Suppose that Aristarchus had found that the interval from third quarter to first

Question

Suppose that Aristarchus had found that the interval from third quarter
to first quarter moon was 13 days, and that the interval from first quarter
to third quarter moon was 15 days. Then what distance would he have
derived for the Sun (in terms of, or relative to, the Moon

Explanation / Answer

That is explained here: http://en.wikipedia.org/wiki/Aristarchus… From your data we conclude that the angle 90-phi corresponds to 0.5 days. So 90-phi = 0.5 * 360 / 28 = 6.42 degree => phi = 83.58 Therefore S/L = 1/cos phi = 8.94 So Aristarchus thinks that the distance to the Sun is about 9 times as far as the distance to the Moon. Note that in reality S/L =400 and third quarter is only half an hour "late" rather than half a day.

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