What determined the three mile limit

[trice]

I cam across this as a footnote at https://2015fallhw.github.io/arcidau/CannonBallistics.html

One of the reasons put forward for the international agreement that territorial waters extend 3 nautical miles from the coast, is that 3 nm was the maximum range of shore battery guns. The accuracy of this assertion can be tested using the software. 3 nm is 6076 yards, it is just possible to achieve this range by using a 64 lb gun with a long barrel (21 calibre) fired with a charge of 64 lb of powder (3 times the normal service charge) at an elevation of 41°, even then the gun must be mounted about 300 ft above the water to achieve the 6076 yd range.​

 

[Mark F. Jenkins]

The original rule was "long cannon shot" from the shore, whether any guns were placed there or not... basically, a country's territorial waters were anything that shore-based guns could theoretically control. The rule solidified into a standard three (nautical) miles sometime in the early 1800s IIRC. There seems to be some argument about exactly when it occurred, but it seems to have been a general evolution of an idea rather than something established at a particular time and place.

ETA: Frankly, it was a rather practical idea as well. In the era of sailing vessels, one wouldn't want to get a whole lot closer than that with a large ship on an unfamiliar coast. The closer you got, the riskier it became unless you had authentic local piloting knowledge available.

 

[Carronade]

Similarly, the 12-mile limit became popular in the early 20th century, as the range of guns increased.