= Aristarchus of Samos
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<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> $\epsilon$ at <Earth> gives
$$
\cos\epsilon=\frac{d_{EM}}{d_{ES}},\qquad \frac{d_{ES}}{d_{EM}}=\frac{1}{\cos\epsilon}.
$$
The $87^\circ$ <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. https://doi.org/10.5802/crphys.246[Siebren van der Werf and Tjeerd van Albada, An explanation for Aristarchus' measurement of the sun's distance, 2025]{ref}.
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