## Concentric Circles

Even in so brief a treatise on perspective it seems necessary for us to make some reference to concentric circles, as they must be frequently drawn and as they often cause trouble. Students sometimes are under the mistaken impression that circles in perspective do not appear as true ellipses. They argue that as the nearer half of the ellipse is not so far from the spectator as the other half it appears larger and hence must lie drawn so. Whereas this may sound logical on the face of it, it is not true. For if you test actual objects you will find the circles always appear in perspective as true ellipses. We can make this more clear lureferring to Sketch 11. We have already mentioned that one cannot see half way around a cylinder. At "A." we have drawn the top view of a cylinder. The spectator is standing at "s." Lines of tangenev from "s" to the cylinder give us at "1" and "2" points representing the extreme limits of the cylin-drical surface visible from "s." If we draw a straight line across from "1" to "2" it marks the greatest width of the cylinder as it appears from "s." This line really does not pass through the true center of the circle, represented at "o," but is between this center and the spectator, and becomes the major axis of the ellipse representing the circle. The shaded portion hack of this line on the sketch will appear from "s" exactly the same size at that portion left white; hence the ellipse must appear truly symmetrical about this line. At "B" the spectator stands closer, and sees less of the cylindrical surface. Now suppose we have two concentric circles representing the tops of two concentric cylinders as indicated at "C." the spectator still standing at "s." If we treat these independently as before, drawing tangents to the curves, these tangents will measure off visible surfaces from 1 to 2 on the larger and from 3 to 4 on the smaller. This shows that the eye will see relatively more of the cylindrical surface of the smaller cylinder. Line 3-4 is nearer the center "o" than line 1-2 but does not pass through it. Now the easiest way for the student to draw such circles in perspective is to assume that they are inscribed in squares. At "D" two squares having a common center are shown in perspective. The crossing of the diagonals gives us the true center of the circle at "o," correctly located in perspective. At 1. 2. 3 and 4 are points through which the larger ellipse must pass. Line "x." just half way from points 1 and 3 will be the long axis of the large ellipse, which will be drawn symmetrically about this line, passing through points 1, 2, 3 and 4. The smaller ellipse will be drawn in exactly the same way. passing through points 5, 6, 7 and 8, and drawn symmetrically about axis "v." which is half way from 5 to 7.

Figure 9. Illustrating Sonic Further Perspective Considerations.

Study these circles at "D" and examine objects in which other concentric circles are found. Is it not true that foreshortened concentric circles appear as ellipses? And would not the short axis lines of these ellipses coincide? It will he noticed, too, at "D," that distances 3-7, 7-0, 0-5 and 5-1 on the short axis seem to decrease gradually though actually the same as the unforeshortened distances on the long axis, 2-6, 6-0, 0-8 and 8-4. So in drawing such ellipses remember to have the space between them widest at the ends as at 2-6 and 8-4, and a little wider between the near curves as at 3-7 than at the farther side as at 5-1.

When one feels able to do all the more common of the geometric forms individually in every possible position let him draw combinations of several. This work should be followed by a practical application of the same principles to the drawing of objects of all sorts and sizes based on the same forms, as discussed in the chapter on object drawing. And as one draws he should analyze and memorize.

And one should attempt to make free-hand perspective sketches from memory or the imagination or from actual working drawings prepared instru-mentallv such as a front and side and top view.

In the chapter on object drawing some of the advantages of studying certain things by drawing them on glass have been pointed out and we have also described the glass invented by Mr. Cross specially for this purpose. Either the common or the patented glass might be of great help to the student in his perspective studies, particularly if this subject proves difficult. Training in instrumental perspective is often of help. too. though instrumental perspective sometimes shows apparent distortions which mislead one. A certain amount of help is gained from it. however, and students who are familiar with the instrumental work usually advance more rapidly in free-hand work because of the training. Likewise the student who understands freehand perspective will find a great deal in the subject to help him to do instrumental problems more artistically than he otherwise could.

We have several times mentioned that once skill is gained in drawing cubes and other simple forms such as we have just described, it is not difficult to apply the knowledge acquired to the representation of more complex subjects.

The architectural student desires to sketch buildings and so let us consider the application of the principles stated above to work of this nature.

Let us assume that we are to draw a house, for example, which is twenty feet wide and forty feet long, and twenty feet from the ground to the eaves, the house being so turned that we look more directly at the long face than at the end. The land is assumed to be level. At Sketch 1, Figure 10, such a house has been drawn. As the eye is usually from four to five feet above the ground, the horizon line has been drawn one-quarter of the way up on the building. The nearest cube was worked over first until its proportion and perspective convergence seemed satisfactory. Then lines "D" and "E" were produced indefinitely (See "A," Sketch 1) and a diagonal line AC was carried through point "B,"' exactly half way from the ground to the eaves, thus automatically marking off at "C," the end of a second cube. When the two cubes were completed the roof was added. By crossing the diagonals of the square ends of the house proper, centers "o" and "p" were located and through these, vertical lines "s" and "t" were erected, and on these points were taken to mark the height of the ridge "F," which was converged towards its correct vanishing point at the right. Sketch "B" is the same with the exception of the roof which is here hipped instead of gabled. The ends of the ridge were located by erecting "A" and "B" perpendicularly through the points of intersection of the diagonals of the tops of the two cubes forming the main house. And Sketch "C" shows a different roof of the gambrel type, the gable having been drawn first just as at "A" as a guide.