In an important contribution Powell has suggested an approach for determining the unconstrained minimum of a function of several variables, and determining it without calculating derivatives. This paper studies his approach in some detail. It is first shown by counter-example that his basic method for minimizing a quadratic function in a finite number of iterations contains an error. His modification of his basic method is then simplified, and the simplification proven to converge for strictly convex functions. Finally, we pose a new method not only which converges in a finite number of iterations for a quadratic, but also for which theoretical convergence is established in the strictly convex case.