Once the diameter of a 1 pound lead ball (1 gauge) is known it is a simple matter to find the diameter of any gauge. You simply cube the 1 gauge size, divide by the gauge desired & then take the cube root of the result. One day some years back at work I was running a CNC milling machine on some airfoils which had cuts running for several hours each. I had to stay with the machine but had virtually nothing to to unless some unexpected event were to occur. I had copied off the chart of British proof gauge sizes. I had a programmable Casio Calculator. After putting in the program using a given size for 1 gauge I only had to enter a different gauge & it would give me the answer.
I very quickly discovered that using 1.669" for the base that all sizes did not fit, some came out too small. Stepping up to 1.6695 some sizes cam out too large. It proved out that to get a perfect match for the charts the 1 gauge ball had to be carried out to 6 decimal places. There was a very narrow range which would fit by rounding the end result to 3 decimal places. The figure I settled on was 1.6699285" for the 1 gauge ball.
As stated using this diameter fits all gauges listed in British, German & I believe Belgian charts from the proof houses from 1 gauge down to 172.28 gauge (.300 caliber).
These sizes have been accepted as standard since well back into the 1800's at least. Using .665" for the 16 would give .732" for the 12. It has always been listed as .729". Sizes for the common gauges are;
10=.775"
12=.729"
16=.662"
20=.615"
28=.550"
These sizes all have a perfect fit to "The Formula".
I also discovered that to obtain the "Tween" gauges as in 13/1 you average 12 & 13 then drop to three decimal places, the tween sizes are not rounded as are the primary gauges. On the larger gauges with two tween sizes as 8/1 & 8/2 you use third & two thirds & drop beyond three decimal places.
I have absolutely No Idea as to why SAAMI opted to change the size of the 16 but the .662" size has been nominal for "Well Over" a century.
As has been noted though individual makers have always bored guns to whatever they desired. The proof houses used the actual bore size to determine the proof charge, not the chambering. Thus if you had a gun chambered 12ga, but bored for brass shells with an enlarged bore which came up to 10 gauge diameter the gun would be proofed as a 10, not a 12.
