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 third rod. Let him move this rod until, on looking from A towards B, the rod at A hides from view both the others; the third rod is then 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 them 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

FIG. 14

outside to outside of the handles (from g to h, Fig. 14). 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stands until the follower reaches him; the latter holds the end of the chain against the last arrow, which is then withdrawn.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.

DEFINITIONS.

1. Geometry is the science which treats of the properties of space.

2. A portion of space limited in all directions is termed a solid.

3. The boundaries of a solid are termed surfaces.
4. The boundaries of a surface are termed lines.
5. The extremities of a line are points.

6. The intersection of two lines is a point.

7. A solid has length, breadth, and thickness.

8. A surface has length and breadth.

9. A line has length.

10. A point has only position.

II. A straight line is a line which has the same direction throughout its entire length.

12. Two straight lines having two points in common lie wholly in the same straight line.

This property of a straight line may also be expressed in the following different ways:

a. Two straight lines cannot enclose a space.

b. Two points determine a straight line.

c. Only one straight line can be drawn to join two given points.

13. A plane is a surface, such that the straight line joining any two points in it lies wholly in the surface.

EXERCISE.

Take a sheet of paper, and with pencil, ruler, and compasses, work the following exercises :

1. Mark the position of a point, and letter it A.

2. Mark the position of a second point B.

3. Draw a straight line joining the points A B.

4. Produce the line A B to the point C, so that B C shall be equal to A B.

5. Produce the line B A to the point D, so that D A shall be equal to A B.

6. Draw two straight lines on paper, and then from the larger mark off a part equal to the smaller.

Easy Exercises.

(a). Draw a straight line, and, from the scale in Fig. II, transfer to the line a length, A B, of two inches and a half.

(b). Produce the line A B to a point C, and make B C equal to one inch and three-eighths.

(c). Produce B A to D, and make D A equal to one inch and seveneighths.

(d). Measure the line DC, and express its length in inches and eighths.

(e). Again draw a line, M N, and make it two inches and five-eighths long; then produce M N to a point, P, such that N P shall be one inch and seven-twelfths; and finally measure the line M P.

(f). What measurements would you take to express the size of a book?

(g). A workman wishes to see whether the line formed by the edge of a piece of brickwork is straight or not: how can he do it? What properties of the straight line will he employ?

(h). Draw two straight lines from the same point, but not so as to lie in the same straight line. Join the extremities of these straight lines.

(i). Measure the lengths of the three straight lines, and express them as nearly as possible in inches and parts of an inch.

CHAPTER II.

ANGLES.

IN the previous Chapter, the properties of a straight line are investigated; we have now to consider different ways in which two straight lines can be drawn from the same point.

Magnitude of Angles.—Mark on a sheet of paper the position of a point, A, and from A draw a straight line in any direction, A B. Take a fine thread, AC, and attach one end of it by a drawing-pin to the point A. Stretch the thread first along the line A B, and then gradu. ally move it round the point A. When the thread lies along AB, the inclination of the two lines A B and

Α

B

FIG. 15.

AC is nothing, or, in mathematical language, zero. As the thread revolves, the inclination of AC to AB, or the amount of turning of A C, gradually increases. The inclination of any two straight lines (which meet in a point,) as AC and A B, is termed an angle. The two straight lines forming the angle are termed its sides, and the point in which the sides meet is termed the vertex. To measure the magnitude of an angle, we have simply to consider what part of a whole revolution is required to carry one of the sides to the other. This has nothing whatever to do with the length of the sides. In Fig. 16, for instance, although the sides of B are longer than those of A, the angles A and B are equal, because the same part of a whole revolution will be required, in either case, to make one side of the angle, by moving round its extremity, fall upon the other.

FIG. 10.

B

D

Designation. When there is only one angle at a point it is desig. nated by a letter placed at the vertex, as, for example, the angle A (Fig. 18). If several lines are drawn through the point B A, several angles are formed: it is necessary then to place letters at the ends of the lines also. Thus we say "the angle BA C," to indicate the angle of which A B and A C are the sides (Fig. 17).

A

FIG. 17.

Equality of Angles.-This may be proved by placing one upon the other. Let B A C and E D F be two angles; place the angle D on the angle A, so that the point D is on the point A, and the

side D E on the side A B; then, if D F lies on A C, it will be evident that the angle D is cqual to the angle A.

Again, if it be known that the angle A is equal to the angle D, it will be evident that when the point D is placed on A, and the line DE on the line A B, the line D F will fall on the line A C.

FIG. 19.

adjacent angles.

FIG. 20.

D

B

B

Adjacent Angles.-Draw a straight line A B, and from a point C in A B draw a straight line CD. Two angles are formed at the point C, on the same side of A B, having a side CD belonging to both. Two such angles are termed

The Right Angle.-Suppose CD to be a thread attached by a drawing-pin to the point C in the straight line A B. At first let the thread CD lie upon C B. Then, after it has made a whole revolution, it will again lie upon CB. When half a revolution is made, the revolving line C D will lie upon C A, the continuation of C B; for the same amount of revolution would evidently carry CD from the line CA to its continuation C B. When the straight line C D has arrived at that position in which it makes with A B an angle on one side equal to the angle on the other, C D is said to be at right angles to A B (Fig. 20). CD, when in this position, is also called a perpendicular to A B. The two equal adjacent angles then formed are termed right angles. From this explanation it will be seen, therefore, that half a revolution is the measure of two right angles, and consequently a fourth of a revolution is the measure of one right angle. Before the moving line reaches the position C D, the angle it makes with C B is less than a right angle, and when the moving line has passed this position it makes with C R an angle greater than a right angle. There is, therefore, only one position of the revolving line in which it makes right angles or is per pendicular to A B.