We reconsider the ellipsoid method for linear inequalities. Using the ellipsoid representation of Burrell and Todd, we show the method can be viewed as coordinate descent on the volume of an enclosing ellipsoid, or on a potential function, or on both. The method can be enhanced by improving the lower bounds generated and by allowing the weights on inequalities to be decreased as well as increased, while still guaranteeing a decrease in volume. Three different initialization schemes are described, and preliminary computational results given. Despite the improvements discussed, these are not encouraging.