He figured out how to measure the relative distances from the Earth (E) of the Sun (S) and the Moon (M). When the Moon is exactly half full, the angle E-M-S must be exactly 90 degrees. Therefore, a measurement of the angle M-E-S when the Moon is half full will give the ratio of the Earth-Moon distance to the Earth-Sun distance.

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Aristarchus measured the angle M-E-S to be 87 degrees, giving the ratio to be 1/19. Actually, the angle is 89 degrees, 51 minutes, giving an actual value of 1/400, that is, the Sun is 400 times further away from the Earth than the Moon is.