On the Sizes and Distances (Aristarchus)

On the Sizes and Distances of the Sun and Moon is the only surviving work attributed to the ancient Greek astronomer and mathematician Aristarchus of Samos (c. 310 – c. 230 BCE). The treatise attempts to determine the sizes of the Sun and Moon relative to the Earth, and their distances from the Earth, using geometric arguments based on observed angles during a lunar eclipse and the phase of the Moon.

The work relies heavily on observation and geometric reasoning. Aristarchus's method involves measuring the angle between the Sun and the Moon when the Moon appears half-illuminated (the first or last quarter moon). He then uses this angle to form a triangle with the Earth, Sun, and Moon, and applies geometric principles to estimate the ratio of their distances. Similarly, he uses observations of lunar eclipses to estimate the relative sizes of the Earth and Moon.

It is important to note that the measurements Aristarchus obtained were significantly inaccurate due to limitations in observational instruments and techniques available at the time. As a result, his conclusions regarding the relative sizes and distances were also incorrect. For instance, he concluded that the Sun was only about 18 to 20 times farther away than the Moon, and similarly small factors off for the size calculations. In reality, the Sun is roughly 400 times farther away.

Despite the inaccuracies in its numerical results, On the Sizes and Distances is a significant contribution to the history of astronomy and science because it represents one of the earliest known attempts to determine the sizes and distances of celestial bodies using mathematical and observational methods. It also demonstrates Aristarchus's ingenuity in applying geometry to astronomical problems, even with imperfect data. The work provides valuable insights into the scientific thinking and astronomical knowledge of the Hellenistic period. Furthermore, it demonstrates his heliocentric viewpoint, as assuming the Sun and Moon orbit the Earth in his calculations would render the attempt useless. It is this treatise that provides evidence of Aristarchus's heliocentric theories, though this theory is not explicitly argued for within On the Sizes and Distances. The text is preserved primarily through later commentaries and references made by other ancient authors.