when the point is at E' in the plan, its vertical projection
will be the point of intersection B of the perpendicular
drawn through B' and the horizontal drawn through the
first point of division. Also when the point arrives at C',
in the plan, its vertical projection is the point C where
the perpendicular drawn from C' cuts the horizontal passing
through the second point of division, and so on for all the
remaining points. The curve A B C F A3, drawn through
all the points thus obtained is the helix required.
Figs. 1 and 2.—To draw the vertical elevation of the
solid contained betiveen two helical surfaces, and tivo
A helical surface is generated by the revolution of a
straight line round the axis of a cylinder; its outer end
moving in a helix, and the line itself forming with the
axis a constant and invariable angle.
Let A' C' F' and K' M' O' represent the concentric bases
of the cylinders, whose common axis S T is vertical; the
curve of the exterior helix A C F A3 is first to be drawn
according to the method pointed out above. Then having
set off from A to A2 the thickness of the required solid,
draw through A2 another helix equal and similar to the
former. Now construct, according to the method given
above, another helix K C 0 of the same pitch as the last,
but on the interior cylinder; as also another K2 C2 O2,
equal and parallel to the former. The lines A' K', B' L',
C' M', &c., represent the horizontal projections of the vari-
ous positions of the generating straight line, which, in the
present example has been supposed to be horizontal; and
these lines are projected vertically at A K, B L, &c.
It will be observed that in the position A K, the gene-
rating line is projected in its actual length, and that, at
the position C' M', its vertical projection is the point C.
The same remark applies to the generatrix of the second
helix. The parts of both curves which are visible in the
elevation may easily be determined by inspection.
Figs. 3 and 4.—To determine the vertical projection of
the solid formed by a sphere moving in a helical curve.
Let A' C' E' be the base of a cylinder upon which the
centre point, C, of a sphere whose radius is a C, describes
a helix, which is projected on the vertical plane in the
curve A C E F. After determining, as above, the various
points A, B, C, D, &c., in this curve, draw from each of
these points as centres, circles of the radius a C; the cir-
cumferences of these circles will denote the various posi-
tions of the sphere during its motion round the cylinder;
and if lines be drawn touching these circles, the curves
thereby formed will constitute the figure required. One
of these curves will disappear at 0, which is its point of
contact with the circle described from the point E, the
intersection of the helix with the perpendicular E E'. It
will again re-appear at the point I, where it becomes a
tangent to the circle described from the point J, in the
prolongation of the line A A'. The exterior and interior
circles in Fig. 4, represent the horizontal projection of the
solid in question.
Our design is, in the first place, to show by simple ex-
amples, how different objects, by their various necessities,
require various treatment for their complete illustration;
and to lead the way to the more complicated treatment
and illustration of machinery in larger masses.
DRAWING OF SCREWS.
The screw is a cylindrical piece of wood or metal, in the
surface of which one or more helical grooves are formed.
The thread of the screw is the solid portion left between
the grooves; and the pitch of the screw is the distance,
measured on a line parallel to the axis of the cjdinder,
between the centres of any two contiguous threads.
A screw is said to have a triangidar thread when the
thread is triangular in section.
The screw is usually accompanied by a Nut, which is
a detached piece, formed of suitable material, and perfo-
rated by a cylindrical hole having grooves formed on its
periphery corresponding in all respects to those on the
screw ; so that the threads of the screw shall fit exactly
into the grooves of the nut, and all the corresponding
points in the two surfaces shall coincide.
Projections of a Triangular-threaded Screw and
The ground line A M having been drawn throughout
the entire length of the paper, as also the centre lines C C' of
the figures, as before directed, the next step is to lay
down the circles which are to represent the exterior and
interior cylinders. For this purpose, observing that the
diameter of the exterior cylinder is 4-}-g- inches, the half of
this length is, by the aid of the foot-rule, to be taken up
in the compasses ; and with this radius, at a suitable dis-
tance C' from the ground line, a semicircle A' G' B' is to be