circumference by the diameter, or to the product of 3.1416 by the square of the diameter. 2. The CONTENTS of a sphere are equal to the product of the surface by one third of its radius, or to the product of one sixth of 3.1416 by the cube of the diameter. Exercises. 1. What is the surface of a cannon ball whose diameter is 9 inches? Ans. 254.46+ sq. in. 2. How many cubic meters in a sphere whose diameter is 12 centimeters? Ans. .000904 cu. me. 3. Required the contents of a globe 15 inches in diameter. Ans. 1767.15 cu. in. 4. What is the surface of the earth, allowing it to be a sphere 7912 miles in diameter? Ans. 196663355.75+ sq. m. SIMILAR FIGURES AND VOLUMES. 431. Two Figures, or two Volumes, are similar, when they exactly correspond in form, without regard to size. From the relations of similar figures, or of similar volumes, to each other, which may be proved by Geometry, we have the following PRINCIPLES. 1. The areas of similar figures and volumes are to each other as the squares of their corresponding dimensions. Hence, 2. The corresponding dimensions of similar figures and volumes are to each other as the square roots of their areas. 3. The contents of similar volumes are to each other as the cubes of their corresponding dimensions. Hence, To what is the surface of a sphere equal? The contents of a sphere? When are two Figures or two Volumes similar? What is the first Principle? The second? The third? 4. The corresponding dimensions of similar volumes are to each other as the cube roots of their contents. Exercises. 1. If a triangle whose base is 20 feet has an area of 200 feet, what is the area of a similar triangle whose base is 10 feet? SOLUTION. 20a: 10a:: 200 : 50, Ans. 2. If a circle whose diameter is 12 feet has an area of 113.09 square feet, what is the area of a circle whose diameter is 15 feet? Ans. 176.70 sq. ft. 3. If it costs $125 to pave a rectangular court whose width is 40 feet, what will it cost to pave a similar court whose width is 30 feet? Ans. $70.31. 4. If a triangle whose altitude is 40 feet has an area of 1000 square feet, what is the altitude of a similar triangle whose area is 900 feet? Ans. 37.947 ft. 5. If the weight of a cannon ball 8 inches in diameter is 36 kiloe, what is the weight of a similar ball 9 inches in diameter? Ans. 51.25 kilos. 6. If a sphere of silver 1 inch in diameter be worth $6, what must be the diameter of another sphere to be worth $10368 ? Ans. 12 inches. 7. A bushel measure is 18 inches in diameter; what must be the diameter of a half bushel measure of similar form? Ans. 14.68 inches. 8. If the side of a cubical box is 2 feet, what must be the side of a similar box which shall contain 3 times as much? Ans. 2.88 feet. 9. If a cylindrical pipe 20 centimeters in diameter will fill a cistern in 114 minutes, how long will it take a similar pipe 30 centimeters in diameter to fill it? Ans. 5 minutes. 10. If two men own together a conical stack of hay, which REVIEW QUESTIONS. What is a Sphere? (428) The Radius of a sphere? (429) The Diameter of a sphere? (430) is 16 feet in hight, how far down from the top must one of them take off for his part, if it is of the whole? Ans. 8 feet? BOARD MEASURE. 432. Lumber, or sawed timber, as boards, planks, joists, and beams, are usually measured by board measure. In Board Measure 1 foot is reckoned 1 foot long, 1 foot broad, and 1 inch thick. Hence, 433. To find the contents of boards, planks, joists, etc. Multiply the product of the length and breadth, each taken in feet, by the number denoting the thickness in inches. When the boards, planks, etc., are tapering, take half the sum of the breadth of the two ends for the breadth. Since 1 foot board measure is 1 foot or 12 inches long, 1 foot or 12 inches broad, and 1 inch thick, it must be equal 12 X 12 X 1 = 144 cubic inches. 144 cubic inches are contained in 1728 cubic inches, or in 1 cubic foot, 12 times. Hence, 12 board feet1 cubic foot. Exercises. 1. What are the contents of a board 20 feet long and 16 inches broad? Ans. 263 bd. ft. 2. How many square feet in 2 planks, each 16 feet long, 18 inches wide, and 3 inches thick? Ans. 144 bd. ft. 3. What are the contents of 6 joists, 14 feet long, and 4 inches square? Ans. 112 bd. ft. 4. What is the cost of a stick of timber 24 feet long, 10 inches wide, and 6 inches thick, at 3 cents a square foot? Ans. $3.60. 5. What are the contents of a plank 22 feet long, and 3 inches thick, the width of the ends being 16 and 20 inches respectively? Ans. 115 bd. ft. How is lumber usually measured? How do we find the contents of boards, planks, etc.? GAUGING. 434. Gauging is the process of finding the capacity of casks. The Mean Diameter of a cask is very nearly equal to the head diameter increased by two thirds of the difference between the bung and head diameters, or by three fifths when the staves are but slightly curved. The capacity of a cask is that of a cylinder of the same length and mean diameter. Hence, 435. To find the capacity of casks, Multiply the product of the square of the mean diameter and the length, expressed in inches, by .0034 for liquid gallons, or by .0129 for liters. Multiplying by .0034 is the same as multiplying by .7854 and dividing by 231, and by .0129 the same as by .7854 and dividing by 61.022. Since a liter is equal to one cubic decimeter, there will be in a cask 1000 times as many liters as cubic meters. Exercises. 1. How many gallons in a cask whose mean diameter is 18 inches, and whose length is 30 inches? Ans. 33+ gallons. 2. How many gallons in a cask 36 inches long, 22 inches bung diameter, and 16 inches head diameter ? Ans. 48.96 gallons. 3. How many gallons in a cask whose length is 60 inches, bung diameter 36 inches, and head diameter 32 inches? 4. How many liters in a cask 1 meter long, and whose mean diameter is 60 centimeters? Ans. 282.744 liters. What is the capac- What is Gauging? The Mean Diameter of a cask? ity of a cask? How can we find the capacity of casks? pacity of a cask in liters be found, when its contents in cubic meters are known? MEDIAL PROPORTION.* 436. Medial Proportion, or Average, treats of mixing different articles. This subject has sometimes been called Alligation, from the mechanical method formerly adopted of linking or tying together figures by a line, in the process of solving its questions. CASE I. 437. To find the average value of given quantities of different values. 1. Let it be required to find the average value of a mixture of 8 lb. of sugar, worth 10 cents a pound, with 12 lb., worth 15 cents a pound. OPERATION. $.10 X 8$.80 .15 X 12 20 1.80 ) 2.60 $.13 8 pounds of sugar, at 10 cents a pound, are worth $.80, and 12 pounds of sugar, at 15 cents a pound, are worth $1.80; hence, the whole 20 pounds are worth $.80+$1.80, or $2.60. If 20 pounds of the mixture are worth $2.60, one pound of it is worth of $2.60, which is $.13. RULE. Divide the entire value of the mixture by the entire quantity, and the result will be the average value. Examples. 2. A farmer mixed 8 bushels of oats, worth 50 cents a bushel, 12 bushels of corn, worth 65 cents a bushel, and 10 bushels of barley, worth 60 cents a bushel; what was the average value of the mixture a bushel? Ans. $.59. 3. A grocer mixed together 18 lb. of oolong tea, at $1 a pound, 6 lb. of souchong, at $.60 a pound, and 6 lb. of hyson, at $1.20 a pound. How much a pound is the mixture worth? Ans. $.96. What is Medial Proportion? Explain the operation. What is the Rule? * Optional. |