III. Octonions

Octonions is a name adopted for various reasons in place of Clifford’s Bi-quaternions. Formal quaternions are symbols which formally obey all the laws of the quaternion symbols, q (quaternion), x (scalar), ρ (vector) ϕ (linear function in both its ordinary meanings), ϕ' (conjugate of ϕ), i, j, k, ζ, K_q, S_q, T_q, U_q, V_q. Octonions are in this sense formal quaternions. Each octonion symbol, however, requires for its specification just double the number of scalars required for the corresponding quaternion symbol. Thus, of every quaternion formula involving the above symbols there is a geometrical interpretation more general than the ordinary quaternion one, an octonion interpretation.