Aristarchus On The Sizes and Distances - Lunar Eclipse

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, since he would have lacked a precise value for π. However a simple approximation π = 3 will incur in a relative error smaller than 5%, well below experimental errors in measurements at the time.

These formulae are likely a good approximation to those of Aristarchus.