the first man has ground his share ; this work being repeated by subtracting the saine quotient from the remainder, for every man to the last; extract the square root of the remainder, and subtract those roots from the diameter, one after another, and the several remainders will be the answers. Thus: 60 in. diam. Again : 60 in. diam. of stone. 60 X -51.9615 Or, geometrically, thus : on the radius A B describe a semicircle ; also divide A B into as many equal parts as there are shares, and draw perpendiculars from the points of division to the semicircle on A B; then with the centre B, and radii B D, B C, &c., describe circles, and the question is solved. 50. If a load of wood is 8 feet long and 3 feet wide, how high must it be to contain 1 cord ? Ans. 5} ft. =5 li. 4 in. 51. Required the quantity of stone in a pile 16.5 feet in length, 4.5 feet in width, and 4 feet in height, computed at 16.5 cubic feet per perch. Ans. 18 perches. 52. If the floor of a square room contain 36 square yards, how many feet does it measure on each side ? Ans. 18 feet. 53. What is the difference between 100 square feet and 100 feet square ? Ans. 9900. 54. Suppose a ladder 60 feet long so planted as to reach a window 37 feet from the ground, on one side of the street, and without moving it at the foot, will reach a window 23 feet high, on the other side ; required the breadth of the street ? Ans. 102.64. 55. The Bunker Hill monument in Charlestown, near Boston, is said to be 225 feet high; the ascent is gained by winding stairs on the inside, and also by steam-power, acting on machinery for that purpose; suppose a person standing 121 feet from the base, and the ground level, what length of cord would be required to reach from the ground to the top of the monument, allowing the side, or wall to be perpendicular (which, however, is not the case). Thus: 225 X 225=50625; 1212 = 14641+50625 = 65266; then 65266=255.47 feet length of cord. Ans. 56. Required the length of a rope that will be sufficient for a horse to graze just 2 (circular) acres ? Thus : 2 A. (V)=9680 yards=98.3869+root, then 98.3869 x 1.12837=111.116826353 yards, whole diameter which=2 =55.55+yards. Ans. 57. What is the side of a square piece of land containing 1244 acres ? Ans. 141 rods 58. The height of a tree growing in the centre of a circular island 44 feet in diameter, is 75 feet, and a line stretched from the top of it over the hither edge of the water, is 256 feet; what is the breadth of the stream, provided the land on each side of the water be level ? Thus : 256 X 256=65536 ; and 75 x 75=5625 and 59911 =244.76+ and 244.76-1=(22)=222.76 feet. Ans. 59. What is the side of a square field which contains 584 acres ? 60. What is the width of a piece of land which is 280 rods long, and which contains 77 acres ? Ans. 77 x 160=12320-280=44 rods. 61. The Winchester bushel is a hollow cylinder, 181 inches in diameter, and 8 inches deep; what is its capacity? Ans. 2150.42. 62. What is the solidity of a cylinder, whose height is 424, and circumference 213 ? Ans. 1530791+ 63. What is the convex surface of a right cylinder, which is 42 feet long, 15 inches in diameter ? Ans. 42x1.25 X3.14169=164.933 square feet. 64. What is the solidity of a cylinder, whose height is 121, and diameter 45.2 ? Ans. 45.22 X.7854 x 121=194156.6. 65. What is the square root of 374 ? Ans. 373 X 49+36=18 J=67. 66. In a circular cistern, whose diameter at bottom is 9 feet, and top 7 feet, depth 12 feet, required the number of hogsheads it will contain. Thus: 8x8=64 square, mean diameter, X.7854=50.2656 area base, or mean diameter, X 12, height 603.1872, content in feet, X 1728=1042307.4816 cubic inches, --231=4512131 gallons ---63=71__hogsheads, 39 gallons, 2+ quarts. Ans. 2d. As 7:8 :: 22 : 25.1428--2=12.5714 X 4, half diameter, =50.2856 x 12, length, =603.4272 x 1728 = 1042722.2016= 231=4513.9489 gallons, +63=71 hhds., 40 galls. + Ans. 67. Boston is 6° 40' east longitude, from the city of Washington ; when it is 6 o'clock, P. M., at Washington, what is the hour at Boston ? Thus: 6X4=24' : 40 X4=160"=2' 40" +24=26' 40' past 6. Ans. 68. How many planks 15 feet long, 15 inches wide, will floor a barn which is 60 feet long, and 33} wide ? Ans. 10875 69. If we are 95 millions of miles from the sun, and if the earth revolves round it in 3651 days; how far are we carried in 24 hours ? Ans. 1 million 634 thousand miles. 70. If the diameter of Saturn's larger ring be 205000 and 190000 miles, how many square miles on one side of the ring ? Thus : 395000 x 15000 X.7854–4653495000 miles. Ans. 71. What is the expense of paving a street 20 rods long and 2 rods wide, at 5 cents for a square foot ? Ans. D544) 72. How many hills of corn may be planted on an acre, 3 st. distance ? Thus: 1 acre=43560 square feet = (3x3=9) =4840 hills, Ans. 73. How many trees may be planted on 10 acres, at 6 feet distance ? Ans. 12100. 74. How many rails will it require to enclose a field of 16 acres ? (length of rails 13 feet, and 6 rails high.) 75. What is the side of a square whose area is equal to that of a circle 452 feet in diameter ? Ans. V. (452)X.7854=400.574. 76. What is the diameter of a circle which is equal to a square whose side is 36 feet? Ans. ✓ (36)2 =-0.7854=40.6217. 77. If a carriage-wheel 4 feet in diameter revolve 300 times in going round a circular green, what is the area of the green ? Ans. 4154 square rods, or 25 acres, 3 qrs., 34 rods. 78. What is the solidity of a wall which is 22 feet long, 12 feet high, and 2 feet 6 inches thick ? Ans. 660 cubic feet. 79. If the height of a square prism be 21 feet, and each side of the base 10 feet, what is the solidity ? Ans. 1063 square feet, 240 cubic feet. 80. How many ale gallons are there in a cistern which is 11 teet 9 inches deep, and whose base is 4 feet 2 inches square ? Ans. (=-282) 352500 cubic inches=1250 gallons. 81. The highest point of the Andes is about 4 miles above the level of the ocean. If a straight line from this touch tho surface of the water at the distance of 1781 miles, what is the diameter of the earth? Thus: 178.252-4m.=7939.2656 miles, Ans. 82. If the diameter of the earth be 7939.2656 miles, and Mount Etna 2 miles high, how far can its summit be seen at sea ? Thus : 7939.2656 x 2 miles+22=15882 ; then 15882.5312=126+ miles. Ans. 83. How much water can be put into a cubical vessel 3 feet deep, which has been previously filled with canuon-balls of the same size, 2, 4, 6, or 9 inches in diameter, regularly arranged in tiers, one directly above another? Ans. 96 wine gallons. 84. How many such globes as the earth are equal in bulk to she sun, if the former is 7930 miles in diameter, and the latter 890000 ? Ans. 1413678. REVIEW. What is the rule to measure wood? What is a square ? How will you find the area of a square having equal sides? What is a rectangle? How will you find the area of a rectangle? What is a triangle? How will you find the area of a triangle? What is a circle? What is the diameter of a circle ? When the diameter is given, how will you find the circumference? When the circumference is given, how will you find the diameter? How can you find the area of a circle? If the area is given, how will you find the diameter ? how the circumference ? What is a sphere? How will you find the surface of a sphere? How will you find the soļidity of a sphere? What is a prism? How will you find the solidity of a prism? What is a cylinder? How will you find the convex surface of a cylinder ? How will you find the solidity of a cylinder ? What is a cone? How will you find the solidity of a cone ? What is a pyramid ? How will you find the solidity of a pyramid? What is a parallelopipedon? How will you find the solidity of a parallelopipedon? What are the rules for measuring timber, &c. ? PROMISCUOUS QUESTIONS. 1. What is the interest of D650.75 for 150 days at 7 per cent.? Ans. D18.72. 2. What is the interest of D440 for 30 days at 6 per cent. ? Ans. D2.17. 3. Required the interest of D212 for 60 days at 5 per cent. Ans. 11.74.2. 4. A. sells goods on commission at 2.5 per cent. ; how much will he receive for selling D1800 worth? Ans. D4.50. 5. How much'must a broker receive for discounting D2000 ät 3.5 per cent. ? Ans. 170. 6. What is the commission on D2176.50 at 2.5 per cent. ? Ans. 154.41.2. 7. What is the premium of insuring D1650 at 15.5 per ct. ? Ans. D255.75. 8. How many bushels of wheat can be purchased with D81.76 at D1.12 per bushel ? Ans. 73. |