Eristothenes measured the circumference of the earth by measuring the shadow of two poles, miles apart, at the exact same time. So how did he know the poles were being measured at exactly the same time in an era when people used the sun for timekeeping?
In: Mathematics
He didn’t measure two poles. He measured one pole and knew that on one day out of the year where the sun fell directly in a well in a distant city at solar noon. So on that day, he measured the length of the shadow at solar noon then did his trig calculations.
He knew it was the same time because it was the right day (via a calendar) and the right time (the solar noon/highest point in the sky the sun would reach).
By doing it at local apparent noon. This is, by definition, the time of day when the sun reaches its highest point in the sky. If 2 places are on the exact same line of longitude (i.e., directly north and south of each other), local apparent noon will occur at exactly the same time in both places. You can, of course, measure this with a sundial.
It turns out that while close, Alexandria and Syene (modern day Aswan) are not directly north and south of each other, which introduced some error into his calculations. Still, they’re close enough that his measurement was remarkably close to the correct value.
He didn’t. And he didn’t have to.
North of the tropics, the tip of an object’s shadow is the furthest south at the moment of solar noon. All you have to do is trace the path of the tip of the shadow and note where it is the furthest away.
If you take a stick and pop it on the ground, the length of its shadow will change as the sun moves overhead.
The point where the shadow is shortest is noon (the sun is the most overhead).
So, as the shadow moves just trace its path on the ground.
Do this in your two locations, and take the shortest lengths from both, and you have your two ‘noon’ measurements.
If you’re at the equator and the sun is directly overhead, the shadow length will shrink down to essentially zero. But further away from the equator, the sun will be ‘off-center’ and so it will always have some positive length. This distinction is what Eratosthenes noticed, and the basis for the calculation itself.
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