Fig. 19. Hexagon.

The perspective drawing of this figure will be corrected by giving the proper vanishing to the different sets of parallel lines, and by making the divisions on the diagonal A D perspectively equal.

Draw the long and short diagonals upon a large hexagonal tablet. Place this tablet in a horizontal or vertical position, Fig. 19, and then trace upon the slate its appearance and the lines upon it. The tracing illustrates the following rule:

Rule 22. In a correct drawing of the regular hexagon, any long diagonal when intersected by a long diagonal and two short diagonals, will he divided into four equal parts.

Exercise ii. The Center of the Ellipse Does Not Represent the Center of the Circle. Cut from paper a square of three inches, after having inscribed a circle in the square. Draw the diameters of the square and then place the square horizontally at the middle of the back of the table, with its edges parallel to those of the table. Trace the square, its diameters, and the inscribed circle, upon the slate. The circle appears an ellipse, and as the long axis of an ellipse bisects the short, it is evident that it must come below the center of the square, and we discover that the center of the ellipse does not represent the center of the circle, and that the diameter of the circle appears shorter than a chord of the circle.

Fig. 20. Center of Circle not Center of Ellipse.

Fig. 21. Concentric Circles.

Exercise 12. Concentric Circles. Cut a 4-inch square from practice paper, and draw the diagonals. With the center of the square as center draw two concentric circles, 4 inches and 2 inches in diameter.

Place the card horizontally upon the table, as illustrated, and tract its appearance upon the slate, together with all the lines drawn upon it.

Draw the vertical line which is the short axis of both ellipses. Bisect the short axis of the outer ellipse, and draw the long axis of this ellipse. Bisect the short axis of the inner ellipse, and draw its long axis. It will be seen that the long axes are parallel but do not coincide, and that both are in front of the point which represents the center of the circles.

Each diameter of the larger circle is divided into four equal parts. The four equal spaces on the diameter which forms the short axis appear unequal, according to Rule 9. The diameter which is parallel to the long axes of the ellipses has four equal spaces upon it, and they appear equal. This diameter is behind the long axes, but generally a very short distance; and in practice, if the distance 1 2 between the ellipses measured on the long axis is one-fourth of the entire long axis, then the distance between the ellipses measured on the short axis must be a perspective fourth of the entire short axis. This illustrates the following rule:

Rule 23. Foreshortened concentric circles appear ellipses whose short axes coincide. The distance between the ellipses on the short axis is perspectively the same proportion of the entire short axis, as the distance between the ellipses measured on the long axis, is aeometrically the same proportion of the entire long axis.

Exercise 13. Frames. In the frames are found regular concentric polygons with parallel sides, the angles of the inner polygons being in straight lines connecting the angles of the outer polygon with its center. In polygons having an even number of sides, the lines containing the angles of the polygons form diagonals of the figure, as in the square.

In polygons having an odd number of sides, the lines containing the angles of the polygon are perpendicular to the sides opposite the angles, as in the triangle.

Draw upon large triangular and square tablets the lines shown in Fig. 22. Place the tablets horizontally on the table, or support them vertically, and trace upon the slate the appearance of the edges and all the lines drawn upon them. The tracings illustrate the following rule:

Fig. 22. Frames.

Rule 24. In representing the regular frames, the angles of the inner figure must be in straight lines passing from the angles of the outer figure to the center. These lines are altitudes or diagonals of the polygons.

20. After making the tracings described in the foregoing exercises, draw (not trace) freehand on the slate the various tablets, arranged to illustrate each one of the exercises. This is really drawing from objects, and where the rods are used to connect the tablets the figures are equivalent to geometric solids. After the proportions of the surfaces are correctly indicated, lines connecting the corresponding corners of the tablets should be drawn to complete the representation of solid figures. The lines indicating the rods and those lines which in a solid form would naturally be invisible, may be erased. By the use of the three rods of different lengths, three figures of similar character but different proportions may be obtained. These should each be drawn, but each in a different position.

The following directions, which are based on general principles, apply to all drawing whether from objects or from the flat, for work in pencil or in any other medium; drawing from another drawing, a photograph or a print, whether at the same size or larger, is called working from the flat.