There are corresponding relations of concepts and methods between elastic rod statics and rigid body dynamics about a fixed point according to the thought of Kirchhoff kinetic analogy. A twisted elastic column in equilibrium is corresponding to a Lagrange heavy gyro rotating around a plumb axis. The equilibrium of an elastic rod of tension and twist is corresponding to permanent rotation of Lagrange heave gyro about vertical axis. According to the first approximation theory, the process of building the two stability criteria were examined, which shows that the kinetic analogy between them is in the sense of Lyapunov stability. Based on this, the euler elastic stability of the tension and twist rod with both ends pinned was further discussed and the formula satisfied by critical load was derived by Greenhill firstly. The results show that for the tension and twist rod, the Euler stability is different from Lyapunov stability in the concept. The difference is that the boundary condition is perturbed in Lyapunov stability and not in Euler stability.