Distance To The Sun (lunar Dichotomy)
The only surviving work usually attributed to Aristarchus, On the Sizes and Distances of the Sun and Moon, is based on a geocentric world view. It has historically been read as stating that the angle subtended by the Sun's diameter is 2 degrees, but Archimedes states in The Sand Reckoner that Aristarchus had a value of ½ degree, which is much closer to the actual average value of 32' or 0.53 degrees. The discrepancy may come from a misinterpretation of what unit of measure was meant by a certain Greek term in Aristarchus' text.
Aristarchus claimed that at half moon (first or last quarter moon), the angle between Sun and Moon was 87°. Possibly he proposed 87° as a lower bound since gauging the lunar terminator's deviation from linearity to 1° accuracy is beyond the unaided human ocular limit (that limit being about 3° accuracy). Aristarchus is known to have also studied light and vision.
Using correct geometry, but the insufficiently accurate 87° datum, Aristarchus concluded that the Sun was between 18 and 20 times farther away than the Moon. (The true value of this angle is close to 89° 50', and the Sun's distance is actually about 400 times the Moon's.) The implicit false solar parallax of slightly under 3° was used by astronomers up to and including Tycho Brahe, ca. 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, so that that the diameter of the Sun was between 18 and 20 times larger than the diameter of the Moon.
Read more about this topic: Aristarchus Of Samos
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