Mike;
Your formula is of course correct, & only a .001" diference was the result, which is of no consequence. It simply results from rounding in establishing the .729" dia. Many years ago one evening instead of watching the idiot box I sat down with a programmable calculator & a copy of a British Proof house table of gauge sizes with all whole numbers from 1-50 & those odd ball sizes from .450 down to .300 in .010 increments with the gauge carried to two decimal places. It was soon determined to "Fit" the entire chart the dia of a 1 ga (1 lb lead) ball had to be carried to 6 decimal places. Then using the exact formula you used I settled upon an approximately middle of the road figure for a 1ga ball dia of 1.669285" & recorded it.
A couple of things are worthy of note I think. 1st taking thr volume of 1lb of lead in Cubic Inches of any chart I can come up with does not produce a ball having this dia. It will be "Very Slightly" larger. "If" one adopted the theory they used impure lead in determining their standard, this would have required a "Denser" mat'l which is opposite of what would have been expected. It was once suggested by someone on the board here that perhaps they used some form of rolling to get their sample to pefect roundness & it gave a light compression to the metal. This certainly sounds plausible, think may have been Rocketman who suggested this, but just not positive now.
At any rate, once a base is established, & especially consideringat that point all calculations would have been done sans assistant of computers or calculators, their matamatics are astonishing. Cube roots aren't easily handled longhand. I can recall doing sq roots from my school days & they are time consuming.
Incidently using my baseline 12ga figures to .729127+. Using this figure your formula will give a result which rounds up to .506" rather than down to .505". Your calculations certainly showed that 36gauge is not a correct figure for a bore of .410" & was not intending to disparage your figures, was just stating what the British tables listed.
As I recall if one used 1.669 (1 ga from table) as baseline no gauge when rounded to 3 places would miss by more than .001". With this base 12ga is virtually an exact .729" (.729003) & 36 is .505462" (.505" to 3 places) as you correctly calculated.
Obviously when these tables were run some precision calculating was done, even if they did slightly miss the wt of lead. As I recall using current listings for the wt of lead a number of the gauges would "Grow" by .001" with some of the larger gauges perhaps going up by as much as .002", "Terrible Error", but has been some time since I actually ran the figures & didn't save them all.
I enjoy mathamatics to the extent I know them & have just always found this gauge buisness fascinating.
