Where did Greenhill's twist rule come from?

Greenhill's Rule for estimating rifling twist in artillery and small arms is interesting both because of its simplicity and longevity. This article describes (a) Greenhill's Simplified Rule for estimating the twist of rifling; (b) describes his original, more complete, but much less known Rule; (c) summarizes how he got the Rule; and (d) relates it to the modern stability factor. Some comments about twist, bullet stability, and black powder velocities are presented.