Pascal's ring, cardinal points, and refractive compensation

Pascal's ring is a hexagon each of whose corners represents one of the six cardinal points of an optical system and whose sides represent relationships of relative axial position of the cardinal points. Changes to the ring represent the axial displacements of the cardinal points of the visual optical system of an eye that are caused when a spectacle lens compensates for the ametropia. Pascal's schema was described some 70 years ago with little theoretical justification. The purpose of this paper is to derive expressions for the axial locations of the cardinal points of a compound system consisting of an optical instrument and a visual optical system and for the shift caused by the instrument, and to provide theoretical justification for Pascal's schema. The cardinal points are treated not as separate entities but in a unified manner as special cases of an infinite class of special points. Expressions are derived using Gaussian optics. The results are specialized for the case of the eye's ametropia compensated by optical instruments in general and by spectacle lenses in particular. Pascal's schema is shown to be broadly correct although some modification is necessary for the effects on the incident cardinal points especially for the myopic eye.