3. If a cow is fastened by a chain 10 feet long to a stake in a field, how large an area will be within her reach? Ans. 314.16 sq. ft.

4. There is a circular park 240 rods in diameter, and within it a private garden, also circular, 110 rods in diameter; how much of the park is open to the public?

Ans. 223 A. 55.609 P.

1032. A square is inscribed in a circle when the vertex of each of its angles is in the circumference.

Rule. To find the side of an inscribed square, multiply the diameter by .707106, or multiply the circumference by .225079.

1. The end of a round stick of timber is 3 feet in diameter; what will be the side of the largest square stick that can be hewn from it? Ans. 2.12+ ft.

2. The circumference of a circular garden is 320 feet; what is the area of the largest square garden that can be inclosed in it? Ans. 5187.6 sq. ft.

THE ELLIPSE.

1033. An Ellipse is, a plane figure bounded by a curved line, the sum of the distances from every point of which to A two fixed points is equal to the line drawn through these points and terminated by the curve. The two fixed points are

called foci; the line through the foci is the transverse axis, and a line perpendicular to this passing through the centre and terminated by the curve, is the conjugate axis.

Rule. To find the area of an ellipse, we multiply the semi-axes together, and that product by 3.1416.

1. Required the area of an elliptical mirror whose length. is 7 feet and breadth 3.5 feet. Ans. 19.2423 sq. ft.

2. The axes of an ellipse are 100 inches and 60 inches;

what is the difference in area between the ellipse and a circle having a diameter equal to the conjugate axis? Ans. 1884.96 sq. in.

MENSURATION OF VOLUMES.

1034. A Volume is that which has length, breadth, and thickness. Volumes include the Prism, the Pyramid, the Cylinder, the Cone, the Sphere, etc.

THE PRISM.

1035. A Prism is a volume whose ends are equal polygons and whose sides are parallelograms.

1036. The polygons are called bases, the parallelograms form the convex surface, and the prism takes its name from the form of its bases.

1037. The Parallelopipedon is a prism whose bases are parallelograms. A Cube is a parallelopipedon all of whose sides are squares.

Rnle. To find the convex surface of a prism, multiply the perimeter of the base by the height.

NOTE. To find the entire surface we add the area of the bases.

1. What is the convex surface of a triangular prism, the sides of whose base are 10, 12, and 18 inches respectively, and its height 25 inches? Ans. 1000 sq. in.

2. What is the convex surface of a parallelopipedon, the sides of whose base are 12 and 15 inches, and the height 42 inches? Ans. 2268 sq. in.

3. What is the entire surface of a regular hexagonal prism, one side of the base being 25 inches, and the height 32 inches? Ans. 6423.797+ sq. in.

1038. Rule.-To find the contents of a prism, multiply the area of the base by the altitude of the prism.

1. Required the contents of a triangular prism, the sides of the base being 12, 12, and 9 inches, and the height 36 inches. Ans. 1802.124 cu. in. 2. What are the contents of a parallelopipedon, the side

of whose base is 17 inches, the altitude of the base 13 inches, and altitude of prism 25 inches? Ans. 5525 cu. in.

3. Required the contents of a pentagonal prism, the side of the base being 20 inches and the altitude of the prism being 46 inches. Ans. 31656.784+cu. inches.

THE PYRAMID.

1039. A Pyramid is a volume bounded by a polygon and several triangles meeting in a common point. The polygon is called the base, and the triangles form the convex surface.

1040. The point at the top is called the vertex, the distance from the vertex to the base is the alti

tude, and from the vertex to the middle of a side is the slant height.

Rule. To find the convex surface of a pyramid, multiply the perimeter of the base by one-half of the slant height. 1. What is the convex surface of a triangular pyramid, whose sides are each 16 ft. and slant height 26 ft.?

Ans. 624 ft.

2. Required the convex surface of the pyramid of Cheops in Egypt, one side measuring 763.4 feet, and the slant height being about 612 feet. Ans. 934401.6 sq. ft.

3. What is the entire surface of an octagonal pyramid, the side of the base being 64 feet and the slant height 75 feet? Ans. 38977.237+sq. ft. 1041. Rule. To find the contents of a pyramid, multiply the area of the base by one-third of the altitude.

1. What are the contents of a triangular pyramid, the sides of which are 65, 75, and 85 feet, and the altitude 96 feet? Ans. 75119.904 cu. ft. 2. Required the contents of a heptagonal pyramid, each side of the base being 56.52 feet, and the altitude 19.89 feet. Ans. 76964.825+ cu. ft.

3. Required the contents of a decagonal pyramid, each side of the base being 9 ft. 6 in., and the altitude 52 feet. Ans. 12036.307 cu. ft.

THE CYLINDER.

1042. The Cylinder is a round body of uniform diameter, with circles for its ends. The two circular ends are called bases.

1043. The Altitude of a cylinder is the distance from the centre of one base to the centre of the other.

Rule. To find the convex surface of a cylinder, multiply the circumference of the base by the altitude.

1. What is the convex surface of a cylinder, whose altitude is 15 ft. and diameter of base 9 ft.? Ans. 424.116 sq. ft.

2. The warm air pipes of a furnace are 11 inches in diameter and 246 feet in length; how many square feet of tin do they contain? Ans. 708.43 sq. ft.

1044. Rule. To find the contents of a cylinder, multiply the area of the base by the altitude.

1. Required the contents of a cylindrical stick of wood 2 ft.

6 in. in diameter, and 4 ft. 9 in. long.

2. Required the contents of a wire ter and 20 feet long.

3. Required the number of cubic feet 8 inches in diameter on the inside, and 8 the length of the pipe being 650 yards.

THE CONE.

Ans. 23.316 cu. ft.

of an inch in diame

Ans. 11.78 cu. in. of iron in a water-pipe inches on the outside, Ans. 87.74+cu. ft.

1045. A Cone is a volume whose base is a circle, and whose convex surface tapers uniformly to a point called a vertex.

1046. The Altitude of a cone is the distance from the vertex to the centre of the base, and the slant height is the distance from the vertex to the circumference of the base.

Rule. To find the convex surface of a cone, multiply the circumference of the base by one-half of the slant height. 1. What is the convex surface of a cone, slant height 45 in., circumference of base 72 in. ? Ans. 1620 sq. in.

2. There is a conical haystack whose slant height is 7.6 feet, and the diameter of the base 5.5 ft.; how many square yards of canvas will cover it? Ans. 7.29+ sq. yd.

3. The distance to the top of a certain mountain is 2 miles, and the circumference of its base 7.35 miles; what is its surface, supposing it to be nearly a perfect cone?

Ans. 9.1875 sq. miles.

1047. Rule. To find the contents of a cone, multiply the area of the base by one-third of the altitude.

1. What are the contents of a sugar-loaf, the diameter of whose base is 9 inches and whose height is 20 inches? Ans. 424.116 cu. in. 2. How many cubic feet in a conical hay-stack, 6.6 ft. high and 25 ft. in circumference? Ans. 109.4225 cu. ft.

THE FRUSTUM OF A PYRAMID AND CONE.

1048. The Frustum of a Pyramid is the part of a pyramid which remains after cutting off the top by a plane parallel to the base.

1049. The Frustum of a Cone is the part of a cone which remains after cutting off the top by a plane parallel to the base.

Rule. To find the convex surface of a frustum, take the sum of the perimeters or circumferences of the two bases, and multiply it by one-half of the slant height.

1. Required the convex surface of the frustum of a triangular pyramid, the side of the upper base being 3 ft., of the lower 5 ft., and the slant height 8 ft. Ans. 96 sq. ft.

2. Required the convex surface of the frustum of a cone, the diameters of the bases being 6 and 10 feet respectively, and the slant height 12 ft. 3 in. Ans. 307.8768 sq. ft.

1050. Rule. To find the contents of a frustum, take the sum of the two bases and the square root of their product, and multiply the sum by one-third of the altitude of the frustum.