A and B is evidently less than the slack one. Similarly, if any two points be joined by a straight line, this line will be shorter than any

curved line (fig. 8a) or

Fig. 8 b.

any bent line (fig. 86) connecting the same points.

The measurement of lines.

12. When two straight lines are given, we may find which is the longer by applying one to the other, and in the same way we may find how many times one is contained in the other. The two lines may also be compared by applying to them a third line, and observing how many times it is contained in each. The line which is applied to the others is termed a unit or measure, and the process is termed measuring.

The standard unit of length used in England is the yard, which is the length of a straight line joining two marks on a bar of metal preserved in the Houses of Parliament, at Westminster. Several exact copies of this bar are deposited in secure places throughout the kingdom. Drapers,

surveyors, masons, and others, use rods, equal in length to the standard yard. Carpenters, smiths, and mechanics, for the measurement of shorter lines, use a measure called a foot-rule, which is equal in length to one third of the

Fig. 9.

yard. The foot is divided into twelve inches, and the inch is subdivided into eighths or twelfths.

Surveyors use a chain, twenty-two standard yards long, consisting of 100 equal links (fig. 9). Since 22 yds. 792 inches, a link=7.92 inches.

Fig. 10.

Builders use a tape (fig. 10) of the same length as the surveyor's chain, but subdivided into feet and inches. So that the length of lines is estimated in England by comparing them with the stan

the same.

To measure lines on paper an ivory scale, divided into inches or parts of an inch, is used. A line shorter than the scale may be conveniently measured by placing the points of a pair of compasses (fig. 11) at the two extremities, and carrying the compasses to the scale. If one point be placed at the beginning of the scale, the numbers opposite the other will give the length of the line. A line longer than the scale may be measured by marking off with the compasses the length of the scale from the line as many times as it is contained therein, and then taking the measure of the remainder as recommended above.

Fig. 11.

To measure a straight line on land.

13. If the line be not very long, we may stretch a cord between the extremities, and measure the cord.

If the line be long, it will be necessary to find intermediate points. This may be done thus: fix vertical rods at the extremities A and B (fig. 12), then send an assistant to some point between A and B with a

second rod. Let him move this rod until, on looking from A towards B, the rod at A may hide from view both the others; the rod is on the straight line. Any number of intermediate points may be found in this way.

The chain which the surveyor uses to measure the line is accompanied by ten iron skewers or arrows, having a point at one end and a large ring at the other, marked with a piece of red cloth to make it visible from a distance.

The chain is carried by two persons, called respectively leader and follower. The follower holds one end of the chain at the commencement of the line, and the leader, carrying with him all the arrows, fixes his eyes on the line of rods, and walks straight along it, dragging the other end of the chain with him.

When the chain is tightened, the follower sees that it is straight and not entangled, and calls out to the leader to "Mark." The latter then sticks an arrow into the ground in an upright position and exactly at the end of the chain. The length of the chain is

thus marked from outside to outside of the handles (from G to H (fig. 13). Both men now rise and advance until the follower reaches the arrow, when they mark off a second length, and so on. The follower picks up the arrows as he advances, so that he knows by the number of arrows in his hand how many chains have been measured. On fixing the tenth arrow, the leader cries out "Change." The surveyor marks the fact in his book, and the leader stands until the follower reaches him; the latter holds the end of the chain

against the last arrow, which is then withdrawn, and the whole ten taken by the former. In this way they proceed to the end of the line.

CHAPTER II.

ANGLES.

14. If two threads be attached to the same point, and stretched in the same direction, and then while one is held fixed, the other is turned about the point, the figure formed by the two threads is termed an angle (fig. 14). The extent of the angle depends on the amount

point, they include an angle. The point through which they pass is termed the apex or vertex of the angle, and the straight lines are its sides.

An angle is designated by a letter placed at its apex, as, for example, the angle A (fig. 15). If several lines are drawn through the point A, c several angles are formed; it is necessary then to say the angle BA C, to indicate the angle of which A B and A C are the sides.

Fig. 15.