The Round, Pythagorean Earth

My good news today is that the orthopedist agreed to let me go completely free - no cast or even a splint. I have a sling that I should wear most of the time, but in general my orders are to take it easy, be careful with my arm, but move it around a little bit now and then. I'll see him again after a week. I shouldn't regain full range of motion in that time, but I'm already up to 75%. I can't run or exercise yet, but I can type.

Some time ago I wrote in a piece for English class that if you go to the beach and hold a postcard up against the horizon, you can see the curvature of the Earth. I didn't really fact-check this beforehand, though. Now that I think about it some more, I know that not only is it true, but that the ancient Greeks should not have had a very hard time figuring out that the Earth is round, and even could have measured its radius by making simple observations from the shore or from a boat.

The boat's lookout sits in the crow's nest - high above the main deck. This was probably practiced before people knew that the Earth is round. (Although as I Google it now, it turns out that many ancient Greeks did believe the Earth was round, perhaps beginning with Pythagoras, who unfortunately held that belief for retarded reasons. Later Aristotle talked about gravity pulling the Earth into a sphere in and some dude name Eratosthenes actually measured it using the lengths two shadows measured at the same time but at places far away from each other.) Why should the lookout seek a high vantage point? Because he can see farther, of course. This wouldn't be the case with a flat Earth.

On a flat Earth with no atmosphere, you would be able to see all the way to the edge, no matter how high up you are to start. The same is true for the crow's nest. You would have a slightly better vantage point from the crow's nest because objects in the sea far away would subtend a larger angle, but your total distance of visibility would be the same.



On the other hand, on a curved Earth the horizon is in fact further away if you are higher up. The man in the crow's nest can see the mermaid, but the man on the deck cannot.



The first time I sat down to work this out, I calculated the distance to the horizon for a 6-foot tall man standing at the beach, and got something close to an even 5000m. I thought that had to be wrong, since I had seen horizons much further away - when I was climbing mountains.
Only later did I realize that of course you can see further when climbing mountains - you're in a great tall crow's nest.

I originally calculated the distance to the horizon using analytic geometry, but it's an ugly way to do the problem. There's a much cleaner way, as shown here:



So, by the fact that you can see further from a crow's nest, people should have known the Earth is not flat. If you hypothesize a sphere for the Earth's shape, you could measure the radius quite easily by letting a ship sail out to sea, then measuring how far away it is and how high up you have to be to see it. This gives you an estimate of R for every time you measure the distance to the ship. The quality of the estimate will depend on how well you can measure distances out to sea (triangulation from the shore should work well), how accurately you can measure how high you are (probably easy), how spherical the Earth is (spherical enough), whether the ocean is truly flat (over long distances it is), and whether light is refracted on its way from the buoy back to you (variable and hard to control for). But I'd say you should be able to get a pretty good estimate of the radius of the Earth this way, and it can be done with measurements all from one place.


So finally, can you see the curvature of the Earth with your naked eye? Certainly if you go high enough up you can. From far out in space you can see the entire Earth at once. If you want the Earth to curve 1 degree, then you'd need to view about 100km of horizon under your postcard. So the horizon would need to be on the order of 200km away, because if the postcard subtends more than about a 30 degree angle, you can't see both sides of it at once. That means you'd have to be 3000m up. So you probably won't be able to see the curvature of the earth with the naked eye by holding a postcard up at the beach, but you could easily do it from a mountain overlooking the sea.