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Eratosthenes calculated the radius of the Earth from the difference of the lengths of shadows between Aswan and Alexandria were different (see also here). But this could also happen if the Earth were flat and the Sun wasn't really far away. A point source of light close to a flat plane will also cast shadows of different lengths. Did the Greeks already know by that point that the Sun was very far away? What hints in the natural world can lead you to that conclusion?

Moreover, how did he conclude it was a sphere? There could be other shapes that have curvature and lead to different length shadows.

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  • $\begingroup$ This was from a Carl Sagan documentary. But thanks.. I guess there was a lot of compelling evidence before his experiment. $\endgroup$ Commented Aug 21, 2022 at 8:15
  • $\begingroup$ Your link says he assumed the sun was very far away, so its rays were parallel. And this turns out to be a good assumption. What made him think this was the case back then? $\endgroup$ Commented Aug 21, 2022 at 8:21
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    $\begingroup$ "how did he conclude it was a sphere? " Hmm, that round Earth shadow on the moon during a lunar eclipse hint to earth's roundness. Of course the moon/sun look round too form all angles (expect subset). Maybe a sphere, disk, cylinder, etc. $\endgroup$ Commented Aug 24, 2022 at 18:07

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We do not know for sure what Eratosthenes read, but at his time this was a common knowledge. This can be inferred from the book of Archimedes, The sand reckoner (Archimedes was Eratosthenes' contemporary; they corresponded). In his book, Archimedes assumes that the Sun is 30 times larger than the Moon, and he mentions Eudoxus, Pheidias and Aristarchus who gave various estimates (Aristarchus wrote a book on this subject, which survives). Since the Sun and the Moon look about the same size, the conclusion is that the Sun is 30 times further.

I can briefly sketch one of the arguments from Aristarchus book. When we see exactly 1/2 of the Moon, this means that at this moment the triangle Sun-Moon-observer has a 90 degrees angle at the Moon. Measuring the angle at the observer, we can approximately recover the shape of this triangle. This angle is in fact very close to 90 degrees, so we conclude that the Sun has to be much further than the Moon.

That the Moon is quite far is also inferred from simple observations: its parallax is small (about one degree). They did not measure it precisely at the time of Eratosthenes and Archimedes, but knew that it is small.

For the general knowledge at that time of the sizes and distances between Earth, Moon and Sun, the book of Archimedes that I cited is a good source.

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This is a long answer, explaining more or less step to step why the data available to Eratosthenes indicted that the Sun should be a vast distance from the Earth, thus makings its diverging rays almost parallel when they struck the Earth.

Eratosthenes Lived from c. 276 BC to 195/194 BC.

There are (rare) eclipses of the Sun by the Moon.

Even though it is hard to see a thin sliver of the Moon in the daytime sky, especially when it is close to the painful glare of the Sun, people did observe the Moon before the Sun came up or after the Sun went down, approaching the position of the Sun and becoming thinner and thinner until it becomes unseen, and then becoming a thin new moon and the crescent increasing in thickness as the Moon's angular distance from the Sun increased.

Anaxagoras (500 BC - 428 BC) believed that the Sun was larger than the Peloponnese, which is about 80 miles wide. Since the Sun appears to be about half a degree wide as seen from Earth, it should be at least about 114.59 times 80 miles, or 9,167.2 miles, from Earth according to Anaxagoras. And as the ancient Greeks explored the world and found it was larger than they thought, Greek philosophers realized the Sun had to be larger and farther away.

Anaxagoras is said to be the first person to explain that solar eclipses are caused by the Moon passing in front of the Sun. Which of course shows that the Moon is closer than the Sun.

https://www.smithsonianmag.com/science-nature/ancient-greek-philosopher-was-exiled-claiming-moon-was-rock-not-god-180972447/

Anaxagoras was said to be the first person to explain that the Moon shines due to reflected light from the Sun.

https://history.stackexchange.com/questions/7890/who-discovered-first-that-the-moon-does-not-have-its-own-light

And that theory would indicate that the Moon was closer to Earth than the Sun.

I suggest that you make two different diagrams with the Sun and the Moon revolving around the Earth, as most of the ancients believed.

One diagram will show the Moon closer to Earth than the Sun, and one will show the Moon Farther from Earth than the Sun. Make the Sun and the Moon little circles and not mere dots.

Then color the side of the Moon's orb that faces away from the Earth black, and then see how much of the moon's lighted surface is visible from Earth at various angles. The thin crescent phases of the Moon would impossible to see if the Moon was farther away from Earth than the Sun.

Mercury and Venus also show phases like the Moon, visible through telescopes, and there are cases of people observing the crescent phase of Venus without optical aid - I sometimes thought that I could barely see a crescent Venus.

An ancient Greek philosopher who heard that someone saw a thin crescent Venus would deduce that like the Moon, Venus would then be closer to Earth than the Sun is.

And if the Moon was ever observed to occult (pass in front of and hide) Mercury or Venus, a philosopher would know that the Moon was closer to Earth than the planet it occulted. Observations of occultations were probably why the ancients believed that Mars, Jupiter, and Saturn were farther from Earth than the Moon is.

Because Mercury and Venus are actually closer to the Sun than Earth is they never appear more than 28 degrees (Mercury) and 47 degrees (Venus) from the Sun as seen from Earth. Because Mars, Jupiter and Saturn are farther from the Sun than Earth is, they can appear as far as 180 degrees, directly opposite the direction to the Sun as seen from Earth. There is obviously something different about the orbits of Mercury and Venus compared to those of the other 3 planets known in antiquity.

In the 2nd century AD, centuries after Eratosthenes, Ptolemy believed that Mercury and Venus were farther from Earth than the Moon and closer to Earth than the sun. So thee would have to be a large distance between the Moon and the Sun for Mercury and Venus to be between them. But I don't know whether Eratosthenes believed that Mercury and Venus were between the Moon and the Sun.

According to the answer by Alexander Ermenko, Aristarchus believed the Sun was much farther than the Moon.

I can briefly sketch one of the arguments from Aristarchus book. When we see exactly 1/2 of the Moon, this means that at this moment the triangle Sun-Moon-observer has a 90 degrees angle at the Moon. Measuring the angle at the observer, we can approximately recover the shape of this triangle. This angle is in fact very close to 90 degrees, so we conclude that the Sun has to be much further than the Moon.

Aristarchus of Samos c. 310 - c. 230 BC, was an early proponent of heliocentrism instead of geocentrism.

Aristarchus claimed that at half moon (first or last quarter moon), the angle between the Sun and Moon was 87°.[19] He might have proposed 87° as a lower bound, since gauging the lunar terminator's deviation from linearity to one degree of accuracy is beyond the unaided human ocular limit (with that limit being about three arcminutes of accuracy). Aristarchus is known to have studied light and vision as well.[20]

Using correct geometry, but the insufficiently accurate 87° datum, Aristarchus concluded that the Sun was between 18 and 20 times farther away from the Earth than the Moon.[21] (The true value of this angle is close to 89° 50', and the Sun's distance is approximately 400 times that of the Moon.) The implicit false solar parallax of slightly under three degrees was used by astronomers up to and including Tycho Brahe, c. AD 1600. Aristarchus pointed out that the Moon and Sun have nearly equal apparent angular sizes, and therefore their diameters must be in proportion to their distances from Earth.[22]

In On the Sizes and Distances of the Sun and Moon, Aristarchus discusses the size of the Moon and Sun in relation to the Earth. In order to achieve these measurements and subsequent calculations, he used several key notes made while observing a lunar eclipse.[23] The first of these consisted of the time that it took for the Earth's shadow to fully encompass the Moon, along with how long the Moon remained within the shadow. This was used to estimate the angular radius of the shadow.[24] From there, using the width of the cone that was created by the shadow in relation to the Moon, he determined that it was twice the diameter of the Moon at the full, non-central eclipse. In addition to this, Aristarchus estimated that the length of the shadow extended around 2.4 times the distance of the Moon from the Earth.[23]

Using these calculations, along with the aforementioned estimated distances of the Sun from the Earth and Moon from the Earth, he created a triangle. Employing a similar method of geometry that he previously used for the distances, he was able to determine that the diameter of the Moon is roughly one-third that of the Earth's diameter. In order to estimate the size of the Sun, Aristarchus considered the proportion of the Sun's distance to Earth in comparison to the Moon's distance from Earth, which was found to be roughly 18 to 20 times the length. Therefore, the size of the Sun is around 19 times wider than the Moon, making it approximately six times wider than the Earth's diameter.[23]

https://en.wikipedia.org/wiki/Aristarchus_of_Samos

So in the time of Eratosthenes, it was known from traveling and exploration that the Earth was at least a few thousand modern miles or kilometers in diameter.

And for each thousand miles the diameter of the Earth was, the Moon would be about 333.333 miles in diameter and the Sun about 6,000 miles in diameter. If the Sun and the Moon are about 114.59 times as far away as their diameters, for each 1,000 miles the diameter of the Earth was, the Moon would be 38,196.6 miles, and the Sun 687,540 miles, from Earth.

So with the data available to Eratosthenes it was fairly accurate to consider the Sun's rays striking the Earth in different places to be very close to parallel to each other. And as it turned out the Sun is actually many times as far away as was believed in Eratosthenes's era, thus making its rays even closer to being parallel.

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  • $\begingroup$ "Anaxagoras (500 BC - 428 BC) believed that the Sun was larger than the Peloponnese, which is about 80 miles wide." How did he know this? $\endgroup$ Commented Aug 30, 2022 at 19:30

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