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. X=14; that is, primes by primes produce seconds, since seconds are 144ths. 12X144 = 1728; that is, primes by seconds, produce thirds, and so on. This principle will enable the pupil to perform multiplication of duodecimals, in any case. * Length, 5 3' 1 5' 2 3′ 2′′ 5 3' 1. In a board 5 ft. 3 inches (5 3′) in length, and 1 ft. 5 inches (1 5′) 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 decimals, except that we carry for 12, from denomination to denomi nation. Ans. 7 5' 3" 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" 57' 0" 5" 38 3' 8" 10' 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 DENOMINATION OF THE MULTIPLIER; CARRYING FOR 12 FROM DENOMINATION TO DENOMINATION, AND PLACING THE FIRST NUMBER IN EACHI 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′′′′ 6iv 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? 6. What is the solidity of a high, and 2 ft. thick ? Ans. 1,102ft. 10′ 6′′ wall 53ft. 6' long, 10ft. 3′ Ans. 1,0963 ft. floor, 48ft. 6' long, and Ans. 1,176 ft. 7. Required the surface of a 24ft. 3' broad? 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.205§ 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? A. 7,471,442 sq. yds.-$149.42 1427 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′′ 4i by 9 4′ 6′′ Ans. 233 4′ 5′′′ 9′′ 61o 4′ 6'i iv 15. Multiply 44 2′ 9′′ 2′′" 4" by 2 10′ 3′′" NOTE. Ans. 126 2′ 10′′ 8" 10 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 independence, 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 became 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 2 of a pound Sterling. In Canada and Nova Scotia, a pound is of a pound Sterling; in Scotland, and in Ireland, 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. Georgia Currency 4s. 8d.. 5s. Od. New England 6s. Od. Pennsylvania " £2 14s. Od. New-York Scotch We have arranged them in the order of their values, placing highest the currency, whose value is greatest. 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. It ap 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. 14d. and that of the American dollar 4s. id. qrs. 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. 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. Its comparative unimportance will not justify us in devoting to it much space. By means, therefore, of the followiug table, we have thought proper to combine all the cases, of which it admits, under a single rule. 9 370 225 1 3 30 $F.M. 923 4 Suppose I have a sum in £ Sterling, which I wish to reduce to £ New England. I see, by the table, that any sum expressed in £ Ster. is, of the same sum, expressed in N. E. This I find, by looking for the given Currency, (which is Sterling,) in the upper line, and for the required Currency, (which is New England,) in the right hand column. Under the one, and opposite the other, I have the Fraction. Let the given sum be £6. Then I know that 6 is of the answer. If 6 is, 2 is 4. £2 is of 8£. Ans. It will be seen, that, in this case, we divided by 4, and multiplied by 3, which (§LI.) is dividing by the Fraction. Hence, to reduce a sum from one currency to another, DIVIDE BY THE NUMBER FOUND IN THE TABLE, UNDER THE GIVEN, AND OPPOSITE THE REQUIRED CURRENCY. For similar reasons, the number under the required and opposite the given currency may be used as a multiplier: The numbers in the table may often be reduced to decimals with advantage. NOTE. It will usually be best to reduce shillings, pence, and farthings to a fraction of a pound, either vulgar or a decimal. It may perhaps be desirable that the pupil should commit to memory the numbers used in reducing the currencies of the U. S. to Fed. Money, and the contraThe rest is only intended for reference. ry. 7. Reduce $196.00 to N. E. 9. Reduce £35; 6; 8 Sterling, to N. E. 10. Reduce £120 N. E. to Can. 11. Reduce £155; 13 N. E. to Sterling. 16. Reduce 225; 6 N. E. to F. M. Ans. £58; 16. Ans. £47; 2; 2; 2}. Ans. £450; 15; 6; 3. Ans. £56; 3. 17. Reduce £880; 15; 11; 1 Penn. to Ster. Ans. £528; 9; 6; 3. 18. Reduce £6,750 Irish to Geor. 19. Reduce £1,846 Ster. to Irish. Ans. £6,461. 20. Reduce £1,722; 18; 9; 3 N. E. to N. Y. Ans. £2,298; 5; 1. 21. Reduce £2,114; 1; 3 Can to F. M. Ans.$8,456.25. 22. Change £784; 5; 6; 2 Penn. to Geor. A. £487; 19; 10; 2. 23. Change £923 Sterling to Irish. To Penn.. To Penn. To Geor. 24. Change £4,000 Irish to Sterling. 40. What is the value of 1£ Geo. in F. M. Ans. $4.4444. Ans. $0.370. Ans. $4.285. Hence we see that there are but two of the currencies, in which a pound can be expressed in Federal Money by a finite decimal. This is one of the disadvan tages in their use. The following FOREIGN COINS have assigned to them the values, in FEDERAL MONEY, placed opposite to them, respectively. Re, of Portugal, Milre*, " Marc Banco of Hamburgh, $0.222 1.111 £,) 4.444 Livre of France, 0.185+ Pistole of Italy, Franc Rix Dollar of Austria, 0.778 Rix Dollar of Denmark and Switzerland, 1.000 Five franc piece," 0.937 Rix Dollar, of Sweden, 1.037 Real of Plate, of Spain, 0.100 Real of Vellon, 0.050 Rix Dollar*, of Prussia 0.778 0.259+ Ducat, of Sweden and Dollar, Prussia, 2.074 |