And the chord=12,5 : the perpend. 16. · 12,5x16 m =100, rods, area of the triangle. Area of the sector=142 Area of the segment=42 rods. Ans. Note.--A regular polygon, whose sides and angles are all equal, may be measured by dividing it into triangles, finding the area of one, and multiplying this area by the number of triangles contained in the polygon. CASE 8.— To describe, and find the area of, an ellipse or oval. RULE.To describe an ellipse or oval, draw a line, set one foot of the dividers on the line, as a centre, and describe a circle ; move the dividers to some other point on the same line, [but not so far but that the dividers in forming a second circle may extend within the first,] and describe a second circle of the same radius as the former; then, in the two points where the circles intersect, set the dividers to complete the sides of the oval; and through these intersecting points draw the line called the conjugate diameter, crossing the line first drawn called the transverse diameter, in the centre of the oval. Rule.-To find the area of an ellipse, multiply the transverse, or longest diameter, by the conjugate, or shortest diameter, and their product by.,7854; and the last product is the area required. EXAMPLE. If the transverse diameter of an oval fish pond be 34 rods, and the conjugate diameter be 24 rods, what is its area a 34 x 24x,7854=640,8864 rods. Ans. Case 9.—To find the area of a globe or sphere. Definition.—A sphere or globe is a round solid body, in the middle or centre of which is an imaginary point, from which every part of the surface is equally distant. An apple, or a ball used by children in some of their pastimes, may be called a sphere or globe. Rule.-Multiply the circumference by the diameter, and the product will be the area, or surface.. EXAMPLES 1. What is the superficial content of the earth, if it be 360 degrees in circumference, and every degree measure 69} miles ? 360x691-25020 circumf. 355 : 113 :: 25020 : 7964+ diameter. And 25020x7964–199259280 area in squa. miles Ans. 2. If the moon's diameter be 2180 miles, what is her area ? Ans. 14928640+ square miles. Section II.-OF SOLIDS. Solids are measured by the solid inch, foot, or yard, &c. 1728 of these inches, that is 12x12x12, make 1 cubic or solid foot. - CASE 1.-To measure a Cube. Definition.-A cube is a solid of six equal sides, each of which is an exact square. RULE.-Multiply the side by itself, and that product by the same side, and this last product will be the solid content of the cube. EXAMPLES 1. If the side of a cubic block be 18 inches, or 1 foot and 6 inches, how many solid feet does it contain ? ift. 6in.=1,5ft. and 1,5x1,5X1,5=3,375 solid ft. Ans. in. in. in. Or, 18x18x18 -- =3,375 as before. 1728 2. Suppose a cellar is to be dug which shall contain 12 feet every way, in length, breadth, and depth; how many solid feet of earth must be taken out to complete it? Ans. 1728 sol. ft. CASE 2.- To find the content of any regular solid, of three dimensions, length, breadth, and thickness, such as a piece of square timber, whose length is more than its breadth and depth. RULE.-Multiply the breadth by the depth or thickness, and that product by the length; the last product is the solid content. EXAMPLES 1. How many solid feet are there in a piece of square timber that is 1 foot and 6 inches, or 18 inches broad, 9 inches thick, and 9 feet, or 108 inches long? . 1 ft. 6ini=1,5 foot. 75X1,5X9=10,125 sol ft. Ans. 9 inches=,75 foot. in. in. in. Or, 18x9 x 108 =10,125 as before. 1728 In measuring timber, however, you may multiply the breadth in inches by the depth in inches, and that product by the length in feet : divide this last product by 144, and the quotient will be the solid content in feet, &c. 2. How many solid feet does a piece of square timber, or a block of marble, contain, if it be 16 inches broad, 11 inches thick, and 20 feet long? 16x11x20=3520, and 3520-144=24,4+sol. ft. Ans. 3. If a stick ol square timber be 15 inches broad, 8 inches thick, and 25 feet long, how many solid feet are in it ? Ans. 20,8+ feet. Case 3.- When the breadth and thickness of a piece of square timber are given in inches, to find how much in bength will make a solid foot: RULE-Divide 1728 by the product of the breadth and depth, and the quotient will be the length, making a solid foot. EXAMPLES. 1. In a piece of square timber H inches broad and 8 inches deep, what length will make a solid foot ? 11x8=88)1728(19,6+inches. Ans. 2. In a piece of square timber 18 inches broad and 14 inches deep, what length will make a solid foot ? Ans, 6,8+iaches. CASE 4.-To measure a cylinder. Definition -A cylinder is a round body whose bases or ends are circles like a round column or stick of timber, of equal bigness from end to end. RULE.--Multiply the square of the diameter of the base or end by ,7854, which will give the area of the base ; then multiply the area of the base by the length, and the product will be the solid content. EXAMPLES. 1. What is the solid content of a round stick of timber, or a marble column, of equal bigness from end to end, whose diameter is 18 inches, and length 20 feet? 18 inches=1,5ft. 1,5x1,5x,7854=1,76715 area of the base. 1,76715x20 length=35,33 solid feet. Ans. Or, 18x18x,7854=254,4696 inches, area of the base. 254,4696 X20 --=35,343 as before. 144 2. What is the solid content of a round stick of timber, of equal bigness from end to end, whose diameter is 35 inches, and length 35 feet ? Ans. 233,847 feet. Case 5–To find how many solid feet a round stick of timber, equally thick from end to end, will contain, when hewn square. RULE.-Multiply twice the square of its semidiameter, in inches, by the length in feet; then divide the product by 144, and the quotient will be the answer. EXAMPLES. 1. If the diameter of a round stick of timber be 22 inches, and its length 20 feet, how many solid feet will it contain when hewn square ! 11x11x2x20 Half diameter=11, and 33,6+ft. the 144 solidity when hewn square, the answer. 2 If the diameter of a round stick of timber be 24 inches from end to end, and its length 20 feet, how many solid feet will it contain, when hewn square, and what will be the content of the slabs which reduce it to a square ? Am I 40 feet solidity when hewn square, w and 22,832ft. the solidity of the slabs. CASE 6.- To find how many feet of square edged boards, of a given thickness, can be sawn from a log of a given diameter. RULE.-Find the solid content of the log, when made square, by the last Case ; then say, as the thickness of the board, including the saw calf, is to the solid feet, so are 12 inches to the number of feet of boards. . EXAMPLES 1. How many feet of square edged boards, 14 inch thick, including the saw calf, can be sawn from a log feet long, and 24 inches diameter ? 12x12x2x20 --=40ft. solid content when hewn square. 144 As 11 : 40 :: 12 : 384 feet. Ans. 2. How many feet of square edged boards, 11 inch thick, including the saw gap, can be sawn from a log 12 feet long, and 18 inches diameter? Ans. 108 feet. · Note.—A short rule for finding the number of feet of one inch boards that a log will make, is to deduct of its diameter in inches, and of its length in feet; then for each inch of diameter that remainis, reckon 1 board of the same width as this reduced diameter, and of the same length as this reduced length of the log: thus a log 12 feet long, and 12 inches through, gives 9 boards, I feet long, 9 inches wide, or 60 feet-a log 16 feet long, and 16 inches through, gives 12 boards, 12 inches wide, 12 feet long, or 144 feet. Case 7.-The length, breadth, and depth of any cubical box being given, to find how many bushels it will con- . tain. RULE.-Multiply the length, breadth and depth together, in inches, and divide the last product by 2150,425, the solid inches in the statute bushel, and the quotient will be the answer. EXAMPLE There is a square or cubical box; the length of its bottom is 50 inches, breadth of ditto 40 inches, and its depth 60 inches ; how many bushels of corn will it hold ? 50 x 40 x 60 - =55,8+ or 55bush. 3 pecks. Ans. 2150,425 CASE 8.-To find the solidity of a cone or pyramid, , whether round, square, or triangular. Definition.--Solids which decrease gradually from the base till they come to a point, are generally called cones or pyramids, and are of various kinds, according to the |