The force, speed, dexterity, and compact size required of prosthetic hands present extreme design challenges for engineers. Current prosthetics rely on high-quality motors to achieve adequate precision, force, and speed in a small enough form factor with the trade-off of high cost. We present a simple, compact, and cost-effective continuously variable transmission produced via projection stereolithography. Our transmission, which we call an elastomeric passive transmission (EPT), is a polyurethane composite cylinder that autonomously adjusts its radius based on the tension in a wire spooled around it. We integrated six of these EPTs into a three-dimensionally printed soft prosthetic hand with six active degrees of freedom. Our EPTs provided the prosthetic hand with about three times increase in grip force without compromising flexion speed. This increased performance leads to finger closing speeds of ~0.5 seconds (average radial velocity, ~180 degrees second−1) and maximum fingertip forces of ~32 newtons per finger.