horizontal line D E upon fig. 8, and describe from the centre C' (fig. 9) a circle with a radius equal to the half of that line; then draw, through the same centre, a line parallel to the ray of light, which will intersect the plane D E in c; lastly, describe from the point c', as a centre, an arc of a circle with a radius equal to A C; the point of intersection, a', of this arc, with the circumference of the plane D E, will give when projected to a (fig. 8), one of the points in the curve required.

To avoid unnecessary labor in drawing more lines parallel to D E than are required, it is important, in the first place, to ascertain the highest point in the curve sought. This point is the shadow of that marked H on the upper edge of the pulley, and which is determined by the intersection of the ray C'H' with the circumference of that edge in the plan; and it is obtained by drawing through the point A (fig. 8) a straight line at an angle of 35° 16′ with the line A B, and through the point e, striking a horizontal line e f, which by its intersection with the line H h, drawn at an angle of 45°, will give the point sought.

In fig. 9, the pulley is supposed to be divided horizontally in the centre, and the shadow represented is derived from the smaller circle I K, and is easily constructed by methods above described.

Plate IV. To trace the outlines of the shadows cast upon the surfaces of screws and nuts, both triangular and square-threaded.

Figs. 1 and 2 represent the projections of a screw with a single square thread, and placed in a horizontal position, A' a' being the direction of the ray of light. In this example, the shadow to be determined is simply that cast by the outer edge, A B, of the thread upon the surface of the inner cylinder; therefore its outline is to be delineated in the same manner as we have already pointed out, in treating of a cylinder surmounting another of smaller diameter (page 320).

Figs. 3 and 4.-The case of a triangular-threaded screw does not admit of so easy a solution as the above, because the outer edge A CD of the thread, in place of throwing its shadow upon a cylinder, projects it upon a helical surface inclining to the left, of which the generatrix is known. Describe from the centre O (fig. 3) a number of circles, representing the bases of so many cylinders, on the surfaces of which we must suppose helical lines to be traced, of the same pitch with those which form the exterior edges of the screw (see fig. 4). We must now draw any line, such as B' E', parallel to the ray of light, and cutting all the circles described in fig. 3 in the points B', F', G', E', which are then to be successively projected to their corresponding helical lines in fig. 4, where they are denoted by B2, F, G, and E. Then, transferring the point B' (fig. 3) to its appro

priate position B on the edge A CD (fig. 4), and drawing through the latter a line B b at an angle of 45°, its intersection with the curve B2 G E will give one point in the curve of the shadow required. In the same manner, by constructing other curves, such as H2 J K, the remaining points, as h, in the curve may be found.

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Figs. 5 and 6.-The same processes are requisite in order to determine the outlines of the shadows cast into the interior surfaces of the nut corresponding to the screw last described, as will be evident from inspection of figs. 5 and 6. These shadows are derived not only from the helical edge A B D, but also from that of the generatrix A C.

Figs. 7 and 8.—The shadow cast by the helix A B C upon the concave surface of the square-threaded nut is a curve a b C, which is to be determined in the same way as that in the interior of a hollow cylinder. The same observation applies to the edges A A2 and A2 E, as well as to those of the helix F G H and the edge H I. With regard to the shadow of the two edges J K and K L, they must obviously follow the rules laid down in reference to figs. 4 and 6, seeing that it is thrown upon an inclined helical surface, of which A L is the generatrix.

The principles so fully laid down and illustrated in the preceding pages will be found to admit of a ready and simple application to the delineation of the shadows of all the ordinary forms and combinations of machinery and architecture, however varied or complicated; and the student should exercise himself, at this stage of his progress, in tracing, according to the methods above explained, the outlines of the cast shadows of pulleys, spurwheels, and such simple and elementary pieces of machinery. It must be observed, that the student should never copy the figures as here represented, but should adopt some convenient scale somewhat larger than our figures, and construct his drawings according to the description, looking to the figures as mere illustrations; in this way the principles of the construction will be more surely understood, and more firmly fixed in his mind.

MANIPULATION OF SHADING AND SHADOWS.-METHODS OF TINTING.

The intensity of a shade or shadow is regulated by the various peculiarities in the forms of bodies, and by the position which objects may occupy in reference to the light.

Surfaces in the light.-Flat surfaces wholly exposed to the light, and at all points equidistant from the eye, should receive a uniform tint.

In geometrical drawings, where the visual rays are imagined parallel to the plane of projection, every surface parallel to this plane is supposed

to have all its parts at the same distance from the eye; such is the vertical side of the prism a b c d (fig. 4, plate V).

When two surfaces thus situated are parallel, the one nearer the eye should receive a lighter tint than the other. Every surface exposed to the light, but not parallel to the plane of projection, and therefore having no two points equally distant from the eye, should receive an unequal tint. In conformity, then, with the preceding rule, the tint should gradually increase in depth as the parts of such a surface recede from the eye. This effect is represented in the same figure on the surface, a dƒe, which, by reference to the plan (fig. 1), is found to be in an inclined position.

If two surfaces are unequally exposed to the light, the one which is more directly opposed to its rays should receive the fainter tint.

Thus the face e' a' (fig. 1), presenting itself more directly to the rays of light than the face a' b', receives a tint which, although graduated in consequence of the inclination of this face to the plane of projection, becomes at that part of the surface situated nearest to the eye fainter than the tint on the surface a b.

Surfaces in shade.—When a surface entirely in the shade is parallel to the plane of projection, it should receive a uniform dark tint.

When two objects parallel to each other are in the shade, the one nearer the eye should receive the darker tint.

When a surface in the shade is inclined to the plane of projection, those parts which are nearest to the eye should receive the deepest tint.

The face bg nc (fig. 4), projected horizontally at b′ g′ (fig. 1), is situated in this manner. It will there be seen, that towards the line b c the tint is much darker than it is where it approaches the line g h.

If two surfaces exposed to the light, but unequally inclined to its rays, have a shadow cast upon them, that part of it which falls upon the surface more directly influenced by the light should be darker than where it falls upon the other surface.

Exemplifications of the foregoing rules may be seen on various figures in the plates.

In order that these rules may be practised with proper effect, we shall give some directions for using the brush or hair-pencil, and explain the usual methods employed for tinting and shading.

The methods of shading most generally adopted are either by the superposition of any number of flat tints, or by tints softened off at their edges. The former method is the more simple of the two, and should be the first attempted.

Shading by flat tints.-Let it be proposed to shade the prism (fig. 4,