The paper is devoted to the theory of a mechanism called constant force spring system. The system consists of a linear helical spring, a spiral drum, a take-up pulley, and two cords. The spiral drum and the take-up pulley are attached rigidly to each other. One of the cords connects the spring with the spiral drum, and at the initial position of the system fills the entire spiral groove on the drum. During the operation of the system, the spiral drum may rotate about its center, and the cord may gradually unwind from the spiral and wind again. The process of winding/unwinding causes the spring to change its length and change the force it exerts on the spiral drum. Due to the shape of the spiral, the distance of the cord from the center of the drum changes, so that the force in the other cord which is wound on the take-up pulley remains constant. Creation of that constant force is the goal of the system. The heart of the system is a specially designed spiral. The solution to the associated differential equation is provided. The system may allow to eliminate weight towers in exercise machines; eliminate counterweights in elevators, as well as in windows that open by moving upwards. The landing path of fighter planes’ landing on aircraft carriers may be reduced. The spiral of the system exhibits an important property which may interest mathematicians; its behavior is compared with that of the Archimedes’ and logarithmic spirals. Because of this property, the spiral may find other applications.