Abstract: An automatic bicycle transmission includes in combination a sprocket wheel assembly which is responsive to the input torque applied at the pedals, a chain takeup assembly to maintain the chain taut under all operative positions of the sprocket wheel assembly and a hydraulic cylinder operatively connected to the takeup assembly by a control cable to apply hydraulic damping forces upon the sprocket wheel assembly to hydraulically smooth the transition between high gear ratio and low gear ratio positions. The sprocket wheel assembly includes a reference wheel that is rotated by the pedals and a pair of diametrically opposed chain gear sliders and a pair of diametrically opposed chain sliders. All of the sliders are simultaneously radially reciprocal along radial slots in the reference wheel in response to changes of input torque applied at the pedals.