whose base is 4 feet and whose slant height is 20 feet 8 inches. 2. Find the convex surface of a cone, the circumference of whose base is 16 feet and whose slant height is 40 feet. 3. Illustrate by an original problem the method of finding the convex surface of a cone. ART. 506.-To find the contents of a cone.-Multiply the area of the base by one third the altitude. WRITTEN EXERCISES. 1. Find the contents of a cone 2 feet in diameter and 6 feet high. 2. Find the contents of a cone 9 feet high and 12 feet in circumference. 3. Illustrate by an original problem the method of find ing the contents of a cone. THE FRUSTUM OF A PYRAMID OR CONE. ART. 507. The Frustum of a pyramid or of a cone is that portion which remains after cutting off the upper part by a plane parallel to the base. ART. 508.-To find the convex surface of a frustum.Multiply the sum of the perimeters of the two bases by one half the slant height. WRITTEN EXERCISES. 1. What is the convex surface of the frustum of a square pyramid whose slant height is 18 feet, the side. of the upper base 5 feet and of the lower base 12 feet? 2. What is the convex surface of a cone whose slant height is 14 feet, whose upper base is 3 feet in diameter and whose lower base 7 feet in diameter ? 3. Illustrate by an original problem the method of finding the convex surface of a frustum. ART. 509.-To find the contents of a frustum.-To the area of the two bases, add the square root of their product, and multiply the sum by one third the altitude of the frustum. WRITTEN EXERCISES. 1. Find the contents of the frustum of a square pyramid the sides of whose bases are 3 and 4 feet respectively, and whose height is 12 feet. 2. Find the contents of the frustum of a cone 30 feet long, the diameter of one base being 2 feet and of the other 5 feet. 3. A stump 10 feet high was 3 feet in diameter at the base, and 30 inches at the top: find its cubical contents. 4. Illustrate by an original problem the method of finding the contents of a frustum. THE SPHERE. ART. 510.-A Sphere is a volume bounded by a curved surface, every point of which is equally distant from a point within called the cen ter. ART. 511.-The Diameter of a sphere is a straight line beginning at the surface, passing through the center and terminating in the surface on the other side. ART. 512.-The Radius of a sphere is a straight line extending from the center to the circumference. ART. 513.-To find the surface of a sphere.-Multiply the square of the diameter by 3.1416, or multiply the circumference by the diameter. WRITTEN EXERCISES. 1. Required the surface of a sphere whose diameter is 20 inches. 2. Required the surface of a sphere whose circumference is 8 feet. 3. Required the surface of a sphere whose diameter is 1 foot. Illustration.-Take a cube and a sphere made of the same material, and let the edge of the cube equal the diameter of the sphere. Weigh them carefully; divide the weight of the sphere by the weight of the cube, which gives .5236, the ratio of their volumes. ART. 514.-To find the volume of a sphere.-Multiply the cube of the diameter by .5236. WRITTEN EXERCISES. Hence 1. What is the volume of a sphere whose diameter is 30 inches? 2. What is the volume of a globe whose diameter is 18 inches? 3. What is the volume of a sphere whose radius is 10 inches? 4. The diameter of a sphere is 6 feet: how many cubic feet does it contain? 5. The circumference of a sphere is 3.1416 inches: find its cubical contents. 6. Find the cubical contents of a spherical vessel whose diameter is 3 feet. 7. The volume of a sphere is 2 cubic feet: what is its diameter ? 8. Illustrate by an original problem the method of finding the volume of a sphere. GAUGING. ART. 515.-Gauging is the method of finding the capacity of casks, barrels, and similar vessels. A barrel differs from a cylinder by the expansion in the middle. One half of the middle, or bung diameter, added to one half the end diameter is the mean diameter of the barrel, or of a cylinder of equal capacity. The following rule is in general_use: ART. 516.-Multiply the square of the mean diameter in inches by the length in inches, and the product by .0034. The result will be the capacity in wine gallons. WRITTEN EXERCISES. 1. How many gallons in a cask whose head diameter is 26 inches, bung diameter 32 inches, and length 42 inches? 2. What is the capacity of a cask whose bung diameter is 36 inches, head diameter 30 inches, and length 3 feet 8 inches ? 3. What is the capacity of a cask whose length is 40 inches, and the diameters 24 and 28 inches respectively? 4. How many gallons will fill a barrel 4 feet long, head diameter 24 inches, bung diameter 28 inches? 5. Illustrate by an original problem the method of gauging. The Metric System. ART. 517.-The Metric System was adopted by France in 1795. The commission to whom was entrusted the work of constructing a system of weights and measures, divided the distance from the equator to the poles into 10,000,000 equal parts, and named one of these parts a meter. This they made the unit of length, and from it the entire system is derived. Multiples and subdivisions are indicated by prefixes to the names of the fundamental units. The prefixes denoting multiples are deca, ten; hecto, hundred; kilo, thousand; myria, ten thousand. Those denoting subdivisions are deci, tenth; centi, hundredth; milli, thousandth. Reduction from one denomination to another is performed by simply moving the decimal point to the right or left as required. The principal metric units are:— ART. 518.-Meter for length. ART. 519.-Square Meter and the Are for surface. ART. 520.—Cubic Meter or Stere for large volumes. ART. 521.--Liter for smaller volumes. ART. 522.-Gram for weight. Measures of Length. ART. 523.—The Meter is the unit of length. 39.37 inches or 3.28 feet. 10 millimeters (mm.) 10 centimeters 10 decimeters 10 decameters It equals = 1 centimeter, cm. = 1 decimeter, dm. |