I haven't done the experiment, but this brings it back to trusting scientists (light bends just above the water, but doesn't 4m above the water). I would need to do additional experiments to understand the refraction.
The simplified method works by considering two cities along the same meridian and measuring both the distance between them and the difference in angles of the shadows cast by the sun on a vertical rod (a gnomon) in each city at noon on the summer solstice. The two cities used were Alexandria and Syene (modern Aswan), and the distance between the cities was measured by professional bematists.[16] A geometric calculation reveals that the circumference of the Earth is the distance between the two cities divided by the difference in shadow angles expressed as a fraction of one turn.
https://en.wikipedia.org/wiki/Eratosthenes#Measurement_of_Ea...
They found a road that was almost due north and south (US 183 from Bassett, NE to 370 miles south in Coldwater, KS). They measured the elevation of Antares at its zenith one night, then again the next. From that they get the change in latitude. Combined with odometer data and some estimates for wiggles, and end with a result about 5% larger than the best-known solution.
Their measuring instrument was a Ryder van dubbed "The Van of Eratosthenes."
As an aside, later in the same episode is when I learned about GPS. https://youtu.be/yRY2SkMTafc?t=2968 . "This is the future of mapping."
1. Find a sufficiently large "shallow" body of water. 2. Build a number a number of masts/poles in a straight line, all with an identical height over the water.
Now when standing at the shore and looking at the masts I predicts one of two things.
a. you can see all masts in their entirety with a telescope, or b. you can see less and less of the masts the further away they are.
There are actual places with power lines that satisfy this requirement. For b. you can use high school math to reason about the curvature needed to explain the results. There is no such sound logic (to my knowledge) explaining this for a flat surface.