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The famous measurements and calculations done by Eratosthenes around 300 BC are very widely known. He concluded correctly that the circumference of the Earth is about $252\,000$ times the length of an athletic stadium.

But what Eratosthenes did would make no sense if the Sun were (for example) only $6000$ miles from the Earth. How did he know it was much farther away than that?

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    $\begingroup$ This reminds me of a physics class where a Greek natural philosopher measured the speed of light to be instantaneous, or almost instantaneous. He signaled to an assistant who was at the time far away (by ancient standards), measuring time by his heartbeat, and found that he saw his assistant signal back the same number of beats later as when he was nearby. Thus, without modern equipment, he concluded that the speed of light was within measurement error of instantaneous. $\endgroup$ – Christos Hayward Sep 17 at 12:41
  • $\begingroup$ @ChristosHayward: Are you sure that was a Greek philosopher? That sounds suspiciously like Galileo's lantern experiments. $\endgroup$ – Sean Sep 18 at 0:59
  • $\begingroup$ you can tell a light source is far away if an object's shadow is the same size as the object. $\endgroup$ – dandavis Sep 18 at 7:55
  • $\begingroup$ @dandavis : That would probably only tell us that the sun is many miles away (100 miles??), but Eratosthenes needed to know that it was immensely farther than that. $\endgroup$ – Michael Hardy Sep 18 at 20:14
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The sun and the moon go around the observer once a day, Eratosthenes knew that the apparent size of moon doesn't change. This must mean that Alexandria is near the centre of the moon's orbit. But the apparent size also doesn't change when viewed from anywhere. So everywhere is close to the centre of the moon's orbit. Thus the moon must be much further than the radius of the Earth. If the moon were 6000 miles from the Earth, then it would seem to grow and shrink in size as it passed by (such an effect can be seen on Mars, where the moon really does orbit close to the planet)

And the Sun is further still. At half moon, the sun seems to be at $90^\circ$ to the moon. This is only possible if the sun is much further away than the moon.

In conclusion, the distance to the sun must be very very large in comparison to the radius of the Earth, and we can assume that that the rays of light from the sun are parallel.

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    $\begingroup$ Solar eclipeses demostrate that the sun is further, but don't say how much further. The 90degree at first quarter argument says that the distance from earth to sun is many times greater than the distance from earth to moon. $\endgroup$ – James K Sep 14 at 18:41
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    $\begingroup$ All good points. According to Wikipedia, Eratosthenes was from Cyrene (in modern Libya) and presumably did spend most of his career in Alexandria. I think we have this picture of the Greeks in the times of Plato and Aristotle, when they were really mainly in the Aegean (and places like Sicily) and we forget that during the Hellenistic age, Greek culture was much more cosmopolitan! Also an important part of Hellenistic astronomy was access to Assyrian astronomical records going back hundreds of years and presumably done in Mesopotamia. $\endgroup$ – Tim Campion Sep 14 at 18:49
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    $\begingroup$ Maybe worth mentioning that Hipparchus measured the distance to the moon by parallax within a century or so of Eratosthenes' work. I don't know if one is dependent on the other though. $\endgroup$ – Tim Campion Sep 14 at 18:52
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    $\begingroup$ "And the Sun is further still. At half moon, the sun seems to be at 90∘ to the moon. This is only possible if the sun is much further away than the moon." It isn't immediately obvious why this should be the case, can you elaborate further? $\endgroup$ – Chris Cooper Sep 15 at 9:59
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    $\begingroup$ Is this how Eratosthenes knew it or is this just one way to know it? $\endgroup$ – user253751 Sep 15 at 17:06
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Exactly how Eratosthenes calculated the radius of the Earth has been lost. What is presently taught as his method is a simplified version described by Cleomedes.

It is unlikely that Eratosthenes assumed the Sun was infinitely distant, since he apparently also estimated the distance to the Sun himself. In any case, his work came after that of Aristarchus who wrote a huge treatise on the distance between the Earth and the Sun and Moon.

Aristarchus concluded that the Sun was much further away than the Moon (by about a factor 20), by claiming that the angle between the Earth, Moon and Sun, when the Moon was half-illuminated, was 87 degrees. He also knew from the angular size of the Moon and the curvature of the Earth's shadow during a lunar eclipse, that the Moon was a lot further away than the radius of the Earth.

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    $\begingroup$ Why would it be lost if Cleomedes reports on how Eratosthenes did it? What measure of historical accuracy do we retain if we exclude those who report on historical events? Cleomedes is known for writing down information from third sources ad verbatim, and while I am aware this is a wikipedia quote: many modern mathematicians and astronomers believe [Cleomedes'] description [of Eratosthenes' method] to be reasonable. $\endgroup$ – Flater Sep 15 at 11:14
  • $\begingroup$ Also, as far as I'm aware, Eratosthenes didn't believe the sun to be infinitely distant, but rather that the sun's light can be assumed to be sufficiently parallel relative to the precision of his other measurements. The distance between Alexandria and the other location was measure by the time it took a camel to travel it, which isn't particularly precise either. $\endgroup$ – Flater Sep 15 at 11:16
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    $\begingroup$ @Flater and many don't.. As for the latter point, that is assuming the Sun to be infinitely distant and we don;t know what Eratosthenes did, because his account has been lost. The distance between Alexandria and Syrene was not measured in the way you describe. $\endgroup$ – Rob Jeffries Sep 15 at 14:24
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    $\begingroup$ @RobJeffries That isn't assuming the sun to be infinitely distant, which in itself doesn't even make sense. It's assuming that the sun is sufficiently far away that the angle between rays from the sun on earth is negligable for this particular computation. $\endgroup$ – Servaes Sep 15 at 14:59
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    $\begingroup$ @Servaes Fair enough. In the same sense as assuming a massless string doesn't mean you think the string has zero mass. But we don't know what was assumed because there is no account of it. Only a second-hand, simplified version. $\endgroup$ – Rob Jeffries Sep 15 at 16:00

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