In this paper I continue the process, begun in (2), of reducing and weakening the axioms of congruence needed for absolute geometry. The congruence axioms Cl*–C4*, C4**, and C5a–C5c (frequently referred to below) can all be found in (2) and will not be quoted again here. (This paper should be read in conjunction with (2); any attempt to make it self-contained would result in the repetition of large parts of (2).) The notation of (2) will be used throughout the paper.

The main result here is that axiom C5c is unnecessary. This is shown in § 1. In § 2 I discuss three other points arising from (2).

Note added in proof.Since writing this paper, I have constructed examples of (a) Archimedean planes satisfying Cl*-C4* in which not all points are isometric, (b) non-Archimedean planes satisfying Cl*-C4*but not C4**,and (c) one-dimensional geometries in which 2.1 (with “plane” replaced by “line“) is false.