PROBLEMS.

PROBLEM I.

To bisect a given line A B.

1. From the points A and B, as centres, with any distance greater than half A B, describe arcs cutting each other in c and d.

2. Draw the line c d and the point E, where it cuts A B, will be the middle of the line required,

PROBLEM II.

From a given point C, in a given right line A B, to erect a perpendicular.

FIGURE 1. When the point is near the middle of the line, 1. On each side of the point C take any two equal distances Cd and Ce.

2. From d and e, with any radius greater than C d, or C e, describe two arcs cutting each other in f.

3. Through the points f C, draw the line f C, and it will be the perpendicular required,

FIG. II. When the point is at, or near, the end of the line. 1. Take any point d above the line, and with the radius or distance d C, describe the arc e Cƒ, cutting A B in e and C. 2. Through the centre d and the point e, draw the line edf, cutting the arc e Cf, in f.

3. Through the points f C, draw the line fC, and it will be the perpendicular required.

PROBLEM

PROBLEM III.

From a given point C, out of a given right line A B, to let fall a perpendicular

1. From the point C, with any radius, describe the arc de, cutting A Bin e and d.

2. From the points e d with the same, or any other radius, describe two arcs cutting each other, in ƒ.

3. Through the points Cf, draw the line C Df, and C D will be the perpendicular required..

At a given point D, upon the right line D E, to make an angle equal to a given angle a B b.

1. From the point B, with any radius, describe the arc a b, cutting the legs B a, B b, in the points a and b.

2. Draw the line D e, and from the point D, with the same radius as before, describe the arc ef, cutting D E in e.

3. Take the distance b a, and apply it to the arc ef, from e to f.

4. Through the points D f, draw the line Df, and the angle e Df, will be equal to the angle b B a, as was required.

PROBLEM V.

To divide a given angle A B C into two equal angles.

1. From the point B, with any radius, describe the arc A C.

2. From A and C, with the same or any other radius, describe arcs cutting each other in d.

3. Draw the line B d, and it will bisect the angle ABC, as was required.