Now in order to convince yourself that these same facts are true of other objects, take a box or any form similar to the cube, and study it in various horizontal positions above and below the eye, keeping the nearest vertical plane so turned that it is always seen in its true shape. When the object is below the eye do not the horizontal receding lines seem to slant upward with an appearance of convergence? And when the object is above the eye do not these horizontal receding lines seem to slope downward in the same way? And whether above or below the eye is it not true that all the horizontal surfaces appear to slope towards the eye level as they recede? It is interesting to note as mentioned above that such parallel edges as recede would, if continued far enough, appear to converge towards the same point on the eye level, exactly opposite the eye itself, this being termed the vanishing point for that set of edges. Such edges as do not recede have, of course, no appearance of convergence and hence no vanishing point.

All the time that you are studying the object ask yourself such questions as the following, for it is by personal observation and analysis that one can best gain a knowledge of perspective appearances. Is it true that every set of parallel receding horizontal lines has a common vanishing point of its own? And that of two parallel lines of same length which do not recede the one nearest the spectator appears the longer? And that any parallel edges which are at right angles to the line of sight actually appear parallel?

When an object is placed like the cube or box which we have mentioned, so its principal face is at right angles to the line of sight from the eye. we say that it is viewed in parallel perspective. Sketch 5 shows cubes in parallel perspective in various relations to the eye level.

## The Cube In Angular Perspective

We now purpose to turn the cube into a new position, placing it in a horizontal manner below the eye and turned at an angle with all four of the edges of the top receding. None of the edges now appears horizontal. Now sketch the top of the cube in this position. It will be noticed that if the cube is so turned as to make equal angles with the line of sight as at "A," Sketch 6, Figure 9. we will see equal portions of the lines marked "a" and "b" and they will have equal slant. The same will be true of "c" and "d." Now if we turn the cube so that it makes unequal angles with the line of sight, as at "B," Sketch 6, we find that line "a" will seem shorter and line "b" longer than before.

Now to more firmly fix these thoughts in your mind shift the cube from place to place and question yourself in this way. If two edges of the square top of the cube recede from you at unequal angles, which of the two appears the longer? Which the more nearly horizontal? And considering the complete cube, turned at an angle so that two or more of its faces are visible, can any one of these appear in its true shape? Will all parallel edges receding towards the left appear to converge or vanish to one point and those towards the right to another? And if so will these points be on the eye level?

Continue your analysis in this thorough way and you will observe many interesting things. You will see that such edges of the cube as are truly vertical appear so and hence should be drawn so. You will notice that the nearest vertical edge will be the longest and that the others will decrease in length as they get farther away.

When a cube or other object is so placed that no surface is seen in its true shape, or that its principal planes are at other than a right angle with the line of sight, it is said to be in angular perspective. As it is rather difficult for the beginner to draw in angular perspective well, he should work for some time from a cube itself, placing it in different positions above and below the eye. In drawing such an object it is usually advisable to actually locate and draw a line representing the level of the eye on the paper, making sure that the various receding lines are converging to the proper vanishing points on this eye level. It is sometimes wise in these early problems to actually continue such receding lines indefinitely, allowing them to meet at the proper points, as at "C" and "D," Sketch 6. As an aid in testing for correct drawing of a cube in angular perspective it is occasionally helpful to draw diagonal lines on the top foreshortened square as we have done with the dotted lines at "A" and "B." Sketch 6. At "A" with the cube turned at equal angles, the long diagonal is horizontal, the short perpendicular. Let the cube be swung around as at "B," however, and the diagonals immediately tip. Point "g" drops lower than "e," and "h" moves to the right of "f" instead of remaining above it. If the vertical faces are turned at unequal angles, then. we not only see more of one than of the other but the diagonals of the top plane will always be tipped; never vertical or horizontal. Rules of this sort are of comparatively little help, however; the thing that counts in all these objects is the observation and practice from the things themselves.