A new method for calculating electric and magnetic forces is presented. The method makes it possible to calculate forces directly from scalar and vector potentials rather than from field vectors. Several new force equations are derived. They are grouped in four categories: equations for calculating electric forces from electric scalar potentials; equations for calculating electric forces from electric vector potentials; equations for calculating magnetic forces from magnetic vector potentials; and equations for calculating magnetic forces from magnetic scalar potentials. Since the potentials are usually easier to compute than the corresponding fields, the new equations provide an effective alternative to the previously available techniques for force calculations. From the theoretical point of view, these equations reveal a physical significance of electric and magnetic potentials not heretofore apparent and provide a new insight into the nature of electric and magnetic forces. Illustrative examples on the use of the equations are given.