be 18–20 times larger. Lunar eclipse Aristarchus then used another construction based on a lunar eclipse: By similarity of the triangles, and Dividing these two equations and using the observation that the apparent sizes of the Sun and Moon are the same, , yields The rightmost equation can either be solved for ℓ/t or s/t The appearance of these equations can be simplified using n = d/ℓ and x = s/ℓ. The above equations give the radii of the Moon and Sun entirely in terms of observable quantities. The following formulae give the distances to the Sun and Moon in terrestrial units: where θ is the apparent radius of the Moon and Sun measured in degrees. It is unlikely that Aristarchus used these exact formulae, yet these formulae are likely a good approximation for those of Aristarchus. Results The above formulae can be used to reconstruct the results of Aristarchus. The following table shows the results of a long-standing (but dubious) reconstruction using n = 2, x = 19.1 (φ = 87°) and θ = 1°, alongside the modern day accepted values. The error in this calculation comes primarily from the poor values for x and θ. The poor value for θ is especially surprising, since Archimedes writes that Aristarchus was the first to determine that the Sun and Moon had an apparent diameter of half a degree. This would give a value of θ = 0.25, and a corresponding distance to the Moon of 80 Earth radii, a much better estimate. The disagreement of the work with Archimedes seems to be due to its taking an Aristarchus statement that the lunisolar diameter is 1/15 of a "meros" of the zodiac to mean 1/15 of a zodiacal sign (30°), unaware that the Greek word "meros" meant either "portion" or 7°1/2; and 1/15 of the latter amount is 1°/2, in agreement with Archimedes' testimony. A similar procedure was later used by Hipparchus, who estimated the mean distance to the Moon as 67 Earth radii, and Ptolemy, who took 59 Earth radii for this value. Illustrations Some interactive illustrations of the propositions in On Sizes can be found here: Hypothesis 4 states that when the Moon appears to us halved, its distance from the Sun is then less than a quadrant by one-thirtieth of a quadrant [that is, it is less than 90° by 1/30th of 90° or 3°, and is therefore