Durand's rules for approximate integration

In the late 19th century the engineer William F. Durand (1859-1958) derived a series of approximation rules for use in numerical integration. These algorithms were based primarily on trapezoidal and parabolic interpolating formulas (which correspond to the two simplest formulas of the Newton-Cotes family of quadrature rules). This discussion reviews Durand's derivations, with special attention given to the applied, practical perspective that lay behind development of his new numerical routines. It is proposed that Durand's emphasis was toward practical application of theoretical results. This led him to emphasize simplicity and applicability in his new routines for approximate integration and to consider heightened accuracy as only a secondary concern.