In the first part of the paper, Ernst Schmidt's intuitive argument for choosing the optimum fin shape for least material is translated into analysis using the method of variational calculus. It is shown that even when the thermal conductivity of the fin is a function of temperature, the optimum shape of an individual 'heat tube' of the fin is the uniform shape, i.e. the tube the geometry of which does not vary with the longitudinal position. This generalization of Schmidt's argument is used in the search of optimum shapes for fins the materials of which have temperature-dependent conductivities. In the second part of the paper the analutical generalization of Schmidt's argument is applied to the design of ducts for fluid flow. It is shown that the shape of a duct the flow pattern or temperature of which varies with the longitudinal position can be selected optimally such that the overall flow resistance of the duct is minimized. The optimization of the duct shapes illustrated in this paper is conducted subject to one of two constraints, constant total duct volume or constant total duct wall surface. © 1988.