In Note 1568 (V for Victory) very brief reference was made to Generalised Chess, that is, an expansion of the ordinary game as played on the 8 × 8 board. In that note attention was called to the existence of the 5-leaper, whose moves are from (0, 0) to (5, 0); (0, 5); (3, 4); (4, 3). Here it is proposed to give some account of chess in three and four dimensions. In anticipation it must be pointed out that fractions do not exist in chess geometry—a special kind of lattice geometry; but that lengths such as √(a2 + b2) and √(a2 + b2 + c2), which are not integral, but where a, b, c are, do exist, since they can be constructed from integral lattices.