# Evans' theory on Aristarchos

Illustration of James Evans' theory on how Aristarchos might have reached his famous 87° half moon elongation (in

*History and Practice of Ancient Astronomy*, p. 72).In this illustration, the Sun's and Moon's distances

*S*and*M*are expressed in Earth radii; the time*t*, in hours; the length of the synodic month*T*, in days; and the half moon's elongation*θ*, in degrees. For illustrative purposes, distances are not to scale. For true scale set the sliders to*t*= 0.29 hrs,*T*= 29.53 days, and*M*= 60⅓ Earth radii. For Evans' theory to work, move the sliders to*t*= 6 hrs, and*T*= 30 days. Despite Evans' assumptions (that half moons occur closer to new moons than to full moons and that Aristarchos made a purely indoor mathematical calculation to obtain his 87° angle), the fact is that quadratures (first and last quarters) occur a mean of ten minutes closer to full moons than to new moons (Meeus,*Morsels IV*, p. 12) and, though*geocentric*dichotomies (50% illumination as seen from the Earth’s centre) always precede first quarters and follow last quarters by a mean of about 18 minutes (Meeus,*Morsels V*, p. 22), sometimes things are so arranged that an observer on the Earth’s surface can see*topocentric*dichotomies (half moons) occurring closer to*full*moons than to new moons! Drag the point O (representing the observer's position) along the Earth's surface to see how this is so.