The double-six configuration in classical 3-dimensional projective geometry has been discussed by a number of authors. It consists of two sets a1,…, a6 and b1, …, b6 of six lines such that no two lines of the same set intersect, and ai meets bi if and only if i †j. The existence of a doublesix in the 3-dimensional projective geometry over a field F has been proved by Hirschfeld in [2] for all fields F except those of 2, 3 and 5 elements. For an arbitrary 3-dimensional projective geometry in which the number of points on a line is at least 5 but is not 6, the existence of a double-six follows from the fact that the geometry is a geometry over a division ring D with a subfield F satisfying the conditions of Hirschfeld's theorem.