27. To shew this more clearly, we may multiply an integer by any number we please-2, 3, 4, 5, 6, &c., and we always obtain an exact result, which, for the purposes of convenience, we may reckon by groups of 10. It is, therefore, physically possible to multiply any unit by any number whatever, and obtain an exact result. 28. So we may divide an integer by 2, 3, 4, 5, 6, 7, &c., or any number, and obtain an exact result. Hence, division by the ordinary numbers is the correlative of multiplication by them. As we may multiply the unit any number we please, and get an exact result, so we may divide by any number we please, and it is physically possible to obtain an exact result. Therefore, the common fractions are the correlatives of ordinary multiplication in the denary scale. In these the only 29. But in decimal fractions that is not so. divisors allowed are 10, and powers of 10. may multiply by any number whatever, we must only divide by powers of 10. Thus instead of our divisors being unlimited, like our multipliers, they are restricted to a very small number indeed. And this consequence follows, that it is physically impossible to divide a unit exactly into any aliquot parts which are not some powers of the factors of 10. That is, a unit cannot be divided exactly in decimals by any number which is not of the form 2 x 5%. 30. Now, the immense majority of numbers are not of this form at all, and consequently it is a matter of physical impossibility to divide a unit exactly by the immense majority of numbers. 31. To shew how very few they are, we will shew how extremely few there are in the natural numbers up to 1,000, by which a unit can be exactly divided by decimal fractions. Taking powers of 2, we have 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1,024, &c. Taking powers of 5, we have— 1, 5, 25, 125, 625, 3,125, &c. Now a unit cannot be divided exactly in decimals by any number except those in these two series, or those arising from the multiplication of any one in the one series by any one in the other. To shew how extremely few they are, we have only to see how many there are up to 1,000. We shall find that there are only 28 numbers up to 1,000, in which an exact division is possible. They are, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 125, 128, 160, 200, 250, 256, 320, 400, 500, 512, 625, 640, 800, and 1,000. 32. Now, what should we say to a system of multiplication in which it was a physical impossibility to obtain an exact result in the immense majority of cases? What should we say to a system of multiplication in which it was physically impossible to multiply a unit exactly by 3, 6, 7, 9, 11, 12, &c.? It is clear that such a system could not be tolerated for a day. 33. Now, such a system as that would be the correlative of decimal fractions. It would be one in which we were forbidden to multiply by any numbers except 10, and powers of 10; and therefore no multipliers which were not of the form 2o x 5o could bring out an exact answer. 34. Hence we see that the analogy between decimal numbers and decimal fractions entirely fails. In fact, they proceed upon different principles; and it is manifestly the same with any fraction expressed in the radix of the scale of notation. The unit may be multiplied by any natural number whatever. But it can only be divided by powers of the radix. Consequently, it can be divided exactly by no natural numbers whatever, except those composed of powers of the factors of the radix. 35. Hence we see at once, that there is a fundamental distinction between addition or multiplication in the denary scale, and decimal subdivisions, or decimal fractions. For all cases of addition or multiplication, nothing can be better than decimals, but for all cases of subdivision nothing can be worse. 36. The cases, therefore, of a coinage, in which the unit is the lowest possible, and therefore proceeds by multiplication, and that in which the unit is the highest possible, are not only not parallel, but they involve principles which are antagonistic to each other. Where nothing but physical multiplication is wanted, nothing can be better; but where physical division is required, decimal fractions are impracticable. 37. Hence we see at once, that the analogy between coinages of the third system and those of the first entirely fails, and what is the best in the first is impracticable in the third. 38. We have said that the essential peculiarity of decimal fractions is, that the unit cannot be divided into any aliquot parts, except those proceeding by powers of 10. Now this is a restriction that no people would ever submit to in the common affairs of life. We constantly require to divide things exactly into 3, 6, 7, 9, 11, 12 parts. No one would ever dream of proposing that persons should voluntarily preclude themselves from dividing a quantity into any exact parts under 1,000, but those of the 28 numbers above given. But that is what we should do if we were to adopt decimal sub-divisions exclusively. Such a notion is so monstrous, that no one out of Bedlam would propose it. It would be just as rational as to suppose that we should adopt a system of multiplication in which none but these figures should produce exact results. 39. Now it would be the state of greatest perfection if we could imagine the unit of value, such as gold, to be some soft substance like putty, which we could subdivide into any number of parts whenever we pleased. But as that is impossible, the next best thing is to have it divided into that number of pieces which contains the greatest number of divisors possible. Now, 10 is not only not good, but it is extremely bad. 40. Now, considering that the present unit of the English coinage is of gold, and of its existing magnitude, it is quite easy to shew that there is no division of it at all comparable to that of 20, 12, and 4. No other combination within the same compass presents such a richness of factors. For it has no less than 26 factors, namely:-2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, 80, 96, 120, 160, 192, 240, 320, 480; whereas 1,000 has but 14 factors-2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500. Hence the immense superiority of the present division of the pound sterling over that of the millesimal one for all purposes of physical division is manifest. 41. Moreover, every one's daily experience shews that while he naturally uses the decimal scale for multiplication, he never thinks of confining himself to decimal expressions for sub-division. People want, every-day, halves and quarters, and half-quarters of things, and they call them so. But if we are to have decimal fractions exclusively, these expressions must be given up. A snuffy old woman in the Highlands wants a quarter of an ounce of snuff: she must no longer ask for that, but she must ask for a 25-100th of an ounce! And so on. A nation of savans might do that, but common humanity never will. We want a half or a quarter of a thing. The eye performs the work instantaneously. But if we go to decimal fractions, we must first of all divide the whole unit into 10, or 100, and then take 5 or 25 of these parts. Such a statement shews the manifest absurdity of such a thing.. 42. The fact is, the whole confusion is based upon the supposition that decimal fractions are analogous to decimal integers, which is a complete delusion; and if this distinction in principle had been fully considered, the question never would have been agitated at all. 43. Considering, therefore, these fundamental differences of principle between decimal fractions and decimal numbers, and decimal multiplication and decimal division, we may state the following as ascertained principles with respect to a coinage: 1st. Where the unit of account is the lowest coin in common use between man and man, and the whole coinage consists of multiples of that unit, the decimal system is by far the best. 2ndly. Where the unit of account is a coin of some low magnitude, the decimal system will have some conveniences and some inconveniences. And as the unit becomes larger, the practical inconveniences will constantly increase over the advantages. 3rdly. Where the unit of account is very high, and placed far above the immense majority of transactions, the decimal system, which then becomes one of almost entire subdivision, is an intolerable nuisance, which could never subsist for any time at all. 44. From these considerations we see that it would be practically impossible to adopt any system of decimal coinage in this country so long as the pound sterling is the unit of account, and the coinage is one of pure sub-division. Other schemes have been proposed, based upon the penny and the farthing. Of these we shall say something hereafter. 45. It is unquestionable that, for matters of account on paper, especially in large numbers, the decimal system affords an immense superiority. It is no doubt true, that it is physically impossible to divide anything into 3, 6, 7, &c., parts, by decimals. We can, however, carry it as near exactness as we please. The philosopher can afford to balance this inconvenience against the other many advantages, and carry his calculations a few figures further with equanimity, when he knows that the ultimate result will come as nearly true as he pleases. But it is a far different matter with the daily transactions of life, where actual physical sub-division is required, and where the differences which arise from an imperfect division give rise to everlasting and perpetual quarrels. No man who has not studied history can conceive the intolerable practical misery that a depreciated currency causes to a people; and the very same effects are produced by an imperfect system of subdivisions. We shall have ample evidence of the truth of this in the course of this section. We shall now give some historical notices of the adoption of the decimal system of coinage by different nations. Of the Decimal System of Coinage of the United States. 46. The currency of the various American colonies was originally the same as that of the mother country. But we have shewn elsewhere that nearly all the States had issued enormous masses of paper currency, the effect of which had been to depreciate the pound in them. In each State, too, the pound had undergone a different degree of depreciation; hence there was, at the time of the Revolution, an immense confusion between the currencies of the different States. The weight of |