The work of Peierls (1959) and Chan (1961) has shown that the branch points of scattering amplitudes at the thresholds of production of two-particle states have a kinematic origin. The formulation of the scattering problem there involved a momentum variable k which implied that the amplitude when considered as a function of the energy E, must in general have a square-root branch point at threshold, i.e. at k = 0. In the present paper the discussion is extended to similar kinematic branch points of the general n-line graph. Beside the well-known `threshold' branch points for multiple production, another class of kinematic branch points is discovered, the existence of which is due to the necessary linear dependence of any five vectors in the Lorentz space. These new branch points exist only for graphs with six lines or more. Generalized momentum variables (l-variables) are introduced which allows us to find the position of both these types of kinematic branch points in the many-dimensional complex space by simple algebraic computations. The kinematic origin of these branch points as exemplified by this method of treatment also give some indications to the intrinsic nature of these branch points and to the basic structure of the many-dimensional Riemann `surface'.