This video shows the method of constructing equal segments on a line by only using compass operations.

Here, you can see that there is a pattern in the number of required compass operations. For example, if you need to divide the distance by 3, you need 13 circles (or compass operations). This reveals a simple equation of c = 2 * n + 3 where n is the desired number of segments and c is the number of required compass operations. The compass-only constructions are also called Mascheroni constructions. In 1797, Mascheroni proved that all compass-straightedge (Euclidean or geometric) constructions are possible with a compass only.