REVIEW. 1. How has the foot usually been divided? 2. What are the inconveniences of these divisions? 3. What would be a more convenient division? 4. How might these divisions be managed? 5. What name is given to numbers, which express parts in this manner?(114) 6. How are decimals distinguished from integers? What are integers? 7. How would you write 12 feet and 3 tenths? 8. Have figures in decimals a local value? Upon what does it de pend? 9. What is the law by which they diminish?(115) 10. In what does the enunciation of decimals differ from that of whole numbers? 11. Do ciphers on the right hand of decimals alter their value? What does each additional cipher indicate?(116) 12. What effect have ciphers on the left hand of decimals? Why? 13. What are numbers made up of integers and decimals called?(114 14. From what is the word decimal derived? A. From decimus, (Latin) which signifies tenth. 15. What is the rule for the addition of decimals? Where must the · decimal point be placed? 16. What is the rule for the multiplication of decimals? What the rule for pointing? 17. What effect has multiplication by a decimal? Explain by example and diagram. 18. What is the rule for the subtraction of decimals? For the division of decimals? 19. What is the rule for pointing in each? 20. What is to be done if there are not so many figures in the quotient as the number of decimals required? 21. When the decimal places in the divisor exceed those in the dividend, what is to be done? 22. When there is a remainder after division, how do you proceed? 23. What does a vulgar fraction denote?[129] Explain by example. 24. How then can you change a vulgar fraction to a decimal? 25. What is Federal Money? 28. How may the lower denominations be regarded? Explain by example; and also the different methods of reading the same. 29. How then may Federal Money be regarded? 30. How is it denoted? 31. What is the rule for the Addition of Federal Money?-for Multiplication?-for Subtraction?-for Division of Federal Money? SECTION IV. COMPOUND, OR COMPLEX, NUMBERS. 137. Numbers are called Compound or Complex, when they contain units of different kinds, as pounds, shillings, pence and farthings; years, days, hours, minutes and sec onds, &c. 365 d. or 365.25 d. or 365 ds. 6 hrs. 1 year, yr. 31557600 525960 8766 365 52 1 *The above denominations of Federal Money are authorized by the laws of the United States, but in the transaction of business we seldom hear any of them mentioned but dollars and cents. A coin is a piece of money stamped, and having a legal value. The coins of the United States are three of gold; the eagle, half-eagle, and quarter-eagle; five of silver, the dollar, half-dollar, quarter-dollar, dime, and half-dime; and two of copper, the cent and half-cent. Of the small foreign coins current in the United States, the most common are the NewEngland four pence half penny, or New-York sixpence, worth 64 cents; and the New-England ninepence, or New-York shilling, worth 12 cents. The value of the several denominations of English money is different in different places. A dollar is reckoned at 4s. 6d. in England, 58. in Canada, 6s. in New-England, Virginia and Kentucky, 8s. in New-York, Ohio and North-Carolina, 7s. 6d. in Pennsylvania, New-Jersey, Delaware and Maryland, and 4s. 8d. in South-Carolina and Georgia. The year is commonly divided into 12 months, as in the following table, called Calendar months: No. D. July No.Days. January 1 31 April 4 30 February 2 28 May 5 31 March 3 31 June 6 30 Another day is added to February every fourth year, making 29 days in that month, and 366 in the year. Such years are called Bissextile, or Leap year. To know whether any year is a common or leap year, divide it by 4; if nothing remain, it is leap year; but if 1, 2 or 3 remain, it is 1st, 2d or 3d after leap year. The number of days in the several months may be called to mind by the following verse: No. D. Thirty days hath September, years.. All the rest have thirty-one, Which hath twenty-eight, nay more, Hath twenty-nine one year in four. The true solar year consists of 365 days, 5 h. 48 m. 57 s. or nearly to 365 days. A common year is 365 days, and one year is added in Leap years to make up the loss of of a day in each of the three preceding This method of reckoning was ordered by Julius Cæsar, 40 years before the birth of Christ, and is called the Julian account, or Old Style But as the true year fell 11 m. 3 s. short of 365 days, the addition of a day every fourth year was too much by 44 m. 12 s. This amounted to one day in about 130 years. To correct this error, Pope Gregory, in 1582, ordered that ten days should be struck out of the calendar, by calling the 5th of October the 15th; and to prevent its recurrence, he ordered that each succeeding century, divisible by 4, as 16 hundred, 20 hundred, and 24 hundred, should be Leap years, but that the centuries not divisible by 4, as 17 hundred, 18 hundred, and 19 hundred, should be common years. This reckoning is called the Gregorian or New Style. The New Style differs now twelve days from the old style. *The original standard of all our weights was a corn of wheat, taken from the middle of the ear, and well dried. These were called grains, and 32 of them made one pennyweight. But it was afterwards thought sufficient to divide this same pennyweight into 24 equal parts, still calling the parts grains, and these are the basis of the table of Troy weight, by which are weighed gold, silver and jewelry. Apothecaries' weight is the same as Troy weight, only having different divisions between grains and ounces. Apothecaries make use of this weight in compounding their medicines, but they buy and sell their drugs by Avoirdupois weight. In buying and selling coarse and drossy articles, it became customary to allow a greater weight than that used for small and precious articles, and this custom at length established the Avoirdupois, or common weight, by which all articles are now weighed, with the foregoing exceptions. Avoirdupois weight is about one sixth part more than Troy weight, a pound of the former being 7000 grains, and of the latter 5760 grains. In buying and selling by the hundred weight, 28 pounds have been called a quarter, and 112 pounds a cwt. but this practice of grossing, as it is called, is now pretty generally laid aside, and 25 pounds are considered a quarter and 4 quarters, or 100 pounds, a hundred weight. *The original standard of English long measure was a barley com taken from the middle of the ear, and well dried. Three of these in length were called an inch, and then the others as in the table. Long measure is employed for denoting the distance of places, and for measuring any thing where length only is concerned. When measure is applied to surfaces, where length and breadth are both concerned, it is called square measure. A square inch is a square measuring an inch on every side. The table of square measure is made from that of long measure by multiplying the several numbers of the latter into themselves. Thus, 12 inches are a foot in length, a square foot then is a square which measures 1 foot, or 12 inches, on every side, and contains 12X12-144 square inches. 3 feet in length make a yard; a square yard is a square measuring 3 feet on each side: but such a square contains (see figure) nine (3X3=9) squares measuring a foot on each side, or 9 square feet; and when we say that a surface contains so many square feet, or square yards, we mean that the surface is equal to such a number of squares measuring a foot or a yard, on each side. When measure is applied to solids which have length, breadth, and thickness, it is called solid or cubic measure. A solid inch is a body, or block, 3ft. having six sides, each of which is an inch square, and the number of inches in a solid foot is equal to the number of such blocks that would be required to make a pile a foot square and a foot high. Now it would require 144 blocks to cover a square foot one inch high. Hence to raise the pile twelve inches high would require twelve times 144-1728 blocks or inches. In like manner it would require 9 solid blocks, a foot each way, to cover a square yard to the height of one foot, and 3 times 9=27, to raise it three feet, or make one soliá yard. This will be obvious from an inspec X. 1296 SOLID, or CUBIC MEASURE. 1728 inches, in. make 1 foot, ft. 27 feet 128 feet 66 1 yard, yd. acr. sq. mile, mi. in. 1728 feet 1 | yard | cord 46656 27 221184 128 429 40 ft. of round timber, or 50 ft. of hewn timber, make 1 ton, ton. WINE MEASURE.* 9 1 XI. 4 gills, gls. make 1 pint, 2 pints 66 1 quart, qt. in. 57 2 1 gallon, 8 4 1 31 gallons 1 barrel, bar. 2 barrels "1 hogshead, hhd. 7276 252 126,31 14553 504 252 63 2 2 hogsheads 1 pipe, 2 pipes tion of the diagram. The cord of wood is sometimes called eight feet. In this case four feet in length, four in breadth, and one in height=16 solid feet, is called one foot; or eight feet in length, four in breadth, and six inches in height, a foot, that is, 1-8th of a cord is called one foot, 2-Sths, two feet, &c. In measuring lands, roads, &c. the distances are usually taken in chains and links. In ordinary business, feet and inches are the most common measures. Many mechanics, however, now take dimensions in feet and tenths of a foot, instead of inches, and if all would do the same, they would find all their calculations much more simple and easy. By forty feet of round timber, in the table of solid measure, is meant so much round timber, as will make forty feet after it is squared. * Four pounds Troy weight of wheat gathered from the middle of the ear, and well dried, were called one gallon, and this was the original standard of all English measures, both liquid and dry, and this was the same as the present wine gallon. But in time it became customary to use a larger measure in selling cheap liquors, and this custom at length established the beer measure, which bears about the same proportion to wine measure that avoirdupois does to troy weight. The dry measure was also made larger than the wine measure, and was at length established, at about a mean between wine and beer measure. By wine measure are measured wine, all kinds of spirits, cider, vinegar, oil, &c. By beer measure are measured ale and beer, and by dry measure are measured al |