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Greeks and Planets

Ever wondered how on Earth we know how big the Earth is? Or how far away the Sun is? This is a summary of how the Greeks figured it all out (roughly). The information below is taken from Simon Singh's excellent book Big Bang (2004, Fourth Estate). This is third hand information so don't take it as fact, but it does show how the measurements were obtained without advanced scientific instruments, and how close the Greeks were.

Size of the Earth
In the 200s BC, Eratosthenes figured out a very clever method for measuring the size of the Earth. He heard about a well in Syene (in modern Egypt) where the sunshine reached the bottom only at noon in the summer solstice. So he knew exactly when the sun was directly overhead. Further north in Alexandria he put a stick straight up in the ground and measured the angle of its shadow at that precise moment as 7.2°. It meant that the distance from Syene to Alexandria was 7.2/360 of the Earth's circumfrence. Since he knew this distance, he could compute the circumfrence as 39250km, a diameter of about 12500km.
Greeks: 12500km, Nowadays: 12750km

Size of the Moon
It has been observed for millenia that a full Lunar eclipse (when the Earth casts a show on the Moon) can be used to compute the relative sizes of the Earth and Moon. The Moon takes 50 minutes to pass into the Earth's shadow, and 200 mintues to cross the shadow. So the Moon's diameter is 50/200 times (a quarter) as big as the Earth's. Using this ratio and the figures above, the Greeks computed its size (diameter).
Greeks: 3100km, Nowadays: 3480km

Distance to the Moon
If you hold your finger up so that it just covers the glow of a full Moon, then you have created a similar triangle. The ratio between the size of your finger and how far it is away from your eye, is roughly the same as the ratio between the size of the moon and how far it is away from your eye. Assuming you are not a small baby or have massive fingers, this happens when the arm is roughly fully extended, and the ratio is about 1 to 100. So the greeks could estimate the distance to the Moon.
Greeks 310,00km, Nowadays: 384,000km

Distance to the Sun
In the 400s BC Anaxogoras proposed that the Moon glows because it reflects the Sun's light. His peers were not amused, but 200 years later Aristarchus took to the idea. He realised that when the Moon is at half phase, it, the Earth and the Sun form a right angled triangle (90°). If one could draw long imaginary lines in space, then the line from the Earth to the Moon, and the Moon to the Sun would be perpendicular. And there would be a nearly right angle between the line from Earth to Sun and the line from the Moon to the Earth. I imagine it would be quite hard to measure this angle. Aristarchus measured it as 87°. Using trigonometry he computed that the Sun is therefore 20 times more distant than the moon. The angle is actually 89.85° and the Sun is about 400 times more distant.
Greeks: 6,200,000km, Nowadays: 150,000,000km.

Size of the Sun
It has been observed for (probably) an even longer time that the Moon covers the Sun very snugly during a full Solar eclipse (when the Moon passes between the Earth and Sun). This is very similar to the finger trick above, and so the ratio between the Moon's size and distance is approximately equal to the ratio between the Sun's size and distance, and the ratio between your finger's width and arm's length, about 1 to 100. I guess the Greeks didn't like to stick their fingers up at the Sun, so perhaps that's why they had to await an eclipse to figure this one out.
Greeks: 62,000km, Nowadays: 1,390,000km.

Greek translation provided by Zsolt Boros in August 2016.