Aristarchus of Samos’s monograph on the sizes and distances of the sun and moon relative to the earth’s diameter is a major milestone in the history of astronomy. Accounts of Aristarchus’s reasoning can be found in many modern texts, but most of these are renditions of the argument in modern, trigonometric terms. The directness and simplicity of these trigonometric renditions ill-prepare modern readers for the surprisingly complex and convoluted geometrical arguments that they encounter in Aristarchus’s original treatise. It turns out, however, that the sizes and distances sought by Aristarchus can be reckoned very simply without any recourse to trigonometry, using only the classical tools of Greek geometry: the ruler and the compass. That a geometrician of Aristarchus’s stature was unaware of this simple construction is highly unlikely, which leaves us with an intriguing question: why did he produce the complicated argument that we currently possess?