ENGINEER AND MACHINIST’S DRAWING-BOOK.
Supposing it were required to draw these ellipses, not
by means of their axes, but by points, any number of
these may be obtained by making horizontal sections of
the sphere. Thus, for example, if we draw the chord G H,
parallel to A' B', to represent one of these sections, and
from the point a, where it cuts the diameter E2 F2, if we
draw a perpendicular to A' B', the points a a, where it
intersects the circumference of a circle representing the
section G H in plan, will be two points in the line of
shade required. These points may be transferred to the
elevation, by supposing a section g h to be made in fig. 1,
corresponding to G H in fig. 2, and projecting the points
a' a' by perpendiculars to g h, the line representing the
The outline of the shadow cast by the sphere upon the
horizontal plane is also obviously an ellipse; it may be
constructed either by means of its two axes, or by the
help of points, in the manner indicated in the figure.
Figs. 3, 4, and 5.—To draw the line of shade on the
surface of a ring of circular section, in elevation,
plan, and vertical section.
We shall first point out the mode of obtaining those
primary points in the curve which are most easily found,
and then proceed to the general case of determining any
If tangents be drawn to the circles represented in figs.
3 and 5, parallel to the ray of light, their points of con-
tact, a, b, c, d, will be the starting points of the required
lines of shade.
Again, the intersections of the horizontal lines ae,dg,
e f drawn through these points, with the axis of the ring,
will give so many new points in the curve. These points
are denoted in the plan (fig. 4), by setting off the distances
a e and c f upon the vertical line g' D, on both sides of
the centre C'.
Farther, the diameter F' G', drawn at an angle of 45°,
determines, by its intersections with the exterior and
interior circumferences of the ring, four other points
F', t', x, and G', in the curve in question; these points
are all to be projected vertically upon the line A B.
And, lastly, to obtain the lowest points l l, draw tan-
gents to the circles represented in figs. 3 and 5, parallel
to the line o m2 (fig. 6, representing the direction of the
ray of light referred to the vertical plane in which it lies),
and transfer the distances between the points of contact s, s,
and the axis of the ring, to the radius E' O' (fig. 4), where
they are denoted by l' X; these latter points are then to
be projected vertically to l, l, upon the horizontal lines
drawn through the same points s, s (figs. 3 and 5).
The curves sought might now, in most instances, be
traced by the points thus obtained; but should the ring
be on a large scale, and great accuracy be required, it may
be proper to determine a greater number of points. For
this purpose, draw through the centre C', a straight line
T H', in any direction, and, following the method already
explained, draw through O', one of the angular points of
the cube at fig. 6, a straight line parallel to T H', and
from the opposite point m draw a perpendicular m r’
to o' r. Then, having transferred the point r to r- by
means of a circular arc, in order to admit of this last
point being projected to r, we join o r.
Applying this construction to the figures before us, we
now draw tangents to the circles represented in. figs.
3 and 5, parallel to the line o r, and, taking as radii the
distances from their respective points of contact, h and I,
to the axis of the ring, we describe corresponding circles
about the centre C', fig. 4. We thus obtain four other
points in the curves required, namely, I', i, h, and IT,
which may also be projected upon the horizontal lines
drawn through the points h or I (figs. 3 and 4).
By drawing the straight line J' K' so as to form with
F' G' the same angle which the latter makes with the
line H' T, we obtain, by the intersection of that line with
the circles last named, four other points of the curves in
Figs. 7, 8, 9, and 10.-0/ the shadows cast upon the
surfaces of grooved pulleys.
The construction of cast shadows upon surfaces of the
kind now under consideration is founded upon the principle
already announced, that %6hen a circle is parallel to a
plane, its shadoiv, cast upon that plane, is another circle
equal to the original circle.
Take, in the first place, the case of a circular-grooved
pulley (figs. 7 and 8); the cast shadow on its surface is
obviously derived from the circumference of the upper
edge A B. To determine its outline, take any horizontal
line D E upon fig. 7, and describe from the centre C' (fig. 8)
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. 7), one of the points in the
To avoid unnecessary labour 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 inter-
section 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. 7) a straight line at an angle of
35° 16' with the line A B, and through the point e, strik-
ing a horizontal line e f which by its intersection with
the line IT h, drawn at an angle of 45°, will give the
Figs. 9, 10.—In the case of the circular moulding (fig. 7),
which is of frequent occurrence in architecture and ma-
chinery, and exactly corresponds to the lower half of the
pulley (fig. 7), the cast shadow on its surface is evidently
derived from the smaller circle I K; it will, however, be
easily constructed by the method above described, with
the assistance of fig. 8. The triangular-grooved pulley
(fig. 10), should also be treated in the same manner.
The principles we have so fully laid down and illus-