Science, specifically Observing, theorizing and testing.
Most others covered observation, so I’m skipping ahead. Ancient people observed other celestial bodies and realized they may be similar to the Earth under their feet.
So now we assume earth is round, how do we verify this? You test it! This is the easier step, we simply coordinate two people on the same day run the same experiment at different locations.
The specific test was measuring shadows at the same time. The fixed items were time of day, altitude, sticks and longitude. Latitude was the variable. So two people made measurements at two cities and used shadows to triangulate the sun and their two locations. The math showed the person further north was further from the sun, therefore you can conclude the Earth is not flat.
Additionally, if the Earth was flat, we wouldn’t have a day-night cycle. Look at flat earthers model of a disc and a sun spinning circles above. On the same plane, they say half the surface is day and half is night. How does sunlight just stop and not shine on half the planet? Hold a plate under a light bulb and try to make half lit and half shaded on the same face. Impossible.
My understanding is that they used observations of shadow length of obelisk at different locations at specific times of the day on the summer solstice. Obilisks closer to the equater would have smaller shadows than those further away. This can’t happen on a flat surface, but does happen on the surface of a sphere. Using these observations they were able to extrapolate not only the shape of the planet, but the size as well…. to within a working accuracy.
Eratosthenes of Cyrene calculated the radius of the planet pretty accurately by noting that on a given day the sun shown directly down a well in Aswan at noon, and then measured the angle of the sun’s shadow at noon on the same day in Alexandria – about 500 miles away. A bit of math gave a circumference of 25,000 miles. That was ~250 BCE.
Posidonius of Rhodes calculated it by observing the star Canopus, which when it’s on the horizon in Rhodes it was a certain distance above the horizon in Alexandria. His calculations showed a circumference of 18,000 miles.
Columbus knew about Posidonius’ estimate, but not Eratosthenes’ which is why he thought he had travelled farther than he actually had.
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