Precise and convenient force balance has been constructed for direct measurement of the lateral capillary forces between two vertical cylinders and between a cylinder and a wall. The experiments with both hydrophilic and hydrophobic capillaries of submillimeter diameter show that at large distances the capillary forces obey simple universal laws. A theoretical interpretation of this fact is proposed, which is based on a nonlinear superposition approximation for the shape of the liquid meniscus. At separations comparable with the capillary diameter and large slopes of the liquid surface, the nonlinear Laplace equation of capillarity should be solved numerically to describe the measured capillary forces. The obtained results have direct relation to the lateral capillary interactions between spherical particles floating on a single fluid interface or confined in a thin liquid layer.