Knight's Journey

The Brave Knight from the Kingdom of Chessboard has grown tired of the excessive usage of his L-shaped movement in various puzzles has decided to leave forever. However, he quickly realizes that his world is limited to a standard 8 by 8 chessboard. So instead of leaving, he opts to tour as many places on the chessboard without landing on any square twice and without ever crossing his path. The path he takes is defined as a straight line between the starting and ending squares. What path should he take?

This puzzle was originally posted by Laramie on 4/3/2005. The solver only posted the number of moves it took to complete the task and never the route.