In a famous paper [3], Bose, Shrikhande, and Parker proved the existence of a pair of orthogonal Latin squares of order v for all v ≠ 2,6. In the present paper it is shown that there exist three mutually orthogonal Latin squares for all v ≡ 0,1 (mod 4). This result will be needed in several future papers on the covering of pairs by quadruples.