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Guided by the Jacobi's work published the year before his death about therotation of a rigid body, the behavior of the rotation matrix describing thedynamics of the free rigid body is studied. To illustrate this dynamics onedraws on a unit sphere the trace of the three unit vectors, in the body systemalong the principal directions of inertia. A minimal set of properties ofJacobi's elliptic functions are used, those which allow to compute with thenecessary precision the dynamics of the rigid body without torques, the socalled Euler's top. Emphasis is on the paper published by Jacobi in 1850 on theexplicit expression for the components of the rotation matrix. The tool used tocompute the trajectories to be drawn are the Jacobi's Fourier series for {\sltheta} and {\sl eta} functions with extremely fast convergence. The Jacobi's{\sl sn}, {\sl cn} and {\sl dn} functions, which are better known, are usedalso as ratios of {\sl theta} functions which permit quick and accuratecomputation. Finally the main periodic part of the herpolhode curve wascomputed and graphically represented.

Drawing the Free Rigid Body Dynamics According to Jacobi