From: Frank Reed
Date: 2020 Sep 26, 12:43 -0700
Gary, you wrote:
"I've always used 3440 for the earth's radius"
That's more accurate than 3438 for the earth's radius in n.m. by some measures (mean radius is a tricky thing). And here's the great thing: since you already have that committed to memory and backed up by a nice mnemonic, you can do all those angle conversions in your head already. The exact value converting pure angles to minutes of arc (180/pi or 3437.75...) differs from 3440 by 1 part in 1720 which is of no significance for nearly all practical cases.
An example: I am standing on a coastal hill, and I see a ship offshore. It is "beam on" or more or less perpendicular to my line of sight. I know its length from an internet search and it is 620 feet. I grab my sextant and measure the angle from bow to stern. If I find 17 minutes of arc, I can immediately work out its distance: (17 m.o.a.)/3440 = (620 feet)/distance. Therefore (a tiny bit of basic algebra needed here...), distance=620×3438/17. For this example, 3438 (or 3440) is close enough to 3434 and 3434/17 is 202. The distance is then about 125000 feet or 20.5 nautical miles. Notice: no trig needed, no math more complicated than re-arranging a ratio and a little basic multiplication and division.