25 inches; find the expense of paving the court, at 3s. per square foot. (47) The area of a circular pond is 346 sq. yds.; find the expense of fencing it round, at eighteen pence a yard. (48) The radius of a circle is 9 ft. 4 in.; required the number of degrees in the arc whose length is 11 in. (49) What is the length of the side of an octagon whose area is 30.1875 sq. ft.? (50) Find the diameter of a circular surface whose area is 3.9424 sq. yds. (51) The chord is 24 and the height 6; find the area of the segment. (52) A road runs round a circular shrubbery; the outer circumference is 209 ft. and the inner circumference 198 ft.; find the area of the road. (53) A field in the shape of a trapezoid contains 420 sq. yds., and its parallel sides are 80 ft. and 60 ft.; find their perpendicular distance. (54) The south side of a field is 325 lks., the east side is 350 lks., the north side is 100 lks., the west side is 325 lks., and the diagonal from south-west to north-east is 375 lks.; find the area in square chains and links. (55) Find the area of an equilateral triangle whose side is 3.25. (56) The exterior diameter of a metal pipe is 21 in., and the thickness of the pipe in.; what is the area of the circular ring in the section? (57) Find the diameter of a carriage-wheel which is turned round 800 times in two miles and a half. (58) A B C D is a quadrilateral; A C=560, the perpendicular DE upon A C=240, and the perpendicular B F upon A C= 300; required the area. (59) What is the length of the side of a heptagon which has an area of 8.1675 sq. ft. ? (60) Suppose that the distance of the earth from the sun is 91 millions of miles, find the number of millions of miles described by the earth in its revolution round the sun every year. PART III. MENSURATION OF SOLIDS. 128. A solid body has three dimensions, length, breadth, and thickness (height or depth); e.g., a block of wood or stone. 129. The volume, solidity, or solid content of a solid is the quantity of space that it takes up. Volumes are expressed in cubic measure. 130. Parallel planes are flat or even surfaces which are everywhere equally distant from each other. Thus the floor and ceiling of a room are parallel planes. 131. Plane figures which form the boundaries of sides are called faces of the solid; the straight lines which form the boundaries of the plane figures are called edges or sides. THE CUBE. 132. A Cube is a solid bounded by six equal square faces; its length, breadth E and height are all equal. Thus the figure annexed is a cube. ABCD, CDEF, etc., are faces; A B, A D, A G, etc., are edges; A B is the length, A G the breadth, G and AD the height, and these are all equal. 89 A H C 133. To find the volume of a cube when the side is given. RULE. Multiply the area of the base by the height, and the product will be the volume. Suppose we have a cube whose edge measures 3 in. Let it be cut by planes an inch apart parallel to the faces, as in the annexed figure; it is thus divided into 27 equal solids, each of which is a cube, being an inch long, an inch broad, and an inch thick. Such a cube is called a cubic inch; and as the cube contains 27 of them, its volume is said to be 27 cubic inches. The number of cubic units in each horizontal row will obviously be equal to the number of square units in the base; and the number of these rows will be equal to the number of linear inches in the height; therefore the number of cubic inches in the whole figure will be the number of square inches in the base multiplied by the number of linear inches in the height. [As the length, breadth, and height are all equal, this rule may be shortly expressed thus: Cube the side.] Ex. 1. Find the solid content of a cubical block of stone, each side measuring 5 ft. Volume 5x5x5=125 c. ft. Ex. 2. Find the volume of a cube whose edge is 1 ft. 6 in. (i.) By reduction. 1 ft. 6 in. = 18 in. Volume 18 x 18 x 18-5832 c. in.-3 c. ft. 648 c. in. (iii.) By decimals. 1 ft. 6 in. = 1.5 ft. Volume=1.5×1·5×1·5=3·375 c. ft.-3 c. ft. 648 c. in. Ex. 3. Find the value of a cubical block of metal whose edge is 2 ft. 3 in., at 1s. 4d. per solid foot. Solid content = 2}×2×2}=} ×}}=722 c. ft. Value=16d. × 72o =72od.=15s. 21d. Find, by reduction, the number of cubic feet and inches in cubes which have the following lengths:- (7) 2 ft. 6 in. (8) 3 ft. 4 in. (9) 5 ft. 1 in. (10) 5 ft. 6 in. Find, by fractions, the number of cubic cubes having the following lengths: (13) 12 ft. 9 in. (14) 2 ft. 2 in. (15) 8 ft. 4 in. (16) 7 ft. 7 in. (11) 12 ft. 3 in. (12) 18 ft. 9 in. feet and inches in (17) 1 yd. 2 ft. 3 in. following cubes whose (21) 4 ft. 3 in. (22) How many cubic feet of space are there in a room whose length, breadth, and height measure each 15 ft.? (23) How many solid yards are there in a cube whose side is 51 yds.? (24) How many cubic yards of earth were removed in digging a cubical cellar whose length is 6.75 yds.? (25) How many cubic feet of water will a cistern hold which is 4 ft. long, 4 ft. broad, and 4 ft. deep? (26) Find the weight of water in a cubical cistern each of whose sides measures 56 ft. (Express the result in tons.) [A cubic foot of water weighs 1,000 oz. avoirdupois.] (27) Find the weight of a cubical block of marble whose side is 8 ft. (Express the result in tons, etc.) [Marble is 2.8 times heavier than water.] (28) What is the weight, in cwts., etc., of a cubical block of gold whose side is 9 in. ? [Assume that gold is 19'6 times heavier than water.] (29) The side of a cubical mass of lead measures 2 ft. 6 in. find its weight in tons, etc., and its value at 2d. per lb. [Assume that lead is 112 times heavier than water.] (30) What is the weight of a cubical block of marble whose edge is 12.5 ft., at 180 lbs. to a cubic foot? (Express the result in tons, etc.) 134. To find the side of a cube when the volume is given. RULE. Extract the cube root of the number of cubic units in the volume, and the result will give the number of linear units in the side. (See Fig. Art. 133.) The cube of any number is the product of the number multiplied twice by itself; thus the cube of 5 is 125, for 5×5×5=125. The cube root of any number is that number whose cube is equal to the given number; thus the cube of 5 is 125, and therefore 5 is the cube root of 125. Hence we see that as we find the solid content of a cube by cubing the side, so the side may be found by taking the cube root of the solid content. The cube numbers to 1728 and their cube roots, should be committed to memory; they are Cubes 1,8,27,64, 125, 216, 343,512, 729, 1000, 1331, 1728. Cube roots 1,2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. 135. We now proceed to show how to find the cube root of a number exceeding 1728. CASE I. When the cube root has two figures only. RULE 1°. Divide the given number into periods of three figures, beginning with the units-figure. The number of periods shows the number of figures in the root. 2°. From the first period subtract the greatest cube number contained in it, set down its cube root in the quotient, and bring down the next period to the right of the remainder. 3°. Find a trial divisor by multiplying the square of the quotient by 3, find how many times it is contained in the first part of this new dividend, and put the result in the quotient to form the second figure of the root. 4°. Now find the true divisor by adding together (a) the trial divisor, (b) three times the product of the first and |