In preceding parts of this paper the possibility was examined of accomplishing the entirety of physical description by means of mechanics, without help from field-theoretical ideas. Apart from some easily obtained qualitative agreements with general descriptive features of nuclei and elementary particles, we were balked in this program by an inability to handle in purely mechanical terms the relativistic many-body problem. The present paper is addressed directly to the latter problem. No quantitative calculations are attempted, but three suggestions are made (...) — each of a quite radical nature — concerning the reformulation that will be necessary to restore mechanics to the mainstream of physics. (shrink)