annexed to continue the division. Hence, for DIVISION, when the DIVISOR IS FINITE, and the DIVIDEND A REPETEND, I. DIVIDE AS USUAL, ANNEXING THE REPEATING FIGURES OF THE DIVIDEnd, if neCESSARY, INSTEAD OF CYPHERS; AND SO PROCEED UNTIL THE QUOTIENT REPEATS. If the divisor be infinite, the division cannot be performed dectmally. Hence, when the DIVISOR IS A REPETEND, II. CHANGE THE DIVISOR TO ITS EQUIVALENT VULGAR FRACTION; MULTIPLY BY ITS DENOMINATOR, AND DIVIDE BY ITS NUMERATOR. NOTE. For short repetends, the preceding rules are very useful. But when repetends are long, the processes are tedious, and the value of the lower figures so small, that they may be neglected, without occasioning any important error. The same may be remarked of long finite decimals. We have before mentioned that decimals not carried out in full, are marked with + and -, according as they are too small or too great, and called APPROXIMATES. Particular rules may be given for their calculation, which the limits of this work will not permit us to insert. The common rules are sufficient for ordinary purposes; and those who are curious to examine the subject further, must resort to more extensive works. We subjoin a few examples. 1. Add 3.6; 78.3476; 735.3; 375; .27 and 187.4 A. 1,380.0648193 2. Add 5,391.357; 72.38; 187.21; 4.2965; 217.8496; 42.176; .523; and 58.30048 3. Add 9.814; 1.5; 87.26; .083 and 124.09 A. 5,974.1037i A. 222.75572390 4. Add 162.162; 1.5; 134.09; 2.93; 97.26; 3.769230; 99.083 and .814 5. Multiply .136 by .45 A. 501.62651077 A. .0615 10. Multiply 49,640.54 by .70503 A. 25.706 A. 31.791 A. .0929 A. 13.5169533 A. 34,998.4199003 NOTE. The pupil will observe, that as either factor may be made the multi plier, questions like the last may be solved, by rule II. or 111. at pleasure. 11. From 85.62 take 13.76432 12. From 26.43 take 25.2 A. 71.86193 A. 1.2i 13. From 476.32 take 84.7697 14. From 3.8564 take .0382 15. Divide 319.28007112 by 764.5 16. Divide 274.6 by .7 A. 391.5524. A. 3.8i A. .4176325 A. 301.714285 A. 55.69 A. 62.323834196891 A. 1.4229249011857707509881 DUODECIMALS. § LXXII. Decimals, we have seen, consist of a series of numbers decreasing towards the right by tens, as far as we choose to carry them. We are now about to speak of a class of numbers, decreasing in a similar manner by twelves. Of course, the calculations made upon them must be similar, in many respects, to those upon decimals. They are called DUODECIMALS, from the Latin word duodecimus, which signifies twelfth. Duodecimals are commonly used only for measuring length, surface, and solidity. A linear, square or solid foot is, therefore, considered the unit, or whole number. The lower denominations, or orders, are, of course, Fractions of a foot. The twelfth part of a foot, of any kind, whether a solid, a square, or a linear foot, is called a PRIME; the twelfth part of a prime, is called a SECOND; the the twelfth part of a second, a THIRD; the twelfth part of a third, a FOURTH, and so on, as far as we choose to go. Now, as primes are 12ths, seconds, 12ths of 12ths, or 144ths, thirds, 12ths of 144ths, or 1,728ths, fourths, 12ths of 1,728ths, or 20,730ths, &c. it is plain that we might write these numbers, as vulgar Fractions, with their denominators. But, since we know that they increase and decrease regularly by 12, we may, exactly as in decimals, omit the denominators, and write each order in a separate place. This is the mode, in which duodecimals are actually written, and, to distinguish the orders from oneanother, primes are marked with an accent, thus,^; seconds with two accents, thus": thirds with three accents, thus " and so on. Then 3 feet, 2 primes, 10 seconds, 11 thirds, 5 fourths and 6 fifths are written 3 2 10" 11" 5" 6"""" Instead of these marks, the the Roman numerals i, ii, fii, iv, &c. are sometimes employed. These are generally substituted when the number of marks becomes large. Thus the above example may be written, iii 32 10 11 5iv 6 or 3 2 10′′ 11′′ 5iv 6o ADDITION and SUBTRACTION of duodecimals may evidently be performed like addition and subtraction of other compound numbers, or like the same operations in decimals, except that we are to carry and borrow by 12, instead of by 10, from one denomination to another. MULTIPLICATION of duodecimals may be likewise performed like Multiplication of decimals, observing the same rule in carrying. Now, in multiplication of decimals, the pupil will recollect, that there must be as many decimal places, (or places below units) in the product, as there are in both factors. For a similar reason, in multiplication of duodecimals, there must be as many duodecimal places, (or places below units, or feet,) in the product as there are in both factors. This will be evident for the same reasons as in decimals. The same thing may likewise be proved by writing duodecimals, with their denominators like vulgar Fractions. that is, primes by primes produce seconds, since seconds are 144ths. 12×14=1728; that is, primes by seconds, produce thirds, and This principle will enable the pupil to perform multiplication of duodecimals, in any case. so on. 5 3' Length, 2 3′ 2′′ 5 3' Ans. 7 5' 3" 1. In a board 5 ft. 3 inches (5 3') in length, and 1 ft. 5 inches (15') in breadth, what is the amount of surface? We know, in the beginning, that the product is to have two duodecimal places, because there is one place in each factor. We therefore know that the right hand place of the product must be seconds. We commence at the right to multiply as usual, and after having multiplied by the 5', we proceed to multiply by the 1 ft., placing the first denomination of the product one place farther to the left, because we are multiplying by a higher denomination than before. This brings it exactly under the multiplying figure. The process will be seen to be exactly like multiplication of deci. mals, except that we carry for 12, from denomination to denomi nation. 2. In a solid block, the base of which contains 9 sq. ft. 6', 11", and the height of which is 4 ft. 7′ 2′′, what is the solidity? Base, 9 6' 11" Height, 4 7' 2" It will be observed that there are no denominations higher than feet. Therefore, we never carry for 12, after we arrive at feet. We shall always know when this is the case, by observing how many duodecimal places there must be in the answer, and placing the marks, or accents over each denomination of the product as we go on. 1' 7" 1 10'"" 5 70" 5" 38 3' 8" Ans. 44 0' 3" 6" 10" NOTE. The number of any denomination, and of course, the number of accents to be placed over it, will be observed always to be equal to the number of accents over both the numbers multiplied together. Thus the right hand figure of the above product is fourths, which is obtained by multiplying seconds by seconds. There are two accents over each factor, and four over the product. If the higher duodecimal places be wanting, cyphers must be put in their room. From the preceding illustrations, we derive the general rule for multiplication of duodecimals. I. BEGIN ON THE RIGHT AND MULTIPLY THE MULTIPLICAND BY EACH DE NOMINATION OF THE MULTIPLIER; CARRYING FOR 12 FROM DENOMI NATION TO DENOMINATION, AND PLACING THE FIRST NUMBER IN EACH PARTIAL PRODUCT, EXACTLY UNDER THE MULTIPLYING FIGURE. II. THERE WILL BE AS MANY DUODECIMAL PLACES IN THE PRODUCT AS IN BOTH THE FACTORS; WHICH MUST BE MARKED ACCORDINGLY. 3. If a floor be 10ft. 4′ 5′′ long, and 7ft. 8′ 6′′ wide, what is its surface ? Ans. 79ft. 11′ 0′′ 6′′′′ 61o 4. What is the surface of a marble slab, 5ft. 7 long, and 1ft. 10' wide? Ans. 10ft. 2′ 10′′ 5. How many feet of plastering in a ceiling 43ft. 3' long, and 25ft. 6' wide? Ans. 1,102ft. 10′ 6′′ 6. What is the solidity of a wall 53ft. 6′ long, 10ft. 3′ high, and 2 ft. thick? Ans. 1,096 ft. 7. Required the surface of a floor, 48ft. 6' long, and 24ft. 3' broad? Ans. 1,1761ft. 8. The length of a room being 20ft., its breadth 14ft. 6', and its height 10ft. 4', how many yards of painting in its walls, deducting a fire-place 4ft. 4' by 4ft. and two windows, each 6ft. by 3ft. 2? Ans. 73 yds. 9. In a floor 12 8' by 16 3' how many sq. ft. ? A.205ğ NOTE. Some kinds of work are done by the square yard. Such are painting, paving, plastering, &c. 10. A man paved a court 371 2′ 6′′ by 181 1′ 9′′ at 2 cts. pr. sq. yd. How many sq. yds. did he pave, and what did he receive? 10308 A. 7,471, 4427 sq. yds.-$149.42 127 11. How many cord feet in a load 8 ft. long, 4 wide and 3 6' high? A. 7 12. Multiply 4ft. 7 by 6ft. 4' 13. Multiply 39 10′ 7′′ by 18 8′ 4′′ 14. Multiply 24 10' 8" 7" 5 Ans. 29ft. 0′ 4′′ Ans. 745 6′ 10′′ 2′′" 4'▾ by 9 4' 6" Ans. 233 4' 5" 9"" 6iv 4" 6'i 15. Multiply 44 2′ 9" 2" 4 by 2 10' 3"" NOTE. Ans. 126 2′ 10′′ 8" 101o 11′′. It is plain that division, might be performed by duodecimals, but the inconvenience of the process renders it of no particular use. REDUCTION OF CURRENCIES. § LXXIII. The term CURRENCY is applied to any thing which is universally received, in any country, for the payment of debts, or for goods, bought and sold. In other words, the currency of any country is the money of that country. It is often called the circu lating medium, or the medium of trade; because it passes from one individual to another, or is current among all persons; and because, by means of it, trade is carried on. Federal Money is the currency established by law in this country. But before the adoption of this currency, in 1786, all accounts were kept in pounds, shillings and pence. Our money was, originally, the same as that of Great Britain; that is, Sterling Money. But the legislatures of the different states, or colonies, as they were called before our indepen. dence, put bills in circulation, which diminished or depreciated in value. This depreciation was different in different colonies; so that, while the names remained the same, throughout the country, the values became very different. Thus, a pound in the New-England states, and Virginia, became only of a pound Sterling in value. This was also the value of the currencies used in Kentucky and Tennessee. In New-York and North-Carolina, a pound be. came of a pound Sterling. This Currency was afterwards used in Ohio. In New-Jersey, Pennsylvania, Delaware and Maryland, a pound became 3, and in S. Carolina and Georgia 23 of a pound Sterling. In Canada and Nova Scotia, a pound is of a pound Sterling; in Scotland, and in Ireland 1923, nearly. Hence, in thn United States we have four Currencies besides that established by law. These, with the Canada, Scotch, Irish, and Sterling make eight. The names and values, assigned them are as follows: Sterling Money, in which 4s. 6d. make a dollar. 0009 Georgia Currency 4s. 8d. Irish New England Pennsylvania New-York Scotch We have arranged them in the order of their values, placing highest the currency, whose value is greatest. It ap NOTE. We have given above the value of a dollar in Sterling Money as all our Arithmetics have it, and as it is estimated throughout the country. pears, however, by the report in SENATE OF THE U. S., March 29th, 1830, that the value of the Spanish dollar, (generally considered equal to ours,) is only 4s. 1 d. and that of the American dollar 4s. id. 14qrs. nearly. It is to be regretted, that these currencies are still retained and employed by our merchants and tradesmen. The Federal Currency is so much more convenient than any of them, that it must ultimately supersede them all; and the sooner this is the case the better. For, in that event, we shall not only be free from the inconvenience of making calculations in these currencies, but likewise from the greater evil of being obliged to change sums of money from one to another. The making of these changes is what is meant by the REDUCTION of Currencies. Its The intelligent pupil will perceive, that the table of values given above, is sufficient to enable him to make these changes; and that no processes are necessary but those of common Reduction. But, as more concise methods may be suggested, it seems proper to give the subject a particular consideration. comparative unimportance will not justify us in devoting to it much space. means, therefore, of the followiug table, we have thought proper to combine all the cases, of which it admits, under a single rule. By |