Aristarchus On the Sizes and Distances

On the Sizes and Distances (of the Sun and Moon) (Περὶ μεγεθῶν καὶ ἀποστημάτων [ἡλίου καὶ σελήνης], Peri megethon kai apostematon) is widely accepted as the only extant work written by Aristarchus of Samos, an ancient Greek astronomer who lived circa 310–230 BC. This work calculates the sizes of the Sun and Moon, as well as their distances from the Earth in terms of Earth's radius.

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. The earliest Latin translation was made by Giorgio Valla in 1488. There is also a 1572 Latin translation and commentary by Frederico Commandino.

The work's method relied on several observations:

The rest of the article details a reconstruction of Aristarchus' method and results. The reconstruction uses the following variables:

Aristarchus began with the premise that, during a half moon, the moon forms a right triangle with the Sun and Earth. By observing the angle between the Sun and Moon, φ, the ratio of the distances to the Sun and Moon could be deduced using a form of trigonometry.

From the diagram and trigonometry, we can calculate that

The diagram is greatly exaggerated, because in reality, S = 390 L, and φ is extremely close to 90°. 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

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