An old problem of finding the curve of pursuit and the point of intersection, originally proposed by Leonardo da Vinci, is considered in its generality. It is assumed that the pursued moves along an arbitrary straight line and the motion of the pursuer is always directed towards the pursued. Both the pursuer and the pursued move at constant but arbitrarily different speeds. A novel and elegant scheme of calculation is devised here by which the point of intersection can be calculated very simply without any explicit determination of the trajectory of the pursuer. Then the scheme is extended to give the analytical expressions for the actual curve of pursuit. The present method of solution is more general and very considerably simpler than existing solution techniques. Some physics of the problem is discussed..