Figure 8. Illustrating Perspective Principles.

Attention should be called to the fact that we seldom see half way around a sphere. Sketch 2 perhaps explains this more clearly. If "X" represents the top view of the sphere and "Y" the position of the spectator, the lines drawn from "Y" tangent to the sphere, mark at "A" and "B" the limits of the visible portion of the sphere at the plane of its greatest circumference. The larger the sphere or the closer the spectator the smaller this distance becomes.

## The Cylinder. Vertical

Now take a right cylinder and hold it vertically, and with one eye closed raise it until the top is level with the other eye. In this position the top circle will appear as a straight line, the circular plane being so greatly foreshortened that only its edge can be seen. Now lower it a bit. The circular top is now visible but still so much foreshortened that it is elliptical instead of circular in appearance. Lower it still farther and the rounder the ellipse becomes. Now just as this top ellipse appears rounder as it is dropped below the eye, it is evident that if the bottom of the cylinder could be fully seen it would appear still rounder than the top, as it is even farther below the eye. Experience will prove that the degree of roundness of the ellipse will be in proportion to its distance below the eye. Next raise the cylinder vertically until the lower end is at the eye level; this now appears as a straight line just as did the top end before. Raise it still higher and the bottom comes in sight as an ellipse, the top of the cylinder being now hidden. And the higher the cylinder is raised, the rounder the ellipse of the bottom becomes, its fullness being in proportion to its distance above the eye level. If the cylinder is lowered until the bottom and top are both equi-distant from the eve level both will be invisible but the visible edges of each will have like curvature, and if the cylinder were transparent so both the top and bottom could be seen, the ellipses representing both would be identical in size and shape, as both circles are the same distance from the level of the eye.

## Transparent Cylinders

Transparent Cylinders of glass are convenient for such experiments or the student can make one of celluloid or some similar material.

What is true of the perspective appearance of the top or bottom of a cylinder is true of any circle, and if the student wishes to prove this, let him cut a circle from a sheet of heavy paper or cardboard and experiment with this. When held horizontally and level with the eye does it not look like a straight line? And when raised above or dropped below the eye level does it not appear as an ellipse? Note the apparent change in roundness of this ellipse and in the length of its short axis as the circle is raised or lowered. Only the long axis will appear of the same proportionate length regardless of the position of the circle. Is it not true, also, that when a circle appears as an ellipse the ellipse is always perfectly symmetrical about its long and short axis lines, and is it not divided by these axis lines into four quarters which appear exactly equal?

Go back to the cylinder again and see if this, too, does not, when held vertically, appear symmetrical about a vertical central axis line at all times, every element of the cylindrical surface being vertical also? As in the case of the sphere we seldom see half way around the circumference; hence less than one-half of the cylindrical surface is visible at any one time.

Now try a number of sketches of the vertical cylinder and the horizontal circle as viewed from different positions (Sketch 3 shows a few). Practice drawing ellipses, too, until you can do them well; this is no easy matter.

The tipped or horizontal cylinder will be discussed later.

## The Vertical Cone

While we still have the horizontal circle in mind let us consider the right circular cone placed vertically. Sketch 4 shows the cone in this position. It will be seen that the appearance of the circle is the same as in the case of the cylinder. Also that if the apex of the cone is at the top and the cone below the eye, we can see more than half way around the conical surface. If raised above the eye we see less than half way around. And if the cone is inverted the opposite is true. Note also that a right circular cone will always appear symmetrical, the long axis of the ellipse of the base being at right angles to the axis of the cone. Make several drawings of the vertical cone; - the horizontal or tipped cone will be discussed later.

## The Cube In Parallel Perspective

We now turn to the cube. Hold it with the top at the eye level and the nearer face at right angles to the line of sight so it is seen in its true shape. Only one face of the cube is visible now, and that appears as a square. Lower the cube a few inches and the top appears, greatly foreshortened. The farther horizontal edge, being a greater distance away than the nearer one, seems the shorter of the two. The parallel receding edges of the top seem to slant. If these slanting edges were continued indefinitely they would appear to meet at a point and that point would be on the eye level. Lower the cube a few inches farther. The top now appears wider and the two receding edges have still greater slant. If continued they would still meet at a point on the eye level, the same one as before. The front face still appears square. Now raise the cube above the eye, still holding it vertical. The top goes out of sight and the bottom becomes visible. The front face looks square as before. Now the higher the cube is raised the more the bottom shows. The receding1 lines now seem to slant downward towards the eye level; if continued they would meet the very same point on the eye level as when the cube was below the eye.