Finite Galois arithmetics are well-known; finite geometries however, though more interesting to the amateur, have not really acquired professional status and do not appear to any great extent in standard works. The following example arose from a chance remark in Mathematics for T. C. Mits, by L. R. and H. G. Lieber; from this I deduce that a full theory has been worked out, but I have not seen it, and as far as I am concerned what follows is original, and I hope readers of the Gazette may find it new and stimulating.