Let C be a closed convex cone of vertex 0 in a real normed vector space L. We suppose C does not contain an entire straight line, and let _ * be the order induced on L by C. Let S be some subset of L. A map 5p: S--S is said to be order preserving on S if x > y implies so(x) >?p(y). We say a bijection so: S S is regular on S if both so and so-1 are order preserving on S. (We are not assuming so continuous.) Zeeman has shown [2] that if C is a right circular cone in R4, the only regular maps of R4 are affine, with linear part a Lorentz transformation. In this paper we show that similar conclusions can be drawn under more general circumstances. Before proceeding, we would like to record the benefit of several useful conversations with M. Koecher. We also note that, at the suggestion of the referee, we have modified our Proposition 1 to include the infinite dimensional case. As part of the setting for the sequel, we shall insist that C have a compact base. This means there is a continuous linear functional h on L such that h(x) >0 for xeC-0, and the set P=h-'(1)Q\C, called a base of the cone, is compact. Each ray in C intersects P exactly once, and as is customary, we call a ray passing through an extreme (respectively exposed) point of P an extreme (respectively exposed) ray of C. Since P is the closure of the convex hull of its extreme points, C is the closure of the convex hull of its extreme rays. Furthermore, let p be an exposed point of P. There is, by definition, a continuous linear functional s on L and a real number a such that s(x) +a > 0 for x EP,