trated in tlie 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, however varied or complicated; and the stu-
dent 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, spur-wheels, and
such simple and elementary pieces of machinery. Before
quitting this subject, however, we have still to bring under
notice a few particular forms which require, for the deter-
mination of the shadows thrown on their surfaces, certain
peculiarities of construction hitherto unexplained.
Plate LY.—To trace the outlines of the shadows cast
upon the surfaces of screws and nuts, both triangular
Figs. 1 and 2 represent the projections of a screw with
a single square thread, and placed in a horizontal position.
In this example, the shadow to be determined, is simply
that cast by the outer edge, A B D, 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.
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 C D of the thread, in place of throwing
its shadow upon a cylinder, projects it upon a helical sur-
face inclining to the left, of which the generatrix is known.
Describe from the centre 0 (Fig. 3) a number of circles,
representing the bases of so many cylinders, on the sur-
faces 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 cor-
responding helical lines in Fig. 4, where they are denoted
by B2, F, G, and E. Then, transferring the point B'
(Fig. 8) to its appropriate position B, on the edge A C D
(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
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
Figs. 7 and 8.—The shadow cast by the helix ABC
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.
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
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 abed, Fig. 4, PI. 56.
When two surfaces thus situated are parallel, the one j
nearer the eye should receive a lighter tint than the
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 d f e,
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 inclina-