A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2016 Nov 4, 17:55 -0700
David Pike, you wrote:
"3438 is simply the constant needed to convert radians to minutes of arc. Length of vessel divided by distance away comes out in radians. "
Well, yes. Of course. The "magic number" 3438 is just 180·60/pi.
In my original message on this topic, I pointed out that 3438 is also the radius of the Earth in nautical miles (take the average of the equatorial and the polar radii and you will find that it is 3438 nautical miles accurate to one part in 5000). This is because of the definition of the nautical mile which is closely tied to our standard circular measure. In the 180° range from pole to pole there are 180·60 or 10,800 minutes or arc. And a 180° angle is also an angle of pi in "radians" or a pure angle. These numbers aren't a coincidence. They derive from the same mathematical source. I have previously written many times that you can calculate angular sizes of things by using the "magic number" 3438:
angular size (minutes of arc) = 3438 · size (in some units) / distance (in the same units),
and no trig necessary (or appropriate: it is pointless to draw triangles and dig out trigonometry when it's not necessary). Some folks who learn about this 3438 rule will gripe that it's yet another number to memorize, and why bother when the calculator is close at hand with its seductive arcsine and arctangent capabilities? Well, now you have another reason to memorize those four digits! They are, also, and for similar mathematical reasons, equal to the mean radius of the Earth in nautical miles. Fun, fun.
But the really fun, even diabolically entertaining, part is that you now have a way of messing with people --and making them think at the same time. Let's consider a different case. Find some folks that are interested in astronomy. Tell them that you can prove that the Earth is connected to the size of all astronomical bodies (like some crazy pseudo-science theory might claim about the pyramids of Egypt and pi and all that). Show them the following equation for the angular size of every star cluster in the Milky Way Galaxy:
angular size (minutes of arc) = radius of the Earth (in nautical miles) · diameter of cluster (in parsecs) / distance to cluster (in parsecs).
Why on Earth would the angular sizes of all star clusters depend on the radius of the Earth?? And in nautical miles?!? Well, it works. And in fact it's accurate to 99.98% if the radius is calculated as the simple average of the polar and equatorial radii. Why does this work? Well, there's your "teachable moment" as they call it. But before you get to that, you can have them scratching their heads and starting to believe in a higher power out there somewhere. :)
Conanicut Island USA