Aristarchus On The Sizes and Distances - Half Moon

Half Moon

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 Earth, 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

In other words, the distance to the Sun was somewhere between 18 and 20 times greater than the distance to the Moon. This value (or values close to it) was accepted by astronomers for the next two thousand years, until the invention of the telescope permitted a more precise estimate of solar parallax.

Aristarchus also reasoned that as the angular size of the Sun and the Moon were the same, but the distance to the Sun was between 18 and 20 times further than the Moon, the Sun must therefore be 18-20 times larger.