19. What is the weight of a mow of hay 6 feet deep and 200 inches square ? Ans. 4325 lbs. To find the cubic contents of a body, its weight being given : Divide the weight of the body in ounces by its specific gravity, (disregarding the decimal point,) and the quotient will be its cubic contents in feet. EXAMPLES. 1. How many cubic feet in a ton of fir? Ans. 63.036. 2. What is the side of a cube of fine gold, its weight being one ounce avoirdupois ? Ans. 0.44659 of an inch. 3. What is the diameter of a sphere of platinum, its weight being 1 pound? Ans. 1.3492 inches. 4. Find the number of cubic feet in a ton of dry oak. Ans. 38.75 feet. To find the quantity of either of the ingredients in a compound consisting of two ingredients, when the specific gravities of the compound and of the ingredients are given : Multiply each of the three specific gravities by the difference between the other two; then, as the greatest product is to cach of the other products, so is the weight of the compound to the weight of each of the ingredients. EXAMPLES. 1. A composition weighing 56 lbs., and having a specific gravity of 8.784, consists of tin and copper of the specific gravities 7.320 and 9.000 respectively; what are the quantities of the ingredients? Ans. 50 lbs. of copper and 6 lbs. of tin. One statement will be all that is required; for, having found one of the ingredients, if we deduct it from their sum, or the compound, the remainder will be the other. 2. An alloy of the specific gravity of 7.8 weighs 10 lbs., and is composed of copper and zinc of the specific gravities of 9 and 7.2; required the weight of the ingredients. Ans. 3.85 lbs. of copper, and 6.15 lbs. of zinc. 3. An alloy of the specific gravity 7.7 consisting of copper and tin of the specific gravities 9 and 7.3, weighs 25 ounces; what is the weight of each of the ingredients? Ans. 6.87 ounces of copper, and 18.12 ounces of tin. ¶ 70. COINS. The average specific gravity of English gold coins is 17.500. The fineness of gold is estimated by carat grains, equivalent to 2 dwts. Troy, pure gold being 24 carats fine. The purity of the present English gold coins is 11 parts fine gold and 1 part alloy. The sovereign, or twenty-shilling piece, contains 113.001 grains of fine gold, and 123.274 grains of standard gold. The Troy pound of standard gold is coined into 46 sovereigns and of a sovereign, or into £46 148. 6d. The alloy in coins is reckoned of no value; and it is used simply to harden the coins, and to avoid the trouble of refining. 89 The commercial value of a sovereign in the United States is about $4.84. A Troy pound, or 12 ounces of the metal of which English silver coins are made, contains 11 oz. 2 dwts. pure silver, and 18 dwts. alloy. This pound is coined into 66 shillings, each shilling containing 80.727 grains fine silver, and 87.27 grains standard silver. The purity of Amerigan gold coins is the same as that of English gold coins; and the Troy pound of standard gold is coined into 21 eagles, equal in value to $213; or 9 ounces of standard gold are coined into 16 eagles, the value of which is $160; the value of one ounce being $17.773. One pound of fine gold is estimated to be worth 15 pounds of fine silver. The eagle contains 247 grains of fine gold and 270 grains Thus 24 of standard gold, the alloy being 22 grains. grains of fine gold (being one-tenth of the number of grains in an eagle) are worth one dollar; and if multiplied by 15 the product will be 371 grains, grains of fine silver in one dollar. weighs 417 grains; consequently 17 of these dollars will weigh 1 pound 0 oz. 3.2548 dwts. avoirdupois, nearly. which is the number of A Spanish milled dollar In England, a dollar coined in the United States is valued at 4s. 3.68d. sterling. In England the eagle is estimated to contain 246.1 grains of pure gold. I demand the value of a cubic inch of pure gold, admitting that an eagle contains 246.1 grains. Ans. $199.441. 71. PILING OF BALLS AND SHELLS. Balls and shells are usually built into piles of some regular form. The base of a pile is either an equilateral triangle, a square, or a rectangle, and is called accordingly, a triangular, a square, or a rectangular pile. The number of balls in a side of each course diminishes by unity upwards; and the pile is said to be complete or incomplete according as it is or is not finished. The complete triangular and square piles terminate in a single ball; and the complete rectangular pile in a single row. In the complete square and triangular piles, the number of courses is equal to the number of balls in the side of the lowest course; and in the complete rectangular pile, it is equal to the number of balls in the end of the lowest course. 1. To find the number of balls in a triangular pile: To the number of balls in a side of the base add 1, and to this sum add 1; multiply the three numbers together; that is, find their continued product, and one-sixth of this product will be the number required. 2. To find the number of balls in a square pile : To the number of balls in a side of the base add 1, and to twice the number in a side add 1; find the continued product of the three numbers, and one-sixth of this product will be the required number. 3. To find the number of balls in a rectangular pile : From three times the number of balls in the length of the base, subtract the number in its breadth, less one; then find the continued product of the remainder, the breadth, and the breadth increased by 1, and one-sixth of the product will be the number required. 4. To find the number of balls in an incomplete pile : Find the number in the whole pile considered as complete, and the number in the supplementary pile, and their difference will be the number in the incomplete pile. EXAMPLES. 1. How many balls are contained in a triangular pile consisting of 25 courses? Ans. 2,975. Ans. 11,480. 2. How many balls are contained in a triangular pile of 40 balls in one side of the base? 3. How many balls in a square pile of 20 courses? Ans. 2,870. 4. How many balls in a square pile having 15 balls in a side of the base? Ans. 1,240. 5. How many shells are contained in a rectangular pile, the number in the length and breadth of its base being 59 and 20? Ans. 11,060. 6. Find the number of balls in a rectangular pile of 20 courses, the number in the length of its base being 24. Ans. 3,710. 7. Find the number of shot in an incomplete triangular pile, the number in a side of the base and top being 40 and 20. Ans. 10,150. The whole pile considered as complete would contain 11,480; and the supplementary pile (required to complete the imperfect pile or frustum) contains 1,330, which being subtracted from 11,480 leaves 10,150 the number in the frustum. 8. Required the number of shot in an incomplete square pile of 17 courses, a side of the base containing 24. Ans. 4,760. 9. How many shells are contained in an incomplete rectangular pile of 12 courses, the number in the length and breadth of the base being 40 and 20 ? Ans. 6,146. T 72. LAWS OF FALLING BODIES. Every particle of matter in the universe has a disposition or tendency to press towards, and, if not opposed, to approach every other particle. This mutual tendency of all the particles of matter to each other is called the attraction of gravitation. The force of attraction is directly proportional to the masses of the attracting bodies, and inversely proportional to the squares of their distances; that is, the force of attraction diminishes in proportion to the increase of the square of the distance, and increases as the square of the distance diminishes. Terrestrial gravity is the disposition which all heavy bodies manifest, when unsupported, to fall towards the centre of the earth, in consequence of the earth's attraction; which attraction operates in the same manner as if all its matter were condensed into a single point at its centre. Since gravity acts with very nearly the same degree of force on a falling body during the whole time of its descent, it may be regarded as a uniformly accelerating force. Terrestrial gravity acts equally on all bodies; and consequently, were it not for the buoyancy of the atmosphere, a piece of gold and a feather would fall with the same velocity. It has been demonstrated by experiment, that any body (were it not for the resistance of the atmosphere) would fall 16.1 feet the first second, 4 times 16.1 feet in two seconds, |