The Cauchy problem of the Laplace equation is a typical example of ill-posed problems in the sense that the solution is unstable for the Cauchy data. The aim of our research is to solve the Cauchy problem of the Laplace equation numerically. Then we propose a high order finite difference method in which quadrature points do not need to have a lattice structure. We interpret our method from the viewpoint of the exponential interpolation. From numerical experiments, we confirm that our method is effective for solving the two-dimensional Cauchy problem of the Laplace equation.