An introduction to Quaternions and Octonions is given, and the Maxwell Equations are formulated in terms of each. The conventional, source-free relativistic theory of eight dimensional electromagnetism is introduced and examined. Similarly, the source-free Octonionic Maxwell Equations are developed, and it is shown that the seven dimensional electric and magnetic fields pure Octonions each admit plane wave solutions. An Octonionic Faraday tensor is constructed and compared with the conventional Faraday tensor, and it is shown that, in the source-free case, the conventional and Octonionic theories are equivalent
