Every structural member deforms under the forces acting upon it. In this paper the Maximum Principle, recently developed by Pontryagin is applied to the problem of determining that profile of a beam that deforms least under its own weight. The beam considered is a solid horizontal cantilever having constant density, modulus of elasticity and width. The height is taken as an independent variable subject to the restrictions that it lies between an upper and lower bound and is symmetrically distributed about the centre-line. The method is attractive in that it can readily be extended to take account of variable density, width and/or hollow members by extending the control vector u defined in the analysis.