The principle used by a gymnast to increase the number of somersaults, is law of conservation of

When a gymnast jumps onto a rotating apparatus such as a bar or beam, they use their arms to rotate around their axis faster. This increases their angular momentum. As they approach the peak of their rotation, they tuck their body to decrease their moment of inertia, which causes them to rotate even faster. This increase in angular velocity helps them to complete more rotations or somersaults before they land.

The law of conservation of angular momentum states that the angular momentum of an isolated system remains constant unless acted upon by an external torque. As the gymnast's body is isolated and there is no external torque acting on it, the total angular momentum of the system remains constant. 

The gymnast uses this principle to increase the number of somersaults by manipulating their body's moment of inertia and angular velocity.