In this paper, we consider the direct kinematic problem of a parallel robot (called the Stewart platform or left hand) from a mathematical point of view. We do not try to give real time and numerical solutions to this problem but describe tools of effective algebra, which can help us to know a little more about the geometric aspects of the question. A simple proof that the number of solutions we are expecting is actually 40 is then given. We use explicit eliminations techniques, in order to get rid of the solution at infinity and we use Bezout's theorem on surfaces with circularity to conclude. One main obstacle in the natural approach, via symbolic computations, is the great size of the polynomials appearing during the computations. We also try to show how the use of invariant theory and intrinsic approach may "factorize" the results in a more understandable way. In this text, the reader will also find some examples of successful and helful use of a symbolic system (i.e. Maple).