How did Eratosthenes know that the sun is far away?
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?
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.
@ChristosHayward: Are you _sure_ that was a Greek philosopher? That sounds suspiciously like Galileo's lantern experiments.
you can tell a light source is far away if an object's shadow is the same size as the object.
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.
Solar eclipses also demonstrate that the sun is farther away than the moon. For that matter, the apparent size argument can also be applied directly to the sun.
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.
@TimCampion : But possibly only slightly. However, this stuff about the half-moon makes it appear that the distance may be much larger.
I wonder about your comment about Greece. How much of his time did Eratosthenes spend in Greece and how much in Egypt?
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.
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.
"such an effect can be seen on Mars, where the moon really does orbit close to the planet" I had never thought of that! https://space.stackexchange.com/q/46537/6718
"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?
@TimCampion: Does your comment assume already knowing that the sun is many times bigger than the moon? I mean, yes it's farther because it can hide behind the moon during an eclipse, but the same would be true of a moon-sized sun that is only one moon-diameter further from Earth than the moon itself is.
@ChrisCooper The angle between the sun-moon and sun-earth ray must be very small for this observation to work (draw some pictures of triangles and illuminated moons). Because of this the sun-earth distance must be much larger than the earth-moon distance.
@Flater Indeed, as Michael Hardy points out, my first comment about solar eclipses would not prove that the sun is _much_ further than the moon. So maybe there's an interesting question here -- if all you know is that the moon is far enough away that its size appears constant, and that the sun is at least as far away, then do you know the sun is far enough away that the distance can be considered infinite for the purposes of Eratosthenes' measurement? Or is it necessary to know that the sun is _much_ farther away than required to maintain constant apparent size?
In addition to solar eclipses providing evidence the moon is closer than the sun, the phases of the moon are clear proof - we see the moon waning rather than waxing as it draws near the sun in the sky.
"Thus the moon must be much further than the radius of the Earth." -- or at least much further than the radius of the known world, where the apparent size of the moon was recorded?
I think this answer makes a lot of assumptions that we take for granted now, but weren't common - or even settled academic - knowledge back then. For instance, Aristarchus of Samos was the first known person to assume a heliocentric model of the universe, *and he lived at the same time as Eratosthenes*. If you assume other motions of celectial bodies to each other, the measurement of constant size from the earth could mean different things.