The Badal optometer paradox

Badal's optometer is used to change the vergence of the accommodative stimulus without changing retinal image size. It has been accepted that this can be accomplished by putting the optometer's secondary focal point at either the eye's anterior nodal or anterior focal point. I show that there can be only one solution because the simultaneous truth of both solutions would lead to a paradox. The most precise solution is slightly different for the accommodated and the unaccommodated eye, but the nodal point provides the best general solution. For the accommodating eye there is a residual increase in image size with accommodation due to the anterior displacement of the nodal points. For the unaccommodated eye, the correct reference point is its entrance pupil. Use of the nodal point results in accommodation creating an anomalous, but slight, change in retinal image size.