The problems given in Plates IV to VIII inclusive have been chosen because of their particular bearing on the work of the mechanical draftsman. They should be solved with great care, as the principles involved will be used in later work.

## Plate IV

Penciling. The horizontal and vertical center lines and the border lines should be laid out in the same manner as in Plate I. Now measure off 2 1/4 inches on both sides of the vertical center line and through these points draw vertical lines as shown by the dot and dash lines, Plate IV. In locating the figures, place them a little above the center so that there will be room for the number of the problem.

Draw in lightly the lines of each figure with pencil and after the entire plate is completed, ink them. In penciling, all intersections must be formed with great care as the accuracy of the results depends upon it. Keep the pencil points in good order at all times and draw lines exactly through intersections.

Problem 1. To bisect a given straight line.

Draw the horizontal straight line A C about 3 inches long. With the extremity A as a center and any convenient radius - about 2 inches - describe arcs above and below the line A C. With the other extremity C as a center and with the same radius draw similar arcs intersecting the first arcs at D and E. The radius of these arcs must be greater than one-half the length of the line in order that they may intersect. Now draw the straight line DE passing through the intersections D and E. This line will cut A C at its middle point F. Therefore

AF = FC

Proof. Since the points D and E are equally distant from A and C a straight line drawn through them is perpendicular to A C at its middle point F.

Problem 2. To construct an angle equal to a given angle.

Draw the line OC about 2 inches long and the line OA of the same length. The angle formed by these lines may be any convenient size - about 45 degrees is suitable. This angle A 0 C is the given angle.

Now draw F G, a horizontal line about 2 1/4 inches long, and let F, the left-hand extremity, be the vertex of the angle to be constructed.

With 0 as a center and any convenient radius - about 1 1/2 inches - describe the arc L M cutting both O A and O C. With F as a center and the same radius draw the indefinite arc 0 Q. Now set the compass so that the distance between the pencil and the needle point is equal to the chord L M. With Q as a center and a radius equal to L M draw an arc cutting the arc O Q at P. Through F and P draw the straight line F E. The angle E F G is the required angle since it is equal to A 0 C. •

Proof. Since the chords of the arcs L M and P Q are equal, the arcs are equal. The angles are equal because with equal radii equal arcs are intercepted by equal angles.

Problems 3 and 4. To draw through a given point a line parallel to a given line.

First Method. Draw the straight line A C about 3-1/2 inches long and assume the point P about 1 1/2 inches above A C. Through the point P draw an oblique line F E forming any convenient angle - about 60 degrees - with A C. Now construct an angle equal to P F C having its vertex at P and the line E P as one side. (See Problem 2.) The straight line P 0 forming the other side of the angle E P 0 will be parallel to A C.

Proof. If two straight lines are cut by a third making the corresponding angles equal, the lines are parallel.

Second Method. Draw the straight line A C about 3 1/4 inches long and assume the point P about 1 1/2 inches above A C. With P as a center and any convenient radius - about 2 1/2 inches - draw the indefinite arc E D cutting the line A C. Now with the same radius and with D as a center, draw an arc P Q. Set the compass so that the distance between the needle point and the pencil is equal to the chord P Q. With D as a center and a radius equal to P Q, describe an arc cutting the arc E D at H. A line drawn through P and H will be parallel to A C.

Proof. Draw the line Q H. Since the arcs P Q and H D are equal and have the same radii, the angles P H Q and H Q D are equal. Two lines are parallel if the alternate interior angles are equal.