The farther end may appear narrower than the nearer, but must always appear proportionally a wider ellipse than the nearer end.
Rule 18. Vertical foreshortened circles beloin or above the level of the .eye appear ellipses whose axes are not vertical lines.
Rule 19. The long axis of an ellipse representing a ver tical circle below or above the level of the eye is at right angles to the axis ,V a cylinder of which, the circle is an end.
Role 20. The elements of the cylinder appear to converge in the direction of th:e invisible end. This convergence is not represented when the cylinder is vertical.
Note 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 con verge, 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. 1G.) _ Place a circular tablet, Fig. 17, baying 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 appear ance 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 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 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.
Exercise io. 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.
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 be divided into four equal parts.
Exercise xi. 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 discdver 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.
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 trace its appearance upon the slate, together with all the lines drawn upon it.