What were Archimedes's improvements to the screw pump?

In order to understand my motivation to answer this question, I'll remark that Vitruvius said that the screw was inclined by the small angle of a 3:4:5 triangle. So it means that perhaps there was knowledge of how to optimize the design of a screw, and I guess this knowledge is of Archimedean origin. So my question is, is there any evidence that Archimedes improved the screw? It's important, since modern research has found that the proportions described by Vitruvius are near-optimal, so if these proportions were discovered by Archimedes, one can argue that Archimedes developed a complicated geometric theory of the flow of water through the screw.