m B, as a centre, mark the arc I, from D, the arc J, from E, the arc
K, from F, and so on, marking as many arcs as there are points of
division on B C. With the compasses set to the radius of divisions 1, 2,
etc., step off on arc M the five divisions, N, O, S, T, V, and at V will
be a point on the epicycloidal curve. From point of division 4, step off
on L four points of division, as _a_, _b_, _c_, _d_; and _d_ will be
another point on the epicycloidal curve. From point 3, set off three
divisions, and so on, and through the points so obtained draw by hand,
or with a scroll, the curve.

[Illustration: Fig. 239.]

Hypocycloids for the flanks of the teeth maybe traced in a similar
manner. Thus in Figure 239, P P is the pitch circle, and B C the line of
motion of the centre of the generating circle to be rolled within P P.
From 1 to 6 are points of equal division on the pitch circle, and D to I
are arc locations for the centre of the generating circle. Starting
from A, which represents the location for the centre of the generating
circle, the point of contact between the generating and base circles
will be at B. Then from 1 to 6 are points of equal division on the pitch
circle, and from D to I are the corresponding locations for the centres
of the generating circle. From these centres the arcs J, K, L, M, N, O,
are struck. The six divisions on O, from _a_ to _f_, give at _f_ a point
in the curve. Five divisions on N, four on M, and so on, give,
respectively, points in the curve.

There is this, however, to be noted concerning the construction of the
last two figures. Since the circle described by the centre of the
generating circle is of a different arc or curve to that of the pitch
circle, the length of an arc having an equal radius on each will be
different. The amount is so small as to be practically correct. The
direction of the error is to give to the curves a less curvature, as
though they had been produced by a generating circle of larger diameter.
Suppose, for example, that the difference between the arc _a_, _b_, and
its chord is .1, and that the difference between the arc 4, 5, and its
chord is .01, then the error in one step is .09, and, as the point _f_
is formed in five steps, it will contain this error multiplied five
times. Point _d_ would contain it multiplied three times, because it has
three steps, and so on.

The error will increase in proportion as the diameter of the generating
is less than that of