Evidently, if this is to be so, the point _d_ must be on a semicircle, whose diameter is _f b_, for in that case the angle _f d b_ will always be a right angle.
The surface must then be so arranged that, whatever be the position of the cam and of the rod _b d_, the point of contact between them must always be on the semicircle _f c d_; that is to say, as the cam moves round the axis _a_ its shape must be such that a line drawn from _f_ to the point where it cuts the circle _f d b_ is always perpendicular to the curve.
Now suppose that we move a circle whose centre is at _a_, and radius _a f_, so as to roll the circle _f d b_ by simple surface friction round its centre _o_, then any point _d_ on it would mark out a curve on a piece of paper attached to the moving circle whose centre is at _a_, and the direction of motion of the curve would always be such that the point _d_ on it would at any instant be describing a circle round _f_, and the direction of the curve would thus at any point always be at right angles to the line _d f_ for the time being.
This curve, caused by the rolling of one circle on another, is called an epicycloid. Hence, then, for a clock, if we make the pinion wheel with straight spokes and the driving wheel with its teeth cut in the form of epicycloids, caused by rolling a circle with a diameter equal to the radius of the pinion upon the driving wheel, we shall get a uniform ratio of leverage one upon the other.
The circles with radii _a f_, _b f_, are called the “pitch circles,” and these radii are in the ratio of the movement that is required for the wheels, usually six to one or eight to one, as the case may be. The sides of the teeth of the pinion wheels are straight lines radiating from the centre, and rounded off at the ends so as to avoid accidental jambing. The teeth of the cogwheel have epicycloidal sides. The tips are cut off so as to be out of the way, and spaces are left between them for the width of the leaves of the pinion wheel.
Both pinion wheels and cogwheels are cut by cutters rotating at a high speed, about 3,500 times in a minute, the cutters being carefully shaped for the pinion wheels with straight edges, for the cogwheels in epicycloids. It is a pretty thing to see a wheel-cutting engine at work, the cutter flying round with a hum, cutting the rim of a brass wheel into teeth, the brass coming off in flakes thinner than fine hairs and falling in fine dust. When a tooth is cut, the wheel is moved round one division of an apparatus called a “dividing plate,” so as to present a new part of the wheel to the cutter. Of course, the cutter and wheels must all be properly proportioned. Cutters are sold in sets duly shaped for the work they have to do. Wheel-cutting is a special branch of the clockmaking industry. The reason the speed of cutting is so high is because it is desired to take off small portions of metal at a time, and thus not strain the wheel and the cutting machinery. If bigger cuts were made, then the machine would have to go slower, for it is a principle in the construction of cutting machinery that the speed of the cut must always be proportioned to the depth of it. If you want to take deep cuts you must move the cutting edge slowly, and _vice versâ_. The most modern method of making cogwheels of brass, and the best, is to stamp them out of solid sheet metal at a single punch of a punching machine, and cheap watches are always made in this way. In fact, the whole method of watch and clock-making is undergoing a transformation.