Numerical solutions for nonlinear equations and systems of nonlinear equations are always appealing greatly to people.There are many papers discussing the local convergence of Newton method,a universally acknowledged classical algorithm.In the paper,on the basis of the Kantorovich theorem about the classic semi-local convergence of Newton method,affine contravariant conditions were introduced and the semi-local convergence of Newton method under affine contravariant Lipschitz conditions and affine contravariant Holder conditions was analysed.The convergence behavior of Newton method was simplified.The semi-local convergence theorems were presented and the estimations of the iterative residual were obtained.The results extend and improve the results in the related paper,which show that the method is efficient.