The quadrature formulas described by James Gregory (1638--1675) improve the accuracy of the trapezoidal rule by adjusting the weights near the ends of the integration interval. In contrast to the Newton--Cotes formulas, their weights are constant across the main part of the interval. However, for both of these approaches, the polynomial Runge phenomenon limits the orders of accuracy that are practical. For the algorithm presented here, this limitation is greatly reduced. In particular, quadrature formulas on equispaced 1-D node sets can be of high order (tested here up through order 20) without featuring any negative weights.