FIGURE 32.

Other forms of the same are shown in Figs. 32 and 33; in the first two, the point and pen both turn in

AAN

FIGURE 30.

FIGURE 31.

FIGURE 33.

describing the arc; in Figure 32, the rod remains stationary while the pen or pencil turns around it.

COMPASSES (Fig. 34) are used to describe arcs of circles. They are made with joints in the legs so that the steel point may be removed and a line pen, pencil holder, or dotting pen may be inserted in its stead, and in connection with any of these a lengthening bar may be used. It describing arcs it is better to bend the parts at the joints until the points are per

FIGURE

pendicular to the paper; then, holding the top of the compass between the thumb and forefinger, draw the curve with a continuous sweep. The lengthening bar is used between the pen or

FIGURE 35.

pencil holder and one leg of the compass in describing arcs larger than could be reached without it.

Beam Compass (Fig. 35) consists of two point holders for attaching to a bar of some kind to describe arcs with radii greater than the hand compass with lengthening bar will reach. The holders may be set any

distance apart, and in them may be inserted points, line pen or pencil.

PROBLEMS WITH COMPASS.-To erect a perpendicular to a given straight line at a given point on that line (Fig. 36). From the given point a to a center, with any convenient radius, describe arcs intersecting the line at equal distances b and c on each side of the point. From the points of intersection of the arcs and line as centers, with a radius बे greater than the former radius, describe two arcs intersecting each other at d; join this point of intersection by a straight line with the given point, and it will be the perpendicular required.

FIG. 36.

To erect a perpendicular to a given straight line at the end of the line (Fig. 37). Spread the legs of the compass to a convenient radius. Place the pencil on the end of the line a, swing the needle point about half the radius length above the line to c. With this latter point as a center, describe an arc passing

Fic. 37.

through the end of the line and intersecting it in another point, d. Through this latter point and the center of the arc, draw the diameter of the circle and from the other end of the diameter e draw a straight line to the end a of the given line. This will be the perpendicular required.

To draw a perpendicular to a given straight line, from a given point without that line (Fig. 38). From the given point a as a center with a radius greater than

the distance to the line describe an arc cutting the line in two points, b and c.

From these points as centers, with a radius greater than half the distance between them, describe arcs intersecting each other at d on the opposite side of the line from the given point; draw a straight line from the given point to the

Fia. 38.

point of intersection d of the arcs and it will be the perpendicular required.

Tough a given point, to draw a straight line parallel to a given straight line (Fig. 39). From the given point a as a center, with a radius greater than the distance to the line, describe an arc cd intersecting the line at c. From this intersection as a center, with the same radius describe another arc intersecting

FIG. 39.

the line at and passing through the given point a. Take the length of the chord of this arc from the point a to b, in the compass, and from the intersection c of the first arc with the line as a center, describe an arc on the same side of the line as the given point, intersecting at d. Through the point thus found and the given point draw a straight line ad and it will be the parallel required.

To draw a line parallel to a given line (Fig. 40). At any two points of the line, as a and b, erect perpendiculars, and on these lay off equal distances ac and be from the line. Through these points draw a straight line, and it will be a paral

lel to the given line;

40.

or, with any two points of the line as centers, describe arcs on the same side with the same radius af, bg, and draw igoc line tangent to the arcs.

To bisect a gi en straight line (Fig. 41). From the extremities of the line as centers, with radius greater than half the length of the line, describe arcs intersecting on both sides of the line. Join points of intersection by a straight line and it will bisect the given line.

To bisect a given arc (Fig. 42). Draw a straight line joining the extremities of the arc, this will be its chord; then draw a perpendicular bisecting this chord, which will also bisect the arc.

FIG. 41

Fia. 42

To bisect a given angle (Fig. 43). With the vertex a of the given angle as a center, describe an arc with any convenient radius intersecting the sides of the angle at b and c. Join the points of intersection by a straight line, construct the perpendicular ad bisecting it, which, if prolonged, will also bisect the angle.

*

Fic. 43.

To construct an angle equal to a given angle (Fig. 44).

FIG. 44.

With the vertex a of the given angle as a center, with any convenient radius, describe an arc intersecting its sides at Draw a straight line, and, from a radius, describe an arc as before, From this point of intersection as

b and c and draw its chord. point a on it with the same intersecting the line at e. a center, with a radius equal to the chord of the first arc, describe an arc intersecting the last arc at f. Join this point f and the point d on the line first used as a center, and the angle between the lines will be equal to the given angle.

To divide a given straight line into equal parts (Fig. 45). From one extremity ca

of the given line ab, draw

an indefinite straight line

ac, making any conven

FIG. 45.

ient acute angle with it. Set the legs of the com

pass at any convenient distance apart, and from the vertex a of the angle, on the indefinite line, lay off this distance as many times as there are to be equal parts in the given line. Join the last point d so found by a straight line with 6, the other extremity of the given line, and through the points of division on the indefinite line draw parallels to bd. These parallels will divide the given line into the required number of equal parts.

To divide a given straight line into parts proportional to given straight lines (Fig. 46). From one extremity a of the given straight line ab, draw an indefinite straight line ac, making any convenient acute angle with it. Beginning at the vertex of the angle, lay off in succession on the indefinite line the lengths ad, de, ef, fg, and gi of the given

Fig. 46.

A