Aristarchus of Samos

Aristarchus of Samos used the half-lit Moon to estimate the relative distances of the Sun and Moon. In the spherical illumination model a half-lit disk means phase angle $\alpha =\pi /2$, so the Sun–Moon–Earth triangle is right-angled at the Moon. Keeping the Sun at a finite distance, let $d_{EM}$ and $d_{ES}$ be the distances from Earth to the Moon and Sun. Then elongation $ϵ$ at Earth givesThe $87^{\circ }$ elongation attributed to Aristarchus of Samos gives a ratio about $19.1$, consistent with his conclusion that the Sun was between $18$ and $20$ times farther away. The example shows how lunar phase geometry becomes a distance measurement, but the ratio is extremely sensitive to an angle close to $90^{\circ }$. Unlike the distant-Sun approximation, this finite triangle has elongation slightly less than $90^{\circ }$ at half illumination. ^[ref].