 By a total vector ordering on "X" we mean a total order on "X" that is compatible with the vector space structure of "X".The family of total vector orderings on a vector space "X" is in one-to-one correspondence with the family of all proper cones that are maximal under set inclusion.A total vector ordering can"not" be Archimedean if its dimension, when considered as a vector space over the reals, is greater than 1.