of a foot” was described as “ seven of those parts of which nineteen make a foot.” With the beginner in arithmetic there is some trouble: but by practice the two expressions are conjoined. With the uneducated world there is none at all : their wants are supplied by the notion of multiplication, and all that is fractional may be kept out of view. They require to know that 12 pence make a shilling: it matters little whether or not they attach a distinct idea to the statement that a penny is the 12th part of a shilling. For them all tables are con. structed in ascent: they are led up from the farthing to the pound, not down from the pound to the farthing. The advocates of the pound may reason downwards, but they will teach upwards.

The second class of persons to whom we have alluded form a considerable portion, but not the whole, of our opponents. Among these we find some rational arithmeticians who, starting on what we believe to be a mistaken estimate of convenience and inconvenience, find their way to a conclusion opposite to ours, in what we can readily admit to be the proper mode of handling premises to which they have a full right, though we believe them wrong. But there are others who, we feel confident, deserve the following description.

They have some idea of the phraseology of fractions, and employ it in framing arguments against the pound system, on the supposition that its promoters are as much given to abuse the idea of fractions as themselves. They make a dangerous thing of their little knowledge, by assuming that they are fit to discuss the attempts of those who have more to benefit those who have none. They hold that men of long and practised acquaintance with arithmetic cannot communicate with the world at large, except through their own inbroglio of half-understood terms, and their own farrago of doubly-loaded routine. They attribute to the working man their own incapacity to learn, and to the man of knowledge their own inaptitude to teach: and having thus divided themselves, they go to buffets, and call the contest a picture of society. In comparing systems of coinage, they describe what they prefer in the simplicity of ascent by multiplication, and what they oppose in the complexity of descent by division: and this is their only way of intimating that they know the difference. They frighten a poor man with decimal fractions : though in truth there is no

more occasion to tell him that the new system is decimal, than that the old one is quarto-duodecimo vicesimal. They declare that an apple-woman must deal with decimal places ad infinitum, or else have a ready reckoner: they talk of incommensurables, of finite ratios, of reducing vulgar fractions to decimals, &c. They teach a working man that the proper way of representing a thousandth part is .001, and triumph in his perplexity as a thing brought about by the advocates of the pound system. Their arithmetic is never higher than school-boy routine, sometimes lower. We have heard one of them,-a man employed by the country in its arithmetic, at a tolerable salary,-maintain that there is no difference worth speaking of between the trouble of dividing by 10 and dividing by 12: and we saw reason to suspect that his mode of finding out the tens in 287 involved “ 28 times 10 make 280, and 7 over.” Mr. Lowe announced his opinion that it would be a hard thing for members of the House to turn 4 d. into mils; and he was cheered. From the glimpses he gave of his own idea of arithmetical process, and the frequent occurrence of allusion to turning common fractions into decimals, we have no doubt he had in his head the computation on the left, opposite to which we place our own. 410 = 18 farthings

Since a farthing is a mil and one = £958

24th of a mil, 18 farthings is

18 mils and 18-24ths of a mil, 9168 x 1000 mils.

or 18 mils. 960)18000(184 mils.

96

840
768

7 = 1 Mr. Lowe has written his name on the history of this question in legible, and perhaps lasting, characters. He was the only member who made a specific attack upon the proposed system: and, for a few days, he enjoyed the reputation of having done a clever thing. A journalist apologises for him, and condemns the Association for answering, on the ground that he was only attempting to bring some humorous help to the government in delaying the question. This we doubt; there was too much argument in his humour, too much elaboration in his argument: but there can be no objection to his friends putting him to death to save him from slaughter. With great respect for decimals, he denied having any very profound knowledge of them: for a time there were some who thought that this was only modesty. The presumptuous manner in which he tried to raise a laugh at the opinions of those who had studied a question of which he knew nothing, calls for castigation. Let a man who really knows his subject be tolerated when he teaches by ridicule, and be applauded if he do it well; for there is good elucidation in good joking, and a dry discussion is all the better for the introduction. But the sayer of yesterday's lesson, especially when he only grafts the blunders of a novice upon the teaching of an ignoramus, deserves no mercy if he try to be smart upon his betters.

Neither Mr. Lowe, nor any of the minority, ventured to propose any system in opposition to that which ends in the pound. We now take our leave of those who imagine that the technicalities of decimal fractions are part of the proposed plan, and proceed to meet those who insist on it that all sums payable in the old coinage should be capable of exact representation in the

new.

The matter in dispute never amounts to a farthing in calculation, and need not amount to half a farthing in payment, as already seen. Let us first inquire what sort of exactness prevails in actual business. Do men neglect to set down results to the uttermost fraction? Do they ever abandon a farthing for the sake of facility ? Do they ever pay a fraction of a farthing more than the goods ought to cost, because there is no coin less than a farthing? Does every man in business, and every customer, do one or more of these things usually, frequently, day after day, and year after year? If all these questions must be answered affirmatively, it follows that a fraction of a farthing, lost or gained on each transition from old money to new, can only be a great matter to one whose power of judging is a small matter.

And first we take the farthing customer at a chandler's shop. It appears in evidence, that when he buys goods which require the fraction of a farthing to complete the payment, the shopkeeper always takes the whole farthing. This makes a “keen calculator,” as one witness called it, of the customer, who has to ask himself, and settle for himself, whether the next quality will be so much higher in price as to overpass the farthing. For instance, buying half an ounce of three-and-sixpenny tea," by which he will forfeit a fraction of a farthing for want of smaller coin, he finds out whether that fraction would or would not enable him to buy the quantity of three-and-ninepenny tea. What a question for those members of the House of Commons, who, according to Mr. Lowe, would find it hard to turn 4 d. into mils at 25 mils to 24 farthings! We may now guess one reason why neither the small shopkeepers nor their customers would have anything to do with the half-farthings, which the

Mint tried to introduce more than twenty years ago. Precisely the same sort of trouble would have occurred with fractions of this half-farthing, with more elaborate fractions, and for smaller results.

We thus see that the poorest are constantly obliged either to sacrifice a fraction of a farthing, or to make, every now and then, what their betters (but not in arithmetic) would call an intricate calculation. If this calculation were never made, the average loss would be half-a-farthing: probably, the calculation reduces this average to a quarter of a farthing. This is the loss which the necessary subdivisions of retail business impose upon every small purchaser on the average of his small dealings. Now a quarter of a mil is more than the average loss which would be sustained, once for all, on the day of the change, by enacting payment of outstanding copper debts at a mil for the farthing, with a mil additional above 3d.

Is it impossible to make people understand that they can secure a very great advantage to themselves and their children, at no greater cost than running the risk of sustaining, on some one particular day, and on debts below 6d., that loss which they cheerfully sustain on all the days of the year, and which they would rather bear than trouble themselves with coins less than a farthing?

Let us now look at the manufacturer and the tradesman. We need hardly say that they neglect fractions of a farthing in accounts. The more important question with them is the pricing of small articles under the new system. To the wholesale dealer this question does not occur. He sells by the gross or by the thousand, and when he quotes goods at one-32nd or one64th of a penny a piece, it is only a mode of quotation. This wholesale dealer asks nothing of the House of Commons but to give him decimal coinage, and to keep its arithmetic to itself. The retail trader has a harder question; but it is one of policy, not of arithmetic. Goods are priced at three-farthings a piece: what shall he do when the change comes? Sell at 3 mils, and abandon 4 per cent. ? or sell at a penny (4 mils), and take chance of competition? This is no new question for him: the like comes upon him every day of his life; but he is a good administrator, and the details of each hour are linked in his mind to the system of his business. He knows that in very many cases, in a great majority of all, his prices have been adjusted to the farthing or the penny above the result of his calculation of necessary profits; and this to a greater extent than 4 per cent. of the whole. He knows, too, that the facilities of a decimal coinage would be worth to him more than 4 per cent. on his capital.

As an instance of the complicated character of business calculations, we subjoin what is called a cost of a manufactured

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