# Did Aristarchus take the radius of the Earth into account in calculating the distance to the Moon?

My text says that Aristarchus (310 BC – ~230 BC) measured the "angle subtended by the Earth-Moon distance at the Sun" ($\theta$ in the figure below) to establish the relative Earth-Moon and Earth-Sun distances.

I understand that he must, in fact have used the Moon-Earth-Sun angle, and then subtracted that from 90° to arrive at $\theta$; but how did he establish the Moon-Earth-Sun angle? The reference points for all three objects is their centers, yet what Aristarchus must have in fact measured was the angle between the Moon and the Sun at the surface of the Earth.

Did Aristarchus take this discrepancy into account in his calculations? If so, how?

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A diagram would be appreciated, even if Aristarchus  just ignored the issue and it shows that demonstrates that the discrepancy simply didn't matter much; and especially if he used some clever geometric trick that is glossed over in the standard explanation. – raxacoricofallapatorius Sep 11 '12 at 22:15

## 1 Answer

He ignored the radius of the Earth as negligible. His estimates for the angle were from the shape of the shadow the sun casts on the moon, and the difference between this and a straight line when the moon is halfway between full and new is too small to percieve precisely. He fooled himself into thinking he measured a different angle, so his estimate was really only giving a lower bound on the distance to the sun. As a lower bound, it was enough to establish that the sun is larger than the Earth, and this was important, in that it lent strong support to heliocentric models. But it was not an accurate method.

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Yes, it seems more like he described a method, rather than actually "measuring" anything (in the modern sense). Is it fair to say that this was a characteristic (perhaps "weakness") of the ancient Greek style: the focus on reasoning and forms and constructing arguments, with "measurements" often guessed at or very roughly estimated? – raxacoricofallapatorius Sep 12 '12 at 14:01

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@raxacoricofallapatorius - I would say Aristarchos is one of the few Greeks who actually did science = measuring things. Rather than just reasoning from arbitrary ideas of perfect shapes – Martin Beckett Sep 12 '12 at 17:10

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@raxacoricofallapatorius: In this case, he fooled himself into thinking he measured a different angle, so his estimate is really a lower bound on the distance to the sun. As a lower bound, it was enough to establish that the sun is larger than the Earth, and this was important. I don't think he took his estimate too seriously, he knew he needed a better method, but you need something to start, and he did the best he could. I agree with Martin Beckett--- don't assume that the Ancient Greek scientists (Aristarchus/Archimedes/Appolonius) were as stupid as Aristotle. – Ron Maimon Sep 12 '12 at 18:19
Understood, so perhaps "meaningless" is a bit harsh. – raxacoricofallapatorius Sep 12 '12 at 18:28
@raxacoricofallapatorius: You're right, fixed. – Ron Maimon Sep 12 '12 at 18:35
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