I always enjoy Barry Evans' "Field Notes" column.
For last week's -- "Ends of the Earth" -- I anticipated the exceptions as I was reading: Of course the two exceptions are at the two poles. I thought I would be clever and try to figure out the diameter of the area around the poles that would be included in the exception area.
My thinking was that one would have to be far enough away from the pole to point to the intersection of the axis and the surface of the earth. This is assuming the earth is a perfect sphere and that your arm is 5 feet above the surface of the earth. So I was thinking this area of exception would have about a 6 mile radius -- that is, twice the distance of the apparent horizon at sea level. But as I fooled around with my diagrams doubt crept in.
I propose: Given a perfect sphere of an Earth, and standing on the surface of that sphere, if you point with one arm to one pole, and if the other arm is perpendicular to the first arm, then the second arm will be pointing to an area above the surface of the Earth, thereby not intersecting the point where the axis intersects the surface (the pole). This can be proven with the diagram that you show -- the point where the two legs of the right triangle intersect is on the diameter of the circle, not outside the circle.
Is my thinking correct?
Clay Johnson, Trinidad
*Barry Evans responds: Good catch, Clay. The angle between your arms -- one arm pointing to the North Pole, the other to the south pole -- is never going to be exactly 90 degrees since our shoulders aren't at the surface (and, of course, the Earth is way off spherical). *
The way I see it, if you say "at the pole" no one's going to worry if you're a few feet away. Who's checking? Beyond that, the difference gets academic -- 500 feet from the pole, at 4-foot shoulder height, the angle is less than half a degree from 90.