(39) Find a recurring decimal equal to 5·874+1·621423-1748126+2*2428.

(40) Reduce 02391 guineas to the decimal of 5579 of £5.

(41) A book containing 218 leaves is 1.295 inches thick; allowing '08 of an inch for the thickness of the cover, find to 6 places of decimals the thickness of the paper.

(42) Divide the cube of 483 by twice the square of

'4025.

(43) If a sovereign be equal to 25°22 francs, or to 4'85 dollars, what decimal is a franc of a dollar?

(44) 23 coins amount together to £2. II are of one kind, and there are 6 of each of two other kinds whose values are 25 and 125 of that of the former. What are the coins?

(45) At what rate per minute is a point on the equator carried round in the earth's revolution on its axis; the circumference of the earth being 24898.87 miles, and the time of revolution 23 hrs. 56'0682 ?

(46) Express 2 805 florins-1.89 half-crowns as the decimal of £472.

(47) If 327 of some work is done in 3 hrs. 38 min., how long will the whole require?

(48) Cork, whose weight is 24 of that of water, weighs 15 lb. per cubic foot: what is the weight of 6 cubic feet of oak, which weighs 934 times as much as water?

(49) Find the difference between 2 of 1875 of 142857 of £1 158. and •285714 of ·538461 of £1 6s.

(50) A can run a mile in 7·68 minutes, B can run at the rate of 7.68 miles an hour, which is the faster runner? And in a race which the faster wins in six minutes, how far will the loser be behind?

CHAPTER VI.

SQUARE AND CUBIC MEASURE.

46. The units of area, or surface measurement, square inches, square feet. &c., are squares of which one side is a unit of length, an inch, foot, &c. The units of volume, or cubical measurement, are cubes of which one side is a unit of length.

47.-To find the area of a rectangle.

RULE. Express the length and breadth in the same denomination, and multiply them together; the product will be the number of square units of that denomination.

For let ABCD be a rectangle, D
whose length AB is 4 inches, and
breadth AD is 3 inches. Divide
AB into its four inches, and AD
into three; and through the points
of division draw lines

the sides of the figure.

parallel to

The figure

will then be divided into small

C

B

squares, each of whose sides is an inch. Each horizontal row of squares will consist of four, corresponding to the divisions of AB: and there will be three rows, corresponding to the divisions of AD: therefore there will be altogether 4× 3, i. e. 12 square inches in the figure.

It must be noticed that we do not here multiply inches by inches, but merely the numbers together: the former would be unmeaning (12).

Ex. Find the area of a floor 16 ft. 4 in. long, and 12 ft. 6 in. wide.

16 ft. 4 in. 196 in.; 12 ft. 6 in. = 150 in.

Or, 16 ft. 4 in. = 16 ft.; 12 ft. 6 in. = 12 ft.

N.B.-The explanation given above will show the connexion between the numbers given in the tables of measurement of Length and Surface. For instance, a square foot is a square whose sides are each 12 inches, it therefore contains 12 x 12, i. e. 144 square inches; so the square pole contains 5 × 51, i. e. 304 square yards, &c.

Exercise 45.

(1) Find the area of a rectangle 1 ft. 10 in. long, and 1 ft. 3 in. broad.

(2) Of a rectangle 12 ft. 8 in. long, and 7 ft. 6 in. broad. (3) Of a rectangle 2 yds. 8 in. long, and 1 yd. 2 ft. 6 in. wide.

(4) How many square yards are there in a path 36 yds. long, and 4 ft. 2 in. wide?

(5) How many acres are there in a field 385 yds. in length, and 264 yds. in breadth?

(6) Find the area of a field 10 chains 50 links long, and 7 chains 84 links wide.

(7) A wall is 50 yds. long, and 6 ft. 8 in. high; how many square yards of brickwork are there in it, a doorway being left 6 ft. high and ft. broad?

(8) The length and width of a field being 12 chains and 9 chains 75 links, find its area; and its value at £150 per

acre.

(9) How many square yards of carpet will be required for a room 24 ft. 8 in. long, and 18 ft. 9 in. wide?

(10) Find the area of a square whose side is 3 ft. 8 in. (11) What is the area of a plank 7 ft. 53 in. long, and 114 in. wide?

(12) How many square yards are there in a mile of road 21 ft. wide?

48.-There are other surfaces which may be regarded as rectangles, for the purpose of mensuration; as for instance the curved surface of a cylinder, whose circumference and breadth may be taken as the length and breadth of the rectangle. In the same manner, the walls of a room may be considered as together equal to a rectangle whose length is the whole circumference of the room, and breadth the height of the room.

Ex. Required the area of the walls of a room whose length is 18 ft. 4 in., breadth 15 ft. 2 in., and height 10 ft. 6 in.

The circumference=2 (18 ft. 4 in.+ 15 ft. 2 in.)

(1) Find the area of the wall round an enclosure, whose circumference is 520 yards, and height 7 ft.

(2) What is the area of the walls of a room whose length and width are 13 ft. 3 in. and 10 ft. 5 in., and height 9 ft. 3 in. ?

(3) Find the surface of a garden roller whose circumference is 7 ft. 9 in. and width 2 ft. 8 in.

(4) How many square yards of canvas would be required for the sides of a tent 62 ft. 5 in. round, the height of the sides being 6 ft. 9 in.?

(5) How much paper would be required for a room 20 ft. 7 in. long, 14 ft. 2 in. wide, and 11 ft. 2 in. high?

(6) Find the number of square feet of bark on a part of a tree 7 ft. 3 in. long, and a yard round.

(7) Determine the number of square yards of fencing 6 ft. high which would surround a field 210 yds. 2 ft. 7 in. long, and 156 yds. 1 ft. 2 in. wide.

(8) Find the surface of a square tower 27 ft. wide, and 80 ft. high.

(9) Find the surface of an octagonal shaft 9 ft. long, the breadth of each face being 44 inches.

(10) How many square yards of paper will be required for a square room 8 ft. 7 in. high, and 14 ft. 9 in. wide?

(11) A room 27 ft. long and 23 ft. 5 in. wide, has a wainscoting 3 ft. 2 in. high, find the cost of painting it at 38. per square yard.

(12) If the same room be 13 ft. high, what would be the expense of papering the walls above the wainscoting at IS. 4d. per square yard?

49.-Conversely, if the area of a rectangle be given, and one of its dimensions, the other may be found by division. Ex. The length of a rectangular floor which contains 34 square yards is 19 ft. 1 in.; determine the breadth.