For Cauchy boundary condition inverse problems in 2-D elasticity, all the boundary conditions on accessible part of the boundary are known,and the boundary conditions on the rest inaccessible part of the boundary need to be solved. In this paper, based on the boundary element method, and with a polynomial function to approximate the unknown traction boundary conditions, the inverse problem was transformed into a problem with the identification of unknown coefficients of the polynomial. The objective function was defined as the least square error between the calculated values and the given values of the tractions on the measurable part of the boundary. The unknown tractions on the immeasurable boundary were recognized by minimizing the objective function through the cuckoo search (CS) algorithm. Then, the unknown boundary displacements were obtained by solving the direct problem with the inversed tractions and the other known conditions. The calculation results with and without using polynomial approximation were compared, and the influences of nest number, polynomial order and measurement noise on the numerical inversion were also discussed. Numerical examples verify that the CS algorithm combined with polynomial approximation can accurately and effectively solve the Cauchy problem in elasticity.