Determining the Eccentricity of the Moon's Orbit without a Telescope, and Some Comments on "Proof" in Empirical Science

Prior to the invention of the telescope many astronomers worked out theories of the motion of the Moon. The purpose of such theories was to be able to predict the position of the Moon in the sky. These geometrical models implied a certain range of distance of the Moon. Ptolemy's model, in fact, predicted that the Moon was nearly twice as far away at apogee than at perigee. Measurements of the angular size of the Moon were within the capabilities of pre-telescopic astronomers. These could have helped refine the models of the motion of the Moon, but hardly anyone seems to have made any measurements that have come down to us. Using a piece of cardboard with a small hole punched in it which slides up and down a yardstick, we show that it is possible to determine an approximate value of the eccentricity of the Moon's orbit. From 64 observations taken over 14 cycles of the Moon's phases we find find epsilon ~ 0.041 +/- 0.004. A typical measurement uncertainty of the Moon's angular size is +/- 0.7 arcmin. Since the Moon's angular size ranges from 29.4 to 33.5 arcmin, carefully taken naked eye data are accurate enough to demonstrate the periodic variations of the Moon's angular size.