the area of a surface is the number of units of surface it contains. The unit of surface is usually a square each side of which is some unit of length-for example, a square inch or a square foot; hence the name "square measure."

The area of a surface is expressed (in square units) by the product of the numbers (of linear units) that represent its two dimensions.

244. A square is a plane figure bounded by four equal sides and having four right angles. A square inch is a square each side of which is one inch in length. Square feet, square yards, square miles, etc., are similarly formed, with sides respectively of 1 foot, 1 yard, 1 mile, etc.

Studying the table we see that in a square foot we have (12 x 12) or 144 square inches; and (3 x 3) or 9 square feet in a square yard, etc. Figs. 4 and 5 illustrate the reason for this.

Let the side of Fig. 4 be one unit, and that of Fig. 5 two units, in length. Now in Fig. 4 there is one square unit of surface. In Fig. 5, however, in which each side is twice as long, there are (2 X 2) or 4 square units.

245. The following table gives the unit equivalents of square These results should be worked out by the use of the

measure.

table given above, and verified, before being used for reference.

246. Reduction in square measure is an operation similar to

that in linear measure.

Reduce 87 sq. yds. 4 sq. ft. to square feet.

787 number of sq. ft. in 87 sq. yds. 4 sq. ft.

=

Therefore 87 sq. yds. 4 sq. ft. = 787 sq. ft. Ans.

:

Reduce 787 sq. ft. to higher denominations. We find the number of sq. yds. (the next higher denomination) in 787 sq. ft. by dividing by 9, the number of sq. ft. in a sq. yd. 9 is contained 87 times with a remainder of 4.

9) 787
87 4-9

Therefore 787 sq. ft. 87 sq. yds. 4 sq. ft. Ans.

=

NOTE. Do not neglect, in square measure, to prefix the abbreviation sq. to the various denominations-yds., ft., in., etc.

SURVEYORS' MEASURE.

247. Surveyors Measure was obtained by calling one tenth of an acre a square chain. A tenth of an acre is equal to 16 square rods, and is equivalent to a square each side of which measures 4 rods. Thus, since 16 square rods is equal to one square chain, a linear chain is 4 rods or 792 inches. The linear chain is divided into 100 equal parts called links.

248. The measure of volumes or solids is called Cubic Measure.

249. A cube is a body bounded by six equal squares called "faces." The sides of these squares are called the "edges" of the cube; hence a cubic inch is a cube each edge of which is one inch in length.

250. The volume of a solid is the numerical value of its ratio to another solid called the "unit of volume;" or, in other words, the volume of a solid is the number of units of volume it contains The usual unit of volume is a cube each edge of which is some unit of length; hence the name "cubic measure."

251. The volume of a body is expressed (in units of volume) by the product of the numbers (of linear units) that represent Its dimensions.

NOTE. This does not apply to Liquid Measure; for while the volume of a liquid is at all times the same, its form may vary, and hence we use for our unit of liquid measure as much liquid as is contained in a vessel of given capacity, as explained below.

NOTE. 16 cub. ft. =

1 cord foot (cd. ft.). A pile of wood 8 feet long

4 feet wide, and 4 feet high contains a cord. The cord foot is foot of the length of such a pile.

A perch of stone or masonry is 161⁄2 feet (1 rod) long, 11⁄2 feet wide, and 1 foot high. In some localities 22 cubic feet is considered a perch of stone; in others, 18 cubic feet. We shall consider a perch to contain 24 cubic feet.

252. Studying the above table we see that 1,728 is 12 × 12 X 12, and that 27 is 3 x 3 x 3. The following diagrams show the reason for this:

In Fig. 6 we have a block which we shall consider as the unit of volume, each of its edges being taken as a unit of length. Each

edge of Fig. 7 is two units in length, and the figure comprises two layers, an upper and a lower, in each of which there are four units of volume, making a total of eight units.

The volume of Fig. 6 is 1 × 1 × 1, or 1 unit of volume.
The volume of Fig. 7 is 2 x 2 x 2, or 8 units of volume.

NOTE. The use of the multiplication sign in finding the volume has given rise to its use in indicating the dimensions of surfaces and solids. Thus a 28 joist means a joist 2 inches thick and 8 inches wide; or a room 15 X 12 X 10 means a room 15 feet long, 12 feet wide, and 10 feet high.

MEASURES OF CAPACITY.

253. Capacity signifies the extent of volume or space. Measures of capacity are divided into two classes: measures for liquids (liquid measure) and measures for dry substances (dry measure). The units of capacity are the gallon for liquid measure, and the bushel for dry measure.

LIQUID MEASURE.

254. Liquid Measure is used in measuring liquid substances such as water, wine, etc.

NOTE. One U. S. standard gallon contains 231 cubic inches and weighs approximately 8.335 pounds.

One cubic foot of water contains approximately 7.48 U. S. standard gallons and weighs 624 pounds.

255. The following table of equivalents is based on the various values in liquid measure. Before using it for reference, work out carefully the results and verify them.

256. Barrels and hogsheads mentioned in the above table are in practice not fixed values, as they vary when used for commercial purposes. The names of casks that do not indicate any fixed volume are tierce, hogshead, pipe, butt, tun. They are usually gauged and have their capacity in gallons marked on them, Change 1 hhd. 1 bbl. 7 gal. 2 qts. to quarts.

257. Dry measure is used in measuring dry substances such

as grain, fruit, salt, etc.