## The Cylinder, Not Vertical

Now that the drawing of the cube has given one a little knowledge of receding lines, it is well to go back to a consideration of the cylinder, only this time we will not place it vertically. Hold it. instead, in a horizontal position at the level of the eye (closing one eye) and turn it so that the circular end appears in its true shape. In this position nothing is seen but the end. If we then swing it or tip it so that the end and some of the curved surface are both visible, the end will appear as an ellipse. The less of the curved surface shows, the rounder this ellipse will be. Then swing the cylinder until one end appears a straight line. In this position the other end is invisible but if the cylinder were transparent it would be found that this end would appear as an ellipse. Study the cylinder in all sorts of positions above and below the eye, making observations of this sort. Such study and comparison will prove that the right cylinder, regardless of position, will always appear symmetrical about its long axis line; that the long diameters of the ellipses forming the ends will be at right angles to the axis of the cylinder. One will notice, too, that it is never possible to see quite half way around the cylindrical surface. And when the farther end of the horizontal or tipped cylinder is a greater distance from the eye than the nearer end it will appear smaller, which means in turn that the elements of the cylindrical surface will appear to converge, and these elements being all parallel lines they will seem to vanish towards a point. If the cylinder is placed horizontally this point will be on the eye level; if tipped in some other position the point will be above or below the eye. To this same vanishing point the axis of the cylinder will also recede if produced. And it will he noticed, too, that regardless of the placing of the cylinder those elements of the surface which form the straight boundaries will appear tangent to the curves of the bases. At "A." Sketch 7. the cylinder has been drawn within a square prism. To do so gives one a knowledge of the relationship between objects based on the square and the circle.

## The Cone. Tipped

And if we turn to the cone once more for further consideration and look directly at its apex we will find that it appears as a true circle. And when so held that its base becomes a straight line it has the contour of a triangle. The visible curved surface of a cone may range from all to none. The bounding elements of the cone are always represented by straight lines tangent to the ellipse which represents the base. And the right cone, like the cylinder, will always appear symmetrical, being divided by its long axis into two equal parts.

Study the little sketches of cylinders and cones in Sketch 7. Figure 9. Then make many of your own.

Now in just the same way consider other geometric forms, such as the triangular prism placed vertically and horizontally, and the pyramid and the hexagonal prism in various positions. Though our space does not permit full discussion of these here, it seems essential to call attention to a few facts in regard to the appearance of the triangle, the hexagon, etc. Rut first, let us say another word or two about the square. We have drawn a square at Sketch 8 and have crossed its diagonals. Doing this locates the true center of the square "o" as it appears in perspective. It seems more than half" way back, for the farther half of the square, being a greater distance from the eye than the first half, seems smaller For the same reason, line "bo" seems longer than "od," though in top view we know they would be equal. This will perhaps make more clear the fact that equal distances on any receding line seem unequal, the farther seeming the shorter. Now suppose that at the end of this square we draw a triangle, as at "IV Sketch 8. locating its apex b\ drawing a line horizontally from center "o" to line "be." erecting a vertical altitude at the point of intersection "f." choosing point "e" arbitrarily on the altitude and then drawing "ec" and "eh." This triangle illustrates the truth that the apex of a vertical isosceles or equilateral triangle having a horizontal base appears in a vertical line erected in the perspective center of the base. As it is easier to judge the correct proportion of a square in perspective than of a triangle, a square is sometimes drawn first as a guide as in Sketch 9. At Sketch 10 we have shown a hexagon. It will be noticed at "A" that the two short diagonals "bf" and "ce" and the long diagonals "be" and "ef" divide long diagonal "ad" into four equal parts. For in a correct drawing of a hexagon it is always true that any long diagonal when intersected by two short and one long diagonals will be divided into four equal parts.

When a hexagon is sketched in parallel perspective as at "B" they all appear equal. Now in drawing polygons, especially those which are regular such as the hexagon just mentioned, it is often easiest to first draw an ellipse representing a circumscribed circle. In drawing the decagonal prism in Figure 6, for instance, an ellipse was first drawn just as for the cylinders, then the decagon was drawn within it. So try a number of polygons, and later prisms and pyramids built upon polygonal bases.