sun, is assumed to be the ordinary luminary; and for practical purposes, on account of its great distance from our sphere, its rays are supposed to fall parallel to each other, casting thereby the shadow of intercepting objects in a parallel direction on the planes influenced by them.

As that portion of an opaque surface directly exposed to the rays of light is illuminated, that portion which is opposed to it, and consequently deprived of its influence, is said to be "in the shade." The surface or space on which that shade is projected is called the "shadow." The terms "shadow proper," and "cast shadow," are sometimes applied to distinguish these two conditions.

According to nature, and with reference to the co-ordinate planes of projection, the projections of shadows are liable to vary indefinitely; for they may be influenced in plan by the assumed altitude and direction of the luminary, and in elevation by the given angle which the ray of light may happen to form with the vertical plane.

For the sake of uniformity, however, where no particular data is required, a conventional and universal method of determining shadows has been adopted by architects, and will be described hereafter.

The inspection of Diagram, No. 84, will explain the natural method of representing shades and shadows under various conditions.

Let the point a represent the position in plan, and the line a' a" represent the elevation of a stick. Let / determine the vertical position of a luminary, and l its horizontal projection. In this instance it is evident that should the rays of light be parallel to the vertical plane, the shadow of a' a" would be cast to b', which, in plan, would be determined by a line proceeding from the direction of l through a, and determine the cast shadow a b equal in length to a" b' in the elevation.

If, from a as a centre with

radius a b, we describe a circle, its circumference will show, the limits of the length of that shadow, no matter what angle the ray of light may form with the vertical plane on which the elevation is projected. Should the luminary be placed lower, as in l' 2, or higher, as in l' 3, the length of the projected shadow would evidently vary with the altitude of the luminary, for the latter placed in l' 2 would produce the shadow a'-b 2', and, proceeding from 1'3, would determine the shadow a' b3'. Similarly the direction of the shadow in plan must vary with the position of the luminary in reference to the vertical plane. The luminary in 7 will give that shadow parallel to the vertical plane; 72 will cast its shadow a-b 2, proceeding from the left front of the picture towards that plane, and 7 3, proceeding from the left rear, will cast its shadow in a-b 3 towards the spectator. The distances a-b, a-b 2, a-b 3 forming, as we have already observed, the radii of circles determining the horizontal projection of the shadows. It is clear, therefore, that the given angle which the ray of light is to form with the horizon must be determined on the vertical plane, and that the angle which it makes with the vertical plane must conversely be determined on the horizontal plane.

These observations being understood, the following examples will not present much difficulty:

1. Determine the shadow cast by a cube upon a horizontal plane, the rays of light forming an angle of 45° with the horizon, and parallel to the vertical plane. (Fig. 85.)

After having obtained the horizontal and vertical projections of the cube in the usual manner, determine on the vertical plane the angle a" b' a', equal to the given inclination of the ray of light with the horizon: parallel to a" b' draw c' d' and e'f'. On the horizontal plane make a b equal to a'b', and parallel to it draw c d and ef of equal length. Join d bf to complete the shadow required.

N.B. The angle of 45° has the advantage of determining by mere inspection the height of an object, the plan only of which is given; the shadow at that angle being equal in length to the height of the object. The adoption of this property of the angle of 45° may be rendered especially valuable in military topography.

The shadow of an object being determined in plan, and the angle of inclination of the ray of light being known, it is likewise easy to determine its height, which equals that of the perpendicular a c of a right angle triangle a b c, of which the length of shadow a b forms the base, and the hypothenuse b c forming with a b the known angle of inclination, forms the third side.

2. The shadow gb d of the pentagonal pyramid defgh, and the angle of inclination which the ray of light makes with the horizon being given (60°), determine the height of the pyramid (Fig. 86). Join a b, and draw indefinitely the perpendicular a c: at b, make the angle a b c equal to the given angle, and produce b c till it cuts a c: a c will be the height of the pyramid.

3. Determine the shadow of a cylinder lying on its side, the rays of light forming an angle of 60° with the horizon, and of 40° with the vertical plane. (Fig. 87.)

The projections of the cylinder in this position are parallelograms similar to those that would be produced by a parallelopiped whose shadow a' b' c'g' transferred on the horizontal plane, according to the given conditions, would produce the figure c d e f h. But, as this assumed parallelopiped contains the cylinder whose diameters are in plan c c' and in elevation a' a", their projection will be ellipses, the position of whose axis a b, g g" and k, are determined by the direction of the angle of light. Inspection of the diagrams will show how to complete the figure.

front of the picture

(Fig. 88.)

4. Required the shadow of a cone projected on a wall behind it, the ray of light forming an angle of 30° with the horizon, and falling from the left towards the wall at an angle of 55°. Determine the angle a' b a" for the inclination of the ray of light to the horizon, and draw a b' equal to a'' b, forming with x y an angle of 55°. Tangent to the base of the cone, and in the direction of b', draw d e and d' ƒ for the horizontal portion of the shadow. Draw cb" at right angles with x y, and make b' b" parallel to it. Join e b" f for the vertical portion of the shadow cast on the wall.

5. Determine the shadow of a square pyramid whose base is raised at a given height above the horizontal plane, the ray of light falling at an angle of 40° with the horizon, parallel to the vertical plane.

The mere inspection of Diagram 89 should suffice for the solution of this question, as well as that of Fig. 90, which represents

6. A hexagonal prism of 1 inch side, and 1.2 inches edge, laid on one of its faces at right angles to the vertical plane.

PRACTICE.

7. Determine the shadow of a sphere cast on a horizontal plane, the ray of light forming an angle of 40° with the horizon, parallel to the vertical plane.

8. The axis of a cone 1 inch high and .75 inch diameter at its base, is at .65 inch distance from one side of a cubical box and at .6 inch from the other side: show the shadow which would fall both on the bottom and on the sides of the box, supposing parallel rays of light to fall upon the cone at angles of 30° with the horizon, and at 50° with the further side of the box. (Fig. 91.)

The inspection of this diagram will show its analogy

with Fig 88, from which it differs only by the addition of the auxiliary plane x'y', on which a portion of the shadow is projected according to the same method.

9. A line .5 inch long represents the plan of a stick 8 feet long, inclined in the direction of a wall 3 feet distance at an angle of 30°: show its shadow, the ray of light forming an angle of 45° with the horizon and with the vertical plane. (Fig. 92.)

Let a b represent the horizontal projection of the stick, having its upper extremity in a. To obtain its height, erect the perpendicular a c, and with radius equal to the whole length of the stick, from b as a centre determine the point C: transfer the distance a c to a' c' for the height of the vertical projection determined by the line c'b', which line may now be treated as the generatrix of a cone, and its shadow determined accordingly.

Inspection of the diagram will show that the question is but another partial application of question 4. (Fig. 88.) In the conventional method adopted by architects, &c., it is assumed that the rays of light fall parallel to each other, in the direction of the diagonal of a cube, two of whose faces are parallel to the vertical plane, and the traces of which rays form with both planes angles of 45°. This combination of rays forms a real angle of 35° 16' with the horizontal plane, and the shadows in the following diagrams are cast according to those conditions.

Let a b c d (Fig. 93) represent the plan, ef the elevation, and a clm a section taken in the direction of the diagonal of the cube: it is evident that the line mc will show the direction of that diagonal (which has an inclination of 35° 16' with the horizon) and that the ray of light ', parallel to it, will determine on the horizontal plane the length of the shadow cl' cast by the edge of the cube c l. The edges b and d, being of equal height with