To measure stone in a well.

In taking the diameter of the wall, measure from centre to contre, or to the clear; add the thickness of the wall; and when the diameter is taken, or obtained, find the circumference, which is the length of the wall.

RULE.

As 7 is to 22 :: so is the diameter (taken): to the circumfer ence required; having found the length of the wall, and measured its thickness and depth, it is then prepared for calculation as explained above.

35. A well walled with stone 1 foot 2 inches thick, its diameter in the clear 2 feet 4 inches, and depth 50 feet; required the length of the wall, and the quantity of stone.

As 7: 22: 3.5 : 11 feet circumference, or length; then 1 ft. 2 inches 1.1666+6 the thickness, 11 length.

Is what the well measures in the clear (inside the walls); add the thickness of the wall, and you have the diameter of the wall or circle. Ans. 25.925+ perches. 36. Required the number of perches of stone in a well, whose diameter is 4 feet 9 inches in the clear, the depth of the well 20 feet, and the thickness of the wall 15 inches.

Ans. 19.047 perches, or 19+ perches. 37. How many acres are there in a circular island whose diameter is 124 rods? Ans. 75 acres, 76 rods.

38. If the diameter of a circle be 113 and the circumference 355, what is the area? Ans. 10029. 39. What is the diameter of a circle whose area is 380.1336 feet?

Thus: 380.1336.7854-484 and 484-22. Ans. 40. What is the area of a square inscribed in a circle whose area is 159? Thus: .7854: † :: 159 : 101.22. Ans

41. What is the side of a square whose area is equal to that of a circle 452 feet in diameter ?

Ans. √ (452)2 ×.7854=400.574

42. 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. 43. If the height of a square prism be 21 feet, and each side of the base 101, what is the solidity?

Ans, the area of the base 10×10=1067 square feet; and the solid content =1063×21=240 cubic feet.

44. How many gallons ale are there in a cistern, which is 11 feet, 9 inches deep, and whose base is 4 feet, 2 inches square? The cistern contains 352500 cubic inches 282=1250 gallons. Ans.

45. Suppose a cellar dug 16 feet in length, 16 in width, and 6 feet in depth; required the quantity of stone to enclose a wall 1.5 feet thick, and 6 feet in height, in side of said cellar.

Ans. 21 perches. (243 ft.1 perch.) 46. A room is 11 feet high, 21 feet long, and 16 feet wide; how many cubic feet of space in it? Ans. 3696 cubic feet. 47. How many square feet in a stack of 15 boards, 12 feet, 8 inches in length, and 13 inches wide?

Ans. 2055 205 ft. 10 in. 48. Suppose a ship sails from latitude 43o, north, between north and east, till her departure from the meridian be 45 leagues, and the sum of the distance and difference of latitude be 135 leagues; I demand her distance sailed? and latitude come to? Ans. 135x135—45 × 45

135 X2

=60 leagues, and 60×3=180

miles 3 degrees, the difference of latitude 135-60=75 leagues, the distance. Now, the vessel is sailing from the equator, and consequently the latitude is increasing; therefore, to the latitude sailed from 43° add the difference of latitude 3o, and the sum is the latitude come to, that is, 46 degrees.

49. Four men, E, F, G, and H, purchased a grindstone of 60 inches in diameter; how much of its diameter must each grind off, to have an equal share of the stone, if one first grind his share, and then another, till the stone is ground away, making no allowance for the eye.

RULE.

Divide the square of the diameter by the number of men subtract the quotient from the square, and extract the square root of the remainder, which is the length of the diameter after

the first man has ground his share; this work being repeated by subtracting the same 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.

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 si. 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 X225=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. ()=9680 yards=98.3869+root, then 98.3869 ×1.12837=111116826353 yards, whole diameter which÷2 55.55+yards. Ans.

57. What is the side of a square piece of land containing 124 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: 256x256-65536; and 75x75=5625 and 59911 =244.76+and 244.76-2=(22)=222.76 feet. Ans.

59. What is the side of a square field which contains 583

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=12320280-44 rods. 61. The Winchester bushel is a hollow cylinder, 18 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. 42 x1.25x3.14169=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 × 121=194156.6.

65. What is the square root of 3736?

49

Ans. 373×49+36=1849/67. 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, × 12, height 603.1872, content in feet, × 1728=1042307.4816 cubic inches, -231-4512133 gallons÷63=71_hogsheads, 39 gallons, 2+ quarts. Ans.

=

2d. As 7: 8: 22: 25.1428÷÷2=12.5714 × 4, half diameter, =50.2856 × 12, length, 603.4272 × 1728 = 1042722.2016÷ 2314513.9489 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′ : 40×4=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. 10873. 69. If we are 95 millions of miles from the sun, and if the earth revolves round it in 365 days; how far are we carried in 24 hours? Ans. 1 million 634 thousand miles.