r shape of a tooth to act on a pinion with radial
arms and centred at B. In designing a pair of wheels to transmit motion,
which is to be multiplied say 6 times in the transmission (about the
usual ratio for clock wheels), if we take two circles (called the "pitch
circles") touching one another with radii as 1:6, then the circumference
of the smaller will roll 6 times round that of the larger. The smaller
wheel will have a number of teeth, say 8 to 16, each of them being
sectors of the circle (fig. 24). If there are 16 teeth, then on the
surface of the driving wheel there will be 96 teeth. Each of these teeth
will be shaped as the curve of an epicycloid formed by the rolling on
the big circle of a circle whose diameter is the radius of the pitch
circle of the pinion. Points of the teeth so formed are cut off, so as
to allow of the pinion having a solid core to support it, and gaps are
made into the pitch circle to admit the rounded ends of the leaves of
the pinon wheel. Thus a cog-wheel is shaped out.

Clock wheels are made of hard hammered brass cut out by a wheel cutting
machine. This machine consists of a vertical spindle on the top of which
the wheel to be cut is fixed on a firmly resisting plate of metal of
slightly smaller diameter, so as to allow the wheel to overlap. A cutter
with the edges most delicately ground to the exact shape of the gap
between two teeth is caused to rotate 3000 - 4000 times a minute, and
brought down upon the edge of the wheel. The shavings that come off are
like fine dust, but the cutter is pushed on so as to plunge right
through the rim of the wheel in a direction parallel to the axis. In
this way one gap is cut. The vertical spindle is now rotated one
division, by means of a dividing plate, and another tooth is cut, and so
the operation goes on round the wheel.

It is not desirable in clocks that the pinion wheels which are driven
should have too few teeth, for this throws all the work on a pair of
surfaces before the centres and is apt to produce a grinding motion.
Theoretically the more leaves a pinion has the better. Pinions can be
made with leaves of thin steel watch-spring. In this case quite small
pinions can have 20 leaves or more. The teeth in the driving wheels then
become mere notches for which great accuracy of shape is not necessary.
Such wheels are easy to make and run well. Lantern pinions are also
excellent and are much used in American clocks. They are easy to make in