A gain in speed means a loss in power, and _vice versa_. By gearing-up a cycle we are able to make the driving-wheel revolve faster than the pedals, but at the expense of control over the driving-wheel. A high-geared cycle is fast on the level, but a bad hill-climber. The low-geared machine shows to disadvantage on the flat, but is a good hill-climber. Similarly, the express engine must have large driving-wheels, the goods engine small driving-wheels, to perform their special functions properly. In order to travel fast over level country, and yet be able to mount hills without undue exertion, we must be able to do what the motorist does--change gear. Two-speed and three-speed gears are now very commonly fitted to cycles. They all work on the same principle, that of the epicyclic train of cog-wheels, the mechanisms being so devised that the hub turns more slowly than, at the same speed as, or faster than the small chain-wheel,[42] according to the wish of the rider. We do not propose to do more here than explain the principle of the epicyclic train, which means "a wheel on (or running round) a wheel." Lay a footrule on the table and roll a cylinder along it by the aid of a second rule, parallel to the first, but resting on the cylinder. It will be found that, while the cylinder advances six inches, the upper rule advances twice that distance. In the absence of friction the work done by the agent moving the upper rule is equal to that done in overcoming the force which opposes the forward motion of the cylinder; and as the distance through which the cylinder advances is only half that through which the upper rule advances, it follows that the _force_ which must act on the upper rule is only half as great as that overcome in moving the cylinder. The carter makes use of this principle when he puts his hand to the top of a wheel to help his cart over an obstacle. [Illustration: FIG. 223.] [Illustration: FIG. 224.] [Illustration: FIG. 225.] Now see how this principle is applied to the change-speed gear. The lower rule is replaced by a cog-wheel, C (Fig. 223); the cylinder by a cog, B, running round it; and the upper rule by a ring, A, with internal teeth. We may suppose that A is the chain-ring, B a cog mounted on a pin projecting from the hub, and C a cog attached to the fixed axle. It is evident that B will not move so fast round C as A does. The amount by which A will get ahead of B can be calculated easily. We begin with the wheels in the position shown in Fig. 223. A point, I, on A is exactly over the topmost point of C. For the sake of convenience we will first assume that instead of B running round C, B is revolved on its axis for one complete revolution in a clockwise direction, and that A and C move as in Fig. 224. If B has 10 teeth, C 30, and A 40, A will have been moved 10/40 = 1/4 of a revolution in a clockwise direction, and C 10/30 = 1/3 of a revolution in an anti-clockwise direction.