  For a bounded domain $\Omega$ with a piecewise smooth boundary in a complete
Riemannian manifold $M$, we study eigenvalues of the Dirichlet eigenvalue
problem of the Laplacian. By making use of a fact that eigenfunctions form an
orthonormal basis of $L^2(\Omega)$ in place of the Rayleigh-Ritz formula, we
obtain inequalities for eigenvalues of the Laplacian. In particular, for lower
order eigenvalues, our results extend the results of Chen and Cheng \cite{CC}.
