Arithmetic, for schools and colleges. (With key).

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 A B 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.

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 feet.

3 x 3 square feet.

3 × 3 × 3 cubic feet.

1 linear pole

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.

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

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

22 Yards 100 Links

A Palm

1 Chain

used in measuring land.

3 Inches, a Span = 9 Inches, a Cubit = 18

Inches, a Pace = 5 Feet.

REDUCTION.

Ex. 17.

Reduce

(1) To furlongs, 16 miles; 19 miles; 286 miles. (2) To poles, 13 fur.; 178 fur.; 178 ms.; 286 ms.; 136 ms. 6 fur.

(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.

(16) To inches, 7 fur. 16 po. 2 yds. 3 in.; 8 fur. 38 po. 2 ft.; 7 fur. 16 po. 2 ft. 7 in.

(17) To inches, 17 ms. 2 fur. 26 po. 2 yds. 1 ft. 10 in. ; 27 ms. 3 fur. 18 po. 3 yds. 2 ft.

(18) To inches, 196 ms. 7 fur. 34 po. 2 yds. 2 ft. 8 in.; 61 ms. 7 fur. 19 po. 2 yds. 7 in.

101. To reduce yards to poles we must divide by 5. Now from the principle that if the divisor and dividend be multiplied by the same number, the value of the quotient remains unaltered, it follows that we can multiply the number of yards by 2 and divide by 11, which is twice 5. We then obtain the required number of poles in the quotient, but we get twice the correct remainder.

EXAMPLE.-Reduce 3868 yards to perches.