NavList:

Roger P, you wrote:
"360 just happens to be highly divisible"

This is usually described as the value of sexagesimal counting, 60-based, while 360 is its bigger cousin (speculatively attributed to the close match to 365 days in a year). Yet the origin of sexagesimal counting in Ancient Iraq is not usually connected with navigation or even astronomy. The astrologer-astronomers of the Ancient Near East used the sexagesimal counting system, and classsical Greek astronomy inherited it, but its divisibility benefit is (usually) attributed to simple commerce. 

It's interesting that the standard compass rose looks just terrible when expressed in degrees. For example north-northeast is 11.25° toward east from north. And worse still, north-by-east is 5.625° from north. Arguably this is a case where switching from the sexagesimal right angle of 90° to a decimalized right angle of 100 "grads" would have been a bit more compatible with compass points. North-by-east would have been 6.25 grads. It's not much better, but the point is that the 360° system for the compass was not an ideal evolution. 

Finally, the practical value of sexagesimal math, easily dividing by 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 has proved durable for over three thousand years. Nothing to sneeze at! But it doesn't really do anything to guarantee, or even recommend, four primary compass points.

Frank Reed