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Some problems of plane elasticity lead to the solution of biharmonic problem. Many methods have been developped to the solution of this problem (the method of finite differences, the finite element method, classical variational methods, methods based on the theory of functions of a complex variable, etc.). In this paper, the method of least squares on the boundary is presented, having its specific preferences. In the first part, the algorithm of this method and a numerical example are given. This part is mainly intended for "consumers" of mathematics and is written in more detail. In the second part, the proof of convergence of the method is given. This part is mainly intended for mathematicians. Applied to the solution of the biharmonic problem, the method takes an essential use of the form of equation. As to its idea itself, it can be applied - in proper modifications - also to the solution of other problems.
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