(7) Eight iron girders weigh 46 tons 12 cwts. 60 lbs., and three of them weigh 17 tons 3 qrs. 10 lbs. what is the average weight of each of the others?

(8) Show that there are 9 times as many grains in 38 lbs. 10 oz. 19 dwts. 18 grs., as there are ounces in 13 cwts. 3 qrs. 16 lbs. 10 oz.

(9) How many grains make one pound Troy? There are 7000 grains to 1 lb. Avoirdupois: how many pounds Avoirdupois are equivalent to 175 lbs. Troy?

(10) How many pounds Troy are equivalent to 320 lbs. Avoirdupois ?

(11) Write out Apothecaries' weight, and make a list of six things which would be weighed by this weight.

(12) Which is heavier, 1 lb. of gold or 1 lb. of feathers? (13) Reduce 1 lb. Apothecaries' weight to grains, and find how many drams Apothecaries' weight are as heavy as 175 drams Avoirdupois.

(14) Which is greater, 1 lb. of gold or 1 lb of quinine? (15) How many grains of quinine are there in 78 packages, each containing 4 lbs. 6 oz. 2 drams?

(16) How many pounds of Turkey rhubarb would balance 484800 lbs. of sugar?

(17) A passenger on the Great Western is allowed to carry 56 lbs. of luggage, and is charged for excess at the rate of d. per pound for 50 miles. How much would a passenger have to pay who carries 1336 ounces 250 miles ?

(18) How many ounces Troy would be equivalent to 48 lbs. Avoirdupois?

(19) Which is the larger, 1 oz. Troy or 1 oz. Avoirdupois, and by how many grains?

(20) 99 firkins of butter, at 124d. per lb.?

(21) Find the cost of 87 silver ornaments, each weighing 2 lbs. 11 oz. 12 grs., at 34d. per half a dwt.

(22) How many ounces must be added to 276 cwt. 11 lbs.

in order to make 15 tons 2 cwts. 12 oz. ?

(23) A jeweller makes 6 lbs. 6 oz. 2 dwts. 22 grs. of gold into rings, each of which weighs 10 dwts. 2 grs. What will the rings be worth at £1 18s. 6d. each?

(24) In a luggage train there are eight wagons. One weighs 22 tons 1 cwt. 3 qrs. 18 lbs.; two others weigh 15 tons 3 cwts. 2 qrs. 25 ibs. each; three others 51 tons 15 cwts. 19 lbs. 8 oz. each; another 19 tons 19 cwt. 3 qrs.

27 lbs. 15 oz., and the other 17 tons 1 qr. 19 lbs. gine weighed half as much as the rest of the train. was the weight of the train?

The en

What

(25) Three bars of gold weigh 21 lbs. 7 oz. 11 dwts.; the first and second together weigh 15 lbs. 7 oz. 11 dwts. 12 grs., and the first weighs half the third: what does the second weigh?

is the weight of the third ?

(26) Three girders for a railway bridge weigh 54 tons 16 cwt. 11 lbs., and two weigh 18 tons 2 qrs. 8 oz. What (27) Find the cost of three bars of gold weighing respectively 3 lbs. 2 oz. 19 grs.; 1 lb. 10 oz. 15 dwts.; and 4 lbs. 9 oz. 4 dwts. 5 grs., at £3 9s. 91d per oz.

(28) Find by how much the sum of 17 tons 13 cwts 15 lbs., and 19 tons 3 qrs. 18 lbs. exceeds the difference of 20 tons 19 cwts. 18 lbs., and 50 tons 16 cwts. 1qr.

(29) Three parcels consisting of 91 lbs. 1 oz. 5 dwts. 2 scrs. 19 grs. of Peruvian bark (Apothecaries' weight), 75 lbs. of lead (Avoir.), and 91 lbs. 1 oz. 15 dwts. 1 gr. of gold (Troy), are carried by three men. Which will have the heaviest load, and which the lightest ?

(30) How many more ounces are there in 768 tons 11 cwts. 15 lbs. of coals than grains in 1000 lbs. of gold? (31) Divide 187 lbs. 11 oz. 17 dwts. 18 grs. of gold, amongst a number of persons, giving to each 387 grains.

(32) A grocer buys sugar weighing 2 tons 125 lbs., 19 cwts. are spoilt, and the rest is sold at 5d. per lb. The grocer gains £17 18s. 6d. What did the sugar cost? (33) How many tons, qrs., and lbs., of each an equal number, are there in 887 tons 5 cwts. 3 qrs. 8 lbs. ?

(34) If 2 cwts. 5 qrs. 11 lbs. of sugar cost £6 15s. 6d. ; what quantity of the same quality can be bought for £1 13s. 10 d.?

(35) How many parcels can be made out of 38 tons 15 cwts. 77 lbs., if half the number of parcels contain a stone each, and the remainder half as much?

(36) Find the difference of 89 tons 11 cwts. 180 oz. X 98, and 7986 tons 17 cwts. 355 lbs. ÷ 993.

(37) A piece of gold is worth £95680 at £3 17s. 10дd. per oz., what is its weight?

(38) If 9 oz. Avoir. cost 72s., what will 11 lbs. cost ?* (39) If 28 lbs. Avoir. cost £89 14s. 4d., what cost 28 cwt.?

Measures of Extent.

94. DEFINITIONS.-A square is a four-sided figure whose sides are equal and whose angles are right angles.

95. A cube is a solid figure contained by six equal sides. 96. A square inch is a square each of whose sides is an inch in length; a square foot is a square each of whose sides is a foot in length.

97. A cubic inch is a cube each of whose sides is a square inch, and therefore each of its edges is an inch in length; a cubic foot is a cube each of whose sides is a square foot.

The nature of the units of these measures will be best understood by reference to some solid object. Suppose, for example, we desire to know the size of a large block of building stone. The stone has edges the length of which we can find by applying to them a rule or measure of a certain length-say a foot or a yard: such a measure would be a unit of length.

The stone also has sides or surfaces. The amount of surface presented by any one of these may be found by applying to it some other surface of known size; such, for example, as a square foot or square yard. Whatever such surface we might take would be a unit of surface.

Again, we may estimate the solidity of the block, that is, the quantity of stone it contains, by finding how many smaller cubes of known size will form a mass equal to it. The cube selected for comparison would form a unit of solidity or volume.

The mutual relation of the various units of length are shown in the table of Linear Measure; those of surface, in the table of Square Measure; those of solidity or volume, in the table of Cubic Measure.

The units of the same name in the different tables are totally distinct from each other, and must never be confounded, as no multiple of a unit of one kind will make a unit of another kind. For example, no number of linear inches would make a square inch, foot, yard, etc., and no number of square inches would make a cubic inch, foot, or yard.

98. The relation between these measures will appear from the following figures.

Fig. 1.

Suppose each of the divisions in AB to represent a foot of length. Then Fig. 1 represents a yard of length; Fig. 2, a square yard; Fig. 3, a cubic yard.

Fig. 3.

B

A

AB in each figure represents a yard of length, or 3 feet. ABCD is a square yard, each side of which measures 1 yard in length; and it is evident that this square yard contains 3 times 3 square feet. Similarly, each face of the cubic yard contains 9 square feet. This cubic yard can be cut into three layers of cubic feet, with 9 in each layer, so that 1 cubic yard 3 times 9, or 27 cubic feet. Similarly,

=

12 x 12 square inches.
= 12 x 12 x 12 cubic inches.

= 3 x 3 square feet.

= 3 × 3 × 3 cubic feet.

= 3 feet.

1 square yard
1 cubic yard

1 linear pole
1 square pole

=

=

5 yards.

5x 5 square yards = 30 square yards.

LINEAR OR LONG MEASURE.

99. This measure is used to measure distances, lengths, breadths, heights, depths, and the like, of places or

things.

12 Lines make 1 Inch, which is written 1 in.
12 Inches...... 1 Foot

1 ft.

1 yd.

1 fth.

1 Rod, Pole, or Perch... 1 po.

40 Poles

1 Furlong..

8 Furlongs... 1 Mile

3 Miles

1 League

1 fur.

1 m.

1 lea.

100. The following measure may be added as useful in certain cases:

4 Inches make 1 Hand (used in measuring horses.)

1 Chain) used in measuring land.

= 9 Inches, a Cubit = 18

3 Inches, a Span

Inches, a Pace = 5 Feet.

(1) To furlongs, 16 miles; 19 miles; 286 miles. (2) To poles, 13 fur.; 178 fur.;

6 fur.

178 ms.; 286 ms.; 136 ms.

(3) To yards, 166 po.; 178 po. ; 674 po. ; 293 po.; 171 po. 2 yds.

(4) To yards, 756 fur.; 29 fur. 6 po.; 713 fur. 4 po. 3 yds.; 8 fur. 3 yds.

(5) To yards, 713 ms. ; 386 ms.; 417 ms. 6 fur.; 238 ms. 8 fur.

(6) To yards, 9 ms. 7 fur. 6 po.; 7 ms. 6 fur. 27 po.; 371 ms. 6 fur. 19 po.

(7) To yards, 41 ms. 2 fur. 17 po. 2 yds.; 71 ms. 1 fur. 6 po. 2 yds.; 6 ms. 1 po. 3 yds.

(8) To feet, 764 yds. ; 376 yds. ; 274 yds. 1 ft. ; 796 yds. 7 ft.

(9) To feet, 76 po. 2 yds. 1 ft. ; 181 po. 1 yd. 2 ft. ; 186 po. 3 yds. 1 ft.

(10) To feet, 138 fur. 11 po. 2 yds. 1 ft.; 166 fur. 33 po. 3 yds.; 75 fur. 35 po. 1 yd.

(11) To feet, 73 ms. 7 fur. 27 po.; 476 ms. 3 fur. 19 po. 2 yds.; 715 ms. 7 fur. 3 yds.

(12) To feet, 175 ms. 5 fur. 3 yds.; 167 ms. 2 fur. 8 ft.; 12 ms. 3 fur. 19 po. 3 ft.

(13) To inches, 176 ft.; 273 ft.; 1769 ft. 8 in.; 216 ft. 5 in.

(14) To inches, 16 yds. 1 ft. 8 in.; 19 yds. 7 in.; 16 yds. 1 ft. 11 in.; 18 yds. 6 in.

(15) To inches, 7 po. 3 yds. 2 ft. 6 in.; 6 po. 2 yds. 1 ft. 11 in.; 17 po. 2 yds. 2 ft. 8 in.