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some of the most valuable metals ; and I shall presently attempt, in two or three pages, to lay before the reader “the most natural and simple system” in itself, and the one which best fits in with the usages of the two nations which are at the head of the commercial world—England with her colonies, and America. “We are speaking,” continues this author, “of course only with reference to a possible time ; for let that time arrive when it may, the history of the past must be a confused and repulsive subject.” On this important social, commercial, and scientific question, we are now at “fives and sixes” among ourselves, and the whole world is at “loggerheads."
Every one who is conversant with the properties of numbers, knows that the value of any number, as a basis for calculation, depends on its composition, or on the simple lower numbers which enter into it. We reckon by tēns because, before writing was invented, and before the powers of numbers were understood, all counting was done upon the ten fingers and thumbs of the two hands. But the number ten (written 10), has no more virtue as a basis for counting than 8 or 14. Each contains but two lower numbers; 10 contains 5 and 2 ; 8 contains 4 and 2; and 14 contains 7 and 2. There are no other multiples in these three numbers 8, 10, and 14. But there is a number lying between 10 and 14 which contains within it the harmonies and proportions of four other numbers, namely, 12 or the familiar dozen ; and it has worked its way into general use on this very ground. Twelve contains the numbers 2, 3, 4, and 6 repeated, and therefore may be divided by these numbers without leaving fractions. Every mathematician knows the superior value of twelve over ten as a basis for calculation ; but ten has possession of the field. It had not once. I
suppose that the power which brought it into use éan bring in a better number. A world armed with "knowledge,” which is power," must be strong enough to change a custom which was adopted by a world in ignor
Dr. Thompson, in his popular “ Treatise on Arithmetic,” page 232, enters upon a consideration of the value of each number, as a basis for counting, from two to twelve. He observes
“The senary (six) and duodenary (twelve) scales, having each so many integral aliquot parts in proportion to its magnitude, and those of so convenient a kind, give origin to much fewer inter
minate fractions than any others. These two scales are preferable, therefore, in a considerable degree, to any of the others that have been mentioned. The duodecimal has the advantage of expressing numbers concisely, saving one figure in fourteen or fifteen, as compared with the decimal scale. To introduce either of these scales now, however, when men are accustomed to the decimal scale; when the languages of all civilised nations are suited to it; and when so many valuable works, particularly tables, in which it is adopted, would be rendered comparatively useless—would be unadvisable, and perhaps impracticable : but we must regret that the decimal scale was adopted at a time when any other might have been introduced with equal facility.”
I think it is possible both to add two figures to the numerical scale, and to enlarge the English alphabet to the number of distinct sounds that exist in the English language. (See the “Phonetic Journal.”) Twelve is the number for a perfect and easy arithmetic. We can take a third, and especially a fourth of twelve, and keep clear of fractions ; but we cannot take a quarter of ten without a fraction; and we cannot get a third of ten without plunging into the abyss of interminate fractions, nor even then, for it eludes our grasp.
Dr. Thompson thought it would be “unadvisable” to increase the scale of figures from ten to twelve. This is certainly a more reasonable opinion than Lord Brougham's concerning the introduction of gas. He said that if it were brought into London for general consumption, the city and the people would some day be blown to atoms. The difficulties (only the difficulty of labour) that would attend the introduction of a new arithmetic, remind us of Dr. Lardner's opinion on the possibility of navigating a vessel across the Atlantic Ocean by steam ; and of the opinion of the British Houses of Parliament on the introduction of railways, and travelling at the rate of thirty miles an hour. Both projects were pronounced to be impossibilities. But Dr. Thompson, with his characteristic openness to conviction, says that it is "perhaps impracti. cable.” As to the tables now in existence, the alteration of them to suit a new system of money, weights, and measures, would add but little to the labour of printing new editions, which we are doing every day.
If all weights and measurements were reckoned and written by twelves, and if all denominations of money,
weights, and measures, consisted of twelve of the next lower, we should possess all the benefits of a decimal coinage without altering the value of a single coin, or any of the common measures and weights, except the ounce, which would be one-third heavier. I recommend the penny, the pound weight, and the foot measure, as the integers, or roots, or units, on which to base a universal system of money, weights, and measures, which would be gradually adopted by all nations. The yard would, of course, be preserved to us for measuring cloth, &c. I have inquired of drapers whether the English yard or the French metre of about 3 feet 3} inches,
is the most convenient for handling, and they unanimously pronounce the “French yard” to be too long for the arms.
To preserve the pound of twenty shillings intact, and deduce the cent, penny, and mill from it, is like producing the centre of a circle from the circumference. It is the penny that produces the shilling, and the shilling the pound, and not the contrary. We have made twelve pence constitute a shilling because it is a more convenient number for divisions of a shilling than ten would be.
The two new figures necessary for a twelve system of arithmetic might be ő ten, & eleven, something like the writing forms of T and E, recommended by a correspondent of the “Times.” They work well, for I have employed them about five years, and have added them to the figures of all my book fonts, from Nonpareil to, Small Pica. All counting would be done in twelves ; the scale of figures is given in the third line below.
I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII.
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.
1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 9, 10. Duodenary, one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve.
The numeration table would be—Units, dozens, grosses, triples (a new term to signify the third
of twelve), dozens of triples, grosses of triples, sexiads (a new term signifying the sixth power of twelve). Only these two additional words would be required in the place of thousand and million. I recommend that no higher denomination than sexiads (in the place of millions) be employed.
For higher numbers, call over the figures and add the word sexiads. Thus we might say, the American war debt has reached 4 (5, &c.) figures of sexiads of dollars. When we ascend to the region of billions, trillions, and all the other -illions, up to dodecillions, or the twelfth degree above millions, we are lost in a maze of figures and words. Besides, the very meaning of these words is disputed; one method of employing thenı is adopted in this country, and another in France and the States of America. In England we take six figures for each denomination above a million, but in France and America only three are taken. The consequence is, that "
a billion dollars” in America means only the thousandth part of what it means in England. There it means a thousand millions, but here it means a million millions.
As everybody who can cypher has learned the pence table, he may employ this in addition to the ordinary multipli. cation table, in performing multiplication by twelves, until the twelve table shall be learned.
34 | 38 |
6 / 8 / 7 | 10 12 | 14 | 16
| 14 | 16 | 18 | 17 | 20
20 | 23 | 26 29
39 | 4
40 | 48 9 & ] 18 18 | 26 | 34 | 42 | 50 | 57 | 68 | 76 | 84 | 92
29 | 38
20 | 100
This table is to be repeated thus :—Three ones are three, three twos are six, three threes are nine, three fours are a dozen, three fives are one-and-three (that is, one dozen and three), three sixes are one-and-six, &c. ; nine ones are nine, nine twos are one-and-six, nine threes are two-and-three, nine fours are three dozen, &c. Obsolescent numbers may be marked thus ( ), as (1,728) = 1,000, or one triple.
In Money, the only alteration required by this reform would be to replace the ten and twenty shilling gold pieces by others of twelve and twenty-four shillings value. The twelve shilling piece would be the principal or the highest coin of account, and might be named a Mark. There should also be a smaller gold coin of six shillings, about the size of a four-penny piece, to supersede the present lumbering silver coin of 5s., which can scarcely be called “change.” France and America could reconstruct their money on the basis of the English penny, which is equal to two cents in America, and nearly equal to the French penny of ten centimes, 25 French pennies being equal to 24 English ones. The English £5 note would be replaced by one bearing the value of £7. 48., or 100 (one gross) shillings. It inight be called a Banko.
In Weights, I recommend the present pound, and that there be no other higher denomination than a load, or a triple pounds, that is, a dozen gross pounds, or 1,728 pounds, which make a light cart-load of 15 cwt. 1 qr. 20 lb. Intermediate weights would be expressed with sufficient convenience by dozens and grosses of pounds. I would fix the pound at its present weight, and have it registered in several places, rather than introduce a new and different pound. The word pound would thus be properly restricted to the meaning of a weight, and would pass out of use as the name of a coin. The present convenient cwt. (hundredweight) would be replaced by a gross pounds, which would be but thirty-two (or two-and-eight) pounds heavier.
In Liquid Measures the present pint, which weighs about 1 lb. 3ļoz., might be taken as the unit. Dozens and grosses of pints would be sufficient for all higher measures till we reach a dozen gross pints, which might be called a tun; the difference between the old and new tun of wine, being that between (2,016) and (1,7289 pints, or 2 gross pints, or 3 dozen gallons. The word ton or tun (both pronounced tun) would thus signify a liquid measure only, and not 20 cwt. also.
Lineal Nieasures might be—the foot of twelve inches,