On 'Most Perfect' or 'Complete' 8 x 8 Pandiagonal Magic Squares

A particular form of pandiagonal magic squares of doubly even order n defined by Emory McClintock in 1896 but not enumerated (except for the well-known 4 x 4 pandiagonal squares), and described by him as 'squares of best form' or 'most perfect', is discussed. The number of all such squares for n = 8 is found, by the use of symmetries and logical argument only, to be 2¹⁶ x 3^2 x 5 = 2949 120 (that is 2¹³ x 3^2 x 5 = 368 340 essentially different squares). These squares are given in summary form in an appendix.