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. 10). Since 22 yds. 792 inches, a link=7.92 inches.

Fig. 11.

Builders use a tape (fig. 11) 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. 12) 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 Fig. 12. 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.

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, and 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. (See Frontispiece.)

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.

Properties of Lines.

The properties of lines explained in the chapter may be collected as follows:

1. The intersection of two lines is a point.

2. The extremities of a line are points.

3. A point has position, but not magnitude.

4. A straight line marks the shortest path between any of its points.

5. Only one straight line can be drawn to join two given points in other words, two points determine a straight line.

6. If two points in one straight line coincide with two points in another, the first straight line coincides wholly with the other: in other words, two straight lines cannot inclose a space.

7. The intersection of two planes is a straight line.

CHAPTER II.

ANGLES,

14. An angle is the inclination of two straight lines to one another at the point where they meet.

It is frequently convenient to suppose an angle formed by the revolution of a straight line about a fixed point. For example, let two threads be attached to the same point, and stretched in the same direction, and then while one is held fixed, let the other be 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

Fig. 14.

point, they include an angle, the magnitude of which depends on the amount of revolution of a line in passing from the position of one to that of the other. 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 B
drawn through the point A,
several angles are formed: it
is necessary then to say "the
angle BA C," to indicate the
angle of which AB and AC
are the sides.

A

Fig. 15.

15. Two angles, as BA C, CA D, which have the same apex A, and a side, A C, belonging to both, are said to be adjacent (fig. 15).

16. Two angles, BA C, E D F, are equal when one, ED F, can be placed on the other, so that the point D is on A, and the sides DE, DF coincide respectively with A B and A C.

17. Take a sheet of paper, the edge AA' of which is straight (fig. 17), and fold it so that a portion, A O, of the line A A' falls on the other portion, A' O. Let OD be the crease made in the paper. The angle DOA is evidently equal to the angle DO A', for they coincide.

When the adjacent angles formed by two straight lines are equal, they are termed right-angles.

18. Each side of a right-angle is said to be perpendicular to the other.

19. There is evidently only one position of the line CO such that the angles which it makes with A A' at the point O are right-angles.

When the straight line O C, falling on A A', makes the adjacent angles unequal, OC is said to be oblique to A A'.

When one straight line meets another, the sum of the two adjacent angles is equal to two right-angles.

20. Let the straight line OC meet A A' in the point O (fig. 17): then the adjacent angles A'O C, AOC are together equal to two right-angles.

(1.) If the angles A'OC, AOC are equal, each is a rightangle, and the proposition is evident.