  Guided by the Jacobi's work published the year before his death about the
rotation of a rigid body, the behavior of the rotation matrix describing the
dynamics of the free rigid body is studied. To illustrate this dynamics one
draws on a unit sphere the trace of the three unit vectors, in the body system
along the principal directions of inertia. A minimal set of properties of
Jacobi's elliptic functions are used, those which allow to compute with the
necessary precision the dynamics of the rigid body without torques, the so
called Euler's top. Emphasis is on the paper published by Jacobi in 1850 on the
explicit expression for the components of the rotation matrix. The tool used to
compute the trajectories to be drawn are the Jacobi's Fourier series for {\sl
theta} and {\sl eta} functions with extremely fast convergence. The Jacobi's
{\sl sn}, {\sl cn} and {\sl dn} functions, which are better known, are used
also as ratios of {\sl theta} functions which permit quick and accurate
computation. Finally the main periodic part of the herpolhode curve was
computed and graphically represented.
