On the Sizes and Distances (Aristarchus)

On the Sizes and Distances (of the Sun and Moon) (Ancient Greek: Περὶ μεγεθῶν καὶ ἀποστημάτων [ἡλίου καὶ σελήνης], romanized: Perì megethôn kaì apostēmátōn [hēlíou kaì selḗnēs]) is widely accepted as the only extant work written by Aristarchus of Samos, an ancient Greek astronomer who lived circa 310–230 BCE.

The book was presumably preserved by students of Pappus of Alexandria's course in mathematics, although there is no evidence of this.

The editio princeps was published by John Wallis in 1688, using several medieval manuscripts compiled by Sir Henry Savile.

[1] The earliest Latin translation was made by Giorgio Valla in 1488.

Aristarchus determined φ to be a thirtieth of a quadrant (in modern terms, 3°) less than a right angle: in current terminology, 87°.

Trigonometric functions had not yet been invented, but using geometrical analysis in the style of Euclid, Aristarchus determined that In other words, the distance to the Sun was somewhere between 18 and 20 times greater than the distance to the Moon.

Dividing these two equations and using the observation that the Sun and Moon appear the same size to people on Earth,

Then 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.

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.

[citation needed] 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.