Okay.... I've been mucking about with a ball and compass. Here goes!
Diameter of ball = 2.25" therefore,
METHOD
Open your compass to a distance of 1.59099" (or as close to as the human eye can see....) (the Rule-Of-Thumb method may apply here).
1) Find any point on the surface of the sphere, called 'O1', draw a circle, C1' whose diameter is 2.25".
2) From any point on 'C1', called 'O2', draw a circle which intersects 'C1' and 'O3' and 'O4' and whose diameter is 2.25".
3) From either point 'O3' OR 'point 'O4', draw a circle which intersects ''C2' at 'O1' and 'O5' and whose diameter is 2.25".
Point 'O1' = North
Point 'O2' = West
Point 'O3' = East
Point 'O5' = South
