Fig. 10. The Prism.

Rule 12. When two or more faces of a cube are seen, none of them can appear their real shapes.

Place the cubical form on the desk, with the tablets vertical, and one of them seen edgewise (D) and discover that the other tablet does not appear a straight line. This illustrates the following rule:

Rule 13. Only one end of a prism can appear a straight line at any one time.

Exercise 8. The Cylinder. Connect two circular tablets by a 2 1/2-inch stick, to represent the cylinder. Hold the object so that one end only is visible, and see that it appears a circle (Fig. 11).

Place the object on the table, so that its axis is horizontal but appears a vertical line, and trace its appearance. The tracing illustrates the following rule:

Rule 14. When an end and the curved surface of a cylinder are seen at the same time, the end must appear an ellipse (Fig. 12).

Place the object horizontally, and so that one end appears a vertical line, and trace to illustrate the following rule:

Fig. 11. The Cylinder-Circle.

Fig. 12. The Cylinder-Axis Horizontal.

Fig. 13. The Cylinder-One End Straight Line.

Rule 15. When one end of a cylinder appears a straight line, the other appears an ellipse. (Fig. 13.)

Place the object upright on the table, and trace its ends and axis. Draw the long diameters of the ellipse, and discover that they are at right angles to the axis of the cylinder. This illustrates the following rule:

Rule 16. The bases of a vertical cylinder appear horizontal ellipses. The nearer base always appears the narrower ellipse. (Fig. 14.)

Place the object with its axis horizontal and at an angle, so that the surfaces of both tablets are visible. Trace the tablets and the rod, and then draw the long diameters of the ellipses, and discover that they are at right angles to the axis of the cylindrical form. The axes of the ellipses are inclined, and the drawing illustrates the following rules:

Rule 17. The bases of cc cylinder appear ellipses, whose long diameters are at right angles to the axis of the cylinder, the nearer base appearing the narrower ellipse.

Note.-The farther end may appear narrower than the nearer, but must always appear proportionally a wider ellipse than the nearer end.

Fig. 14. The Cyl inder-Upright.

Fig. 15. The Cylinder -Axis Horizontal and at an Angle.

Rule 18. Vertical foreshortened circles below or above the 1evel of the eye appear ellipses whose axes are not vertical lines, Rule 19. The long axis of an ellipse representing a vertical circle below or above the level of the eye is at right angles to the axis of a cylinder of which the circle is an end.

Rule. 20. The elements of the cylinder appear to converge in the direction of the invisible end. This convergence is not represented when the cylinder is vertical.

Note 1-Less than half the curved surface of the cylinder is visible at any one time.

Note 2-The elements of the cylinder appear tangent to the bases and must always be represented by straight lines tangent to the ellipses which represent the bases. When the elements converge, the tangent points are not in the long axes of the ellipses.

See Fig. 12, in which if a straight line tangent to the ellipse be drawn, the tangent points will be found above the long axes of the ellipses.

Exercise 9. The Cone. Hold the cone so that its axis is directed toward the eye, and the cone appears a circle. Hold the cone so that its base appears a straight line, and it appears a triangle. (Fig. 16.)

Place a circular tablet, Fig. 17, having a rod attached, to represent the axis of the cone, so that the axis is first vertical and second inclined. Trace both positions of the object, and discover that the appearance of the circle is the same as in the case of the cylinder. The tracings illustrate the following rule: Rule 21. When the base of the cone appears an ellipse, the long axis of the ellipse is perpen-dicular to the axis of the cone.

Note 1.-More than half the curved surface of the cone will be seen when the vertex is nearer the eye than the base, and less than half will be seen when the base is nearer the eye than the vertex. The visible curved surface of the cone may range from all to none.

Note 2-The contour elements of the cone are represented by straight lines tangent to the ellipse which represents the base, and the points of tangency are not in the long axis of this ellipse.

Fig. 16. The Cone.

Fig. 17. The Cone-Tablet with Rod.

Exercise 10. The Regular Hexagon. In Fig. 18 the opposite sides are parallel and equal. The long diagonal A D is parallel to the sides B C and E F, and it is divided into four equal parts by the short diagonals B F and C E, and by the long diagonals B E or C F.

Fig. 18. Hexagon.