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Eratosthenes measured the radius of the Earth with an incredibly accuracy.

To do it, you need to measure the length of the shadows from 2 different cities at the same time of the day. Then knowing the distance between the cities and a little bit of geometry he calculated the radius of the Earth.

My question is: How did he know when it was "the same time of the day"? There was no long range communications to synchronize two people measuring the shadows.

And if you measure the shadow one day, then travel to the other city to measure the shadow next day, how can you know that you will do it at the same time of the day?

Nowadays with clocks or telephones the experiment is very easy, but back then I can not figure out how he managed to do it.

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    $\begingroup$ See Eratosthenes and the Mystery of the Stades: First, "It is uncertain whether he made the measurements used in the calculation, or if he relied on the information of others." $\endgroup$
    – Mauro ALLEGRANZA
    Jul 13, 2017 at 13:30
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    $\begingroup$ Second: "While working at the library, he learned that on the first day of summer the Egyptian town of Syene cast no shadows. This happens because at noon on the day of the summer solstice the Sun is positioned directly above the town of Syene, near the modern city of Aswan, Egypt. In contrast, on that same day in Alexandria a staff, or gnomon, did cast a shadow." $\endgroup$
    – Mauro ALLEGRANZA
    Jul 13, 2017 at 13:31
  • $\begingroup$ So, he needed at most one measurement at noon. This is "How did he know when it was "the same time of the day" ". $\endgroup$
    – Mauro ALLEGRANZA
    Jul 13, 2017 at 13:32
  • $\begingroup$ The town of Syene cast no shadows (at some specific moment of the day). On the same day (and of course at the same time) in Alexandria, a staff did. But the question is still: How can you know back then that you are talking about the same exact moment of the day? $\endgroup$
    – Sembei Norimaki
    Jul 13, 2017 at 13:36
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    $\begingroup$ You must take care of the fact that ancient Greeks scientists were quite smart: they were not Neandertal men. It can be enough to measure the shadow in different point in time "around noon" and consider the shortest one. $\endgroup$
    – Mauro ALLEGRANZA
    Jul 13, 2017 at 13:39

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The first sentence of the question is not justified. Accuracy of Erathosphenes measurement was much discussed in the literature, and it is certainly not "incredible". What the actual accuracy was is unknown, because we are not sure what his unit "stadium" exactly was.

Considering time measurement, he did not need it. He used two cities on approximately the same longitude and measured shadows at noon. Noon is determined when the shadow is shortest, and one does not need any clock for this.

But the most doubtful thing is how he measured the distance between the two cities, or whether he measured it at all.

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    $\begingroup$ Finding noon this way is a skill that even Boy Scouts learn. $\endgroup$
    – David Smith
    Mar 3, 2022 at 18:06
  • $\begingroup$ Was Erastotenes aware of the need two places at the same longitude, or was it only luck that the Nile has a North-South orientation and so almost all the main cities in Egypt are in approximately the same longitude? $\endgroup$
    – José David
    Jan 2 at 10:23
  • $\begingroup$ He was certainly aware of the need to have cities at (approximately) the same latitude. $\endgroup$
    – Alexandre Eremenko
    Jan 2 at 15:15
  • $\begingroup$ @David Smith: one does not need to find (the moment of) noon. One only need the length of the shadow at noon, that is the shortest length during the day. $\endgroup$
    – Alexandre Eremenko
    Jan 2 at 15:20
  • $\begingroup$ @Alexandre Eremenko: Yes. I thought most people knew this. Certainly people who don't have clocks frequently know this. $\endgroup$
    – David Smith
    Jan 3 at 16:48

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