Euclid is a simple game which can be played with paper and pencil, yet it is a wellspring of interesting ideas and surprises. This paper will explore the many variations of the game which are possible. Many of these variations are restrictions on the moves available to the players. Of particular interest are "dynamic" restrictions which themselves change as the game progresses. It is shown how Euclid can be recast as a take-away game played on the Stern-Brocot Tree, and this transformation is frequently used to find winning strategies in many of Euclid's variations.