 PM ends abruptly with the proof of a theorem (∗375·32) concerning cyclic families, without any concluding remarks or hint of what is to come later. The thought is that further mathematics, including the Volume IV that Whitehead was to write on geometry, would have to be developed piecemeal. First the notions of a given branch of mathematics would have to be defined in terms of earlier notions, such as classes of relations with a given structure, and then the important basic results in that field would be proved one by one, in the style of the work so far. Establishing logicism would be an ongoing project, as open-ended as mathematics itself.