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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.

Elerium115
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    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." – Mauro ALLEGRANZA Jul 13 '17 at 13:30
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    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." – Mauro ALLEGRANZA Jul 13 '17 at 13:31
  • So, he needed at most one measurement at noon. This is "How did he know when it was "the same time of the day" ". – Mauro ALLEGRANZA Jul 13 '17 at 13:32
  • 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? – Elerium115 Jul 13 '17 at 13:36
  • You choose the first day of summer, and thus this is the "same" day. You know that at noon in Syene there is no shadow; thus, no measurement needed. You have to measure at noon of the first day of summer in Alexandria the shadow cast. – Mauro ALLEGRANZA Jul 13 '17 at 13:38
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    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. – Mauro ALLEGRANZA Jul 13 '17 at 13:39
  • You can read all the article linked: Eratosthenes and the Mystery of the Stades - The Basic Problem in order to appreciate the simplicity of the intuition of the genius and at the same time, to appreciate how complex is the process of scientific knowledge: we need a lot of assumptions in order to make a discovery/experiment. – Mauro ALLEGRANZA Jul 13 '17 at 13:44
  • I still don't see how it works! Of course, he can find noon where he is, but how does he know when it is noon at the remote location? – Brian Mc Donagh Oct 15 '19 at 00:52
  • The other "observation" is that Alexandria is "directly north" of Syene, so noon is the same time at the two locations. As Alexandre notes, this is only approximately true. – Gerald Edgar Oct 15 '19 at 10:23
  • I puzzled about this. He knew noon in each city by the fact that the East West shadow was not there. At Syene the longitude of the North South showed no shadow on the solstice, since the tilt of the earth is 23.5° and Syene's latitude is 24°. An assistant in Alexandria could have made the same measurement when the East West shadow was 0, and came up with 7.2° North South. There was no need to make simultaneous time measurements. – Pierre Mar 20 '24 at 15:34

1 Answers1

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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.

Alexandre Eremenko
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    Finding noon this way is a skill that even Boy Scouts learn. – David Smith Mar 03 '22 at 18:06
  • 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? – José David Jan 02 '23 at 10:23
  • He was certainly aware of the need to have cities at (approximately) the same latitude. – Alexandre Eremenko Jan 02 '23 at 15:15
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    @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. – Alexandre Eremenko Jan 02 '23 at 15:20
  • @Alexandre Eremenko: Yes. I thought most people knew this. Certainly people who don't have clocks frequently know this. – David Smith Jan 03 '23 at 16:48