In one 19th-century book, however, I saw a better attempt made; in that book, the author repeated the process for every square on the board, and then calculated the average mobility.
I wish I knew what book it was; I think it was German, from the 1840s. Do you know?
The Knight can move to two squares from a1, to three squares from b1, and so on; this next table shows the Knight's mobility on 16 different squares:
KNIGHT'S MOBILITY ................................. 4 | 4 | 6 | 8 | 8 . |-------|-------|-------|-------. 3 | 4 | 6 | 8 | 8 . |-------|-------|-------|-------. 2 | 3 | 4 | 6 | 6 . |-------|-------|-------|-------. 1 | 2 | 3 | 4 | 4 . |-------|-------|-------|-------. a b c dThus, ( 8+8+8+8+6+6+6+6+4+4+4+4+4+3+3+2 ) divided by 16 gives us a result of 5.25, the average mobility of a Knight.