The "281" isn't so far off from the actual distance from Howland to Gardner, is it?
If you don't have a sextant but you do have an almanac you can get your latitude pretty accurately by simply observing the time of local noon (when the sun is highest). If you know you latitude, you know how far you are from the equator. The spot where we think the Electra was when the "281" message was heard is 280 nautical miles from the equator.
Sorry Ric, that's not how it works. If you could determine the time of noon, when the sun is highest in the sky, then you can easily find your approximate LONGITUDE,
not your latitude. The sun circles around the earth (360 degrees) in 24 hours so it is traveling westward 15 degrees per hour (900 knots at the equator), one degree every four minutes and one nautical mile every 4 seconds. Since "noon" is connected to the sun, "noon" also travels westward at these same rates. This is not a normal technique because there are methods that provide a more accurate longitude. This is because the height of the sun changes very slowly around noon. Looking at noon at Gardner on July 2, 1937, the highest point the sun reaches 62°
18.6' at 2342:00 Z, noon. But, the sun climbs through 62°
17.6' at 2337:59 Z and descends through the same altitude at 2346:06 Z. This means that the altitude of the sun stays within one minute of arc, one-sixtieth of a degree, of its highest point for eight minutes, during which time the sun moves two degrees westward, which makes the longitude determined by this method only accurate within two degrees, 120 NM. And to achieve that accuracy you need a sextant because there is nobody on the planet earth that can see a one minute change in the sum's altitude without a sextant and it is difficult even with a sextant. And that's with a marine sextant which is much more accurate than a bubble sextant.
If you can accurately measure the altitude of the sun at it's highest point, at noon, then you can easily determine your latitude but this also requires a sextant since you can't estimate the height with the naked eye any better than about ten degrees so you can only determine your latitude with a naked eye to a precision of about 10 degrees, 600 NM, so the "281" wasn't determined this way. To get to an accuracy of one mile you need the altitude to be measured to a precision of one minute of arc and the sextant carried by Noonan had a scale marked only every
two minutes of arc.
gl