A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2016 Nov 3, 19:09 -0700
Many navigators do small angle calculations using trigonometric relationships. For example, if they want to know the angular height of a lighthouse, they'll take the height in feet, divide by the distance in feet, and then look up or calculate the inverse sine. Possibly that number then has to be converted to minutes of arc. This is the long way around, and it's an abuse of trigonometry. For small angles, up to a few degrees, you take height (or size, generally) and divide by distance in the same units. Then multiply by 3438. The result is the angular size in minutes of arc. Angles are ratios. The angular size of some object, in "radians" as we're taught to call them in high school, is the ratio of its size to its distance (assuming the resulting angle is fairly small ...smaller than a few degrees). The factor 3438 is just 180·60/pi. That converts from a pure angular ratio (in radians) to minutes of arc. Of course you don't need to know why it works. It's sufficient to remember it as a "magic number" and to know how to use it.
I discovered just now that this number has another interesting place in navigation: it's the radius of the Earth. The mean radius of the Earth in nautical miles is 3438 thanks to the way the nautical mile is defined. So that's another excuse for memorizing this little number.
You can use this latter fact to amaze and mystify your navigation acquaintances. You can explain that you can calculate the angular size of a ship on the horizon using the radius of the Earth:
angular size(minutes of arc) = radius of Earth(nautical miles)·length of ship(feet)/distance to ship(feet)